structural equation modeling (sem): concepts, applications and...

MATHEMATICS RESEARCH DEVELOPMENTS

STRUCTURAL EQUATION

MODELING (SEM)

CONCEPTS, APPLICATIONS

AND MISCONCEPTIONS

No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form orby any means. The publisher has taken reasonable care in the preparation of this digital document, but makes noexpressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. Noliability is assumed for incidental or consequential damages in connection with or arising out of informationcontained herein. This digital document is sold with the clear understanding that the publisher is not engaged inrendering legal, medical or any other professional services.

MATHEMATICS RESEARCH

DEVELOPMENTS

Additional books in this series can be found on Nova‟s website

under the Series tab.

Additional e-books in this series can be found on Nova‟s website

under the e-book tab.

MATHEMATICS RESEARCH DEVELOPMENTS

STRUCTURAL EQUATION

MODELING (SEM)

CONCEPTS, APPLICATIONS

AND MISCONCEPTIONS

LARRY RIVERA

EDITOR

New York

Copyright © 2015 by Nova Science Publishers, Inc.

All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted

in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying,

recording or otherwise without the written permission of the Publisher.

We have partnered with Copyright Clearance Center to make it easy for you to obtain permissions to

reuse content from this publication. Simply navigate to this publication‟s page on Nova‟s website and

locate the “Get Permission” button below the title description. This button is linked directly to the

title‟s permission page on copyright.com. Alternatively, you can visit copyright.com and search by

title, ISBN, or ISSN.

For further questions about using the service on copyright.com, please contact:

Copyright Clearance Center

Phone: +1-(978) 750-8400 Fax: +1-(978) 750-4470 E-mail: [email protected].

NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty

of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or

consequential damages in connection with or arising out of information contained in this book. The Publisher shall not

be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers‟ use of,

or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is

claimed for those parts to the extent applicable to compilations of such works.

Independent verification should be sought for any data, advice or recommendations contained in this book. In

addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising

from any methods, products, instructions, ideas or otherwise contained in this publication.

This publication is designed to provide accurate and authoritative information with regard to the subject matter

covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other

professional services. If legal or any other expert assistance is required, the services of a competent person should be

sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE

AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS.

Additional color graphics may be available in the e-book version of this book.

Library of Congress Cataloging-in-Publication Data Structural equation modeling (SEM) : concepts, applications, and misconceptions / Larry Rivera,

editor. pages cm. -- (Mathematics research developments)

Includes index.

1. Structural equation modeling. 2. Multivariate analysis. I. Rivera, Larry (Mathematician)

QA278.3.S78 2015

519.5'35--dc23

2015016704

Published by Nova Science Publishers, Inc. † New York

ISBN: 978-1-63482-924-3 (eBook)

CONTENTS

Preface vii

Chapter 1 The Determinants of Capital Structure Choice for

Chinese Listed Companies Based on Structural

Equation Modeling Approach 1 Xin-Dan Li, Xiang-Nan Feng,

Bin Lu and Xin-Yuan Song

Chapter 2 An Examination of Predictors and Outcomes Related

to School Climate Using Latent Class Analysis 35 Christine DiStefano, Elizabeth Leighton, Mihaela Ene and Diane M. Monrad

Chapter 3 Assessing Mediation in Simple and Complex Moels 69 Thomas Ledermann and Siegfried Macho

Index 103

PREFACE

Structural equation modeling (SEM) is a general, cross-sectional statistical

modeling technique. The chapters in this book propose a Bayesian approach

based on SEM; an examination of predictors and outcomes related to school

climate using latent class analysis and the testing of specific effects and

contrasts in three types of mediation models followed by a discussion on the

common types of mediation models and their different types of effects.

Chapter 1 – This chapter proposes a Bayesian approach based on

structural equation modeling (SEM) to empirically test the determinants of

capital structure choice for the Chinese listed companies. The chapter

investigates major unobservable theoretical attributes identified by capital

structure theories and constructs proxies for these attributes considering

specific institutional settings in China. The findings suggest that some firm-

specific factors relevant to explaining capital structure in developed economies

are also related to the Chinese economy. Unique determinants of capital

structure choice for Chinese listed companies are also identified, which are

closely related to the special micro and macroeconomic situations in China.

Chapter 2 – A favorable school climate provides the structure within

which students, teachers, administrators, and parents function cooperatively

and constructively. Measures of school climate, however, have received only

passing interest from policy makers as critical elements in accountability

reporting. This study used a state-wide dataset of climate ratings from 610

elementary schools and considered multidimensional information from both

teachers and students to produce latent classes of school climate. Two

variables, school size and a school‟s poverty index, were used as covariates

when creating latent classes. In addition, two measures of school performance

were examined as distal outcomes. The study identified four classes, where

Larry Rivera viii

classes were distinguished based upon school climate scores. Differences in

outcome variables and covariates were observed across the classes. The

information may be used by school personnel in examinations of malleable

factors related to school performance.

Chapter 3 – This chapter addresses the testing of specific effects and

contrasts in three types of mediation models: models with up to four

simultaneous (parallel) mediators, models with two sequential mediators, and

single-mediator models with two initial variables. The authors use the delta

method and provide equations to calculate standard errors for simple and total

indirect effects, total effects, and specific contrasts in each type of model.

They also demonstrate how bootstrap interval estimates of specific effects and

contrasts can be obtained using phantom models and how indirect effects

involving different initial variables can be compared in a scale-free fashion.

Testing contrasts, the authors show how common requirements for complete

mediation can be made stronger. Limitations of both, statistics using standard

errors based on normal theory and bootstrapping to test mediation, along with

new methods are discussed. The methods are illustrated using publicly

available datasets. Supplementary material available online includes Amos,

OpenMx, and Mplus files to estimate the models and an Excel spreadsheet to

calculate the effects.

In: Structural Equation Modeling (SEM) ISBN: 978-1-63482-892-5

Editor: Larry Rivera © 2015 Nova Science Publishers, Inc.

Chapter 1

THE DETERMINANTS OF CAPITAL

STRUCTURE CHOICE FOR CHINESE LISTED

COMPANIES BASED ON STRUCTURAL

EQUATION MODELING APPROACH

Xin-Dan Li1, Xiang-Nan Feng

2, Bin Lu

3

and Xin-Yuan Song2,

1School of Management and Engineering, Nanjing University, China

2Department of Statistics, the Chinese University of Hong Kong, China 3School of Finance, Nanjing University of Finance and Economics, China

ABSTRACT

This chapter proposes a Bayesian approach based on structural equation

modeling (SEM) to empirically test the determinants of capital structure

choice for the Chinese listed companies. The chapter investigates major

unobservable theoretical attributes identified by capital structure theories

and constructs proxies for these attributes considering specific

institutional settings in China. The findings suggest that some firm-

specific factors relevant to explaining capital structure in developed

economies are also related to the Chinese economy. Unique determinants

of capital structure choice for Chinese listed companies are also

Corresponding author: Xin-Yuan Song is Associate Professor, Department of Statistics, the

Chinese University of Hong Kong, Hong Kong, China, [email protected].

Xin-Dan Li, Xiang-Nan Feng, Bin Lu et al. 2

identified, which are closely related to the special micro and

macroeconomic situations in China.

Keywords: capital structure; chinese listed companies; structural equation

modeling

1. INTRODUCTION

As one of the most important areas in corporate finance, capital structure

analysis has attracted significant attention in the literature. Modigliani and

Miller (1958) proposed an M-M theory stating that the capital structure does

not affect firms‟ value in the perfect capital market, which is a very restrictive

assumption. Since then, many efforts have been made to relax the assumptions

of the M-M theory. The development of agency theory (Jensen and Mecking,

1976), coupled with thorough research of bankruptcy costs, suggested that

corporations act as if there is a unique, optimal capital structure. The main

competing theories in explaining firms‟ capital structure choice are the static

trade-off hypothesis (Kraus and Litzenberger, 1973) and the pecking order

hypothesis (Myers and Majluf, 1984). Static trade-off models assume the

optimal capital structure does exist, while the pecking order hypothesis states

that there is no well-defined target debt ratio based on the information

asymmetry. Diverse signaling models have also been proposed to address the

asymmetric information problems (Ross, 1977).

Over the past several decades, numerous research have been conducted to

investigate the determinants of capital structure choice. Bradley et al. (1984)

adopted cross-sectional, firm-specific data to test for the existence of an

optimal capital structure by considering some influential factors. Titman and

Wessels (1988) analyzed the impact of unobservable attributes on the choice

of corporate leverages based on a factor-analytic technique. Homaifar et al.

(1994) applied a general autoregressive distributed lag model to the US data to

estimate the long-run steady state determinants of firms‟ capital structure.

Rajan and Zingales (1995) compared the capital structure decisions across G-7

countries. Wald (1999) examined the factors correlated with the capital

structure in France, Germany, Japan, the United Kingdom, and the United

States. Ozkan (2001) investigated the determinants of capital structure choice

for 390 companies in UK and the role of the adjustment process. Drobetz and

Wanzenriedb (2006) studied 90 Swiss firms‟ data from 1991 to 2001 to

analyze the impact of firm-specific characteristics and macroeconomic factors

The Determinants of Capital Structure Choice for Chinese … 3

on firms‟ speed of adjustment to the target debt ratio. Feidakis and Rovolis

(2007) examined whether there are any robust determinants that affect the

capital structure of the large listed construction firms in the European Union.

Chang et al. (2009) applied a Multiple Indicators and Multiple Causes

(MIMIC) model, with refined indicators, to a pooled sample for the period

1988 to 2003 and found more convincing results than those obtained by

Titman and Wessels (1988). Kayo and Kimura (2011) proposed the use of

hierarchical linear model and random effect model for analyzing the influence

of time-, firm-, industry-, and country-level determinants of capital structure.

Öztekin (2014) analyzed a large sample of firms drawn from 37 countries to

examine the international determinants of capital structure.

Although the major research of the determinants of capital structure

choice have been focused on the developed economies, the works are

becoming noticeable in developing countries and transitional economies in

recent years. Booth et al. (2001) analyzed capital structure choices of firms in

ten developing countries. Deesomsak et al. (2004) investigated the

determinants of capital structure of firms operating in four Asia Pacific

countries. Agarwal and Mohtadi (2004) studied the effect of financial market

development on the financing choice of firms in developing countries using a

dynamic panel approach with aggregate firm level data. Fattouh et al. (2005)

examined the capital structure of listed firms in South Korea from 1992 to

2001. Nivorozhkin (2005) presented evidence on the actual and target capital

structures of firms in five EU accession countries of Central and Eastern

Europe and the former Soviet Union. De Haas and Peeters (2006) examined

the capital structure dynamics of firms in Central and Eastern Europe (CEE).

Nguyen and Ramachandran (2006) identified the determinants influencing the

capital structure of small and medium size enterprises (SMEs) in Vietnam.

Manos et al. (2007) investigated the effect of group affiliation on firm‟s capital

structure decision with data from the Indian economy. Delcoure (2007) also

considered the capital structure determinants in emerging CEE economies to

test whether the traditional theories are still useful. Kim and Berger (2008)

studied the determinants of the capital structure of large companies

headquartered in the United States and the Republic of Korea. Fan et al.

(2012) investigated the influence of the institutional environment on capital

structure and debt maturity choices for firms located in 39 developed and

developing countries. Öztekin and Flannery (2012) showed that certain

associations between institutional arrangements and leverage adjustment

speeds are consistent with dynamic trade-off theory of capital structure choice

with a dynamic panel data set spans 37 developed and developing counties.


China is now the largest developing economy in the world which rapidly

expanding capital market attracts many international investors. As a transition

economy, however, China has distinct institutional features. Therefore, it is

interesting to test whether the capital structure determinants of Chinese firms

differ from those in the western countries. Consequently, studying the effect of

Chinese unique institutional features on its listed companies‟ financing is of

great importance.

This study mainly extends the available empirical works in two ways.

Firstly, we investigate more attributes that have been tested for the firms in

developed countries but not yet for those in China. Secondly, the attributes

identified as the determinants of firms‟ capital structure choice are often not

directly observable; instead they are reflected by multiple indicators.

Therefore, the structural equation modeling (SEM) technique is adopted to

analyze the interrelationships among the latent attributes. Compared to an

ordinary regression analysis, the SEM approach has the following appealing

features. First, through grouping multiple indicators into a few latent

attributes, the SEM reduces the model dimension significantly. Second, based

on the condensed information, the SEM provides clearer and simpler model

interpretation. Finally, by assigning data-driven weights (factor loadings) to

highly correlated predictors via factor analysis, the SEM avoids the

multicollinearity problem encountered in the ordinary regression analysis (see

Section 5.3).

The rest of the chapter is organized as follows. Section 2 presents a brief

discussion of the specific institutional settings in China. Section 3 discusses

the attributes that are identified as the determinants of firms‟ capital structure

choice. Section 4 describes the data set and the methodology. Section 5

presents the empirical results. Section 6 concludes the major findings.

2. CHINESE INSTITUTIONAL SETTINGS

Chinese firms face some complications to achieve the optimum capital

structure under the market inefficiency and institutional constraints. For

example, Chinese banks cannot adequately provide resources to firms,

especially where government‟s credit demand crowds out the private sector, or

where the microeconomic environment is too risky for issuing long-term

loans. Particular macroeconomic conditions in China will also affect the

financing activities of the companies.


Many researchers have made cross-sectional comparisons among

countries and industries around the world. Booth et al. (2001), as pioneers

working on the capital structure in emerging markets, concluded that, to

forecast a firm‟s leverage, it is more important to know the firm‟s country than

the firm‟s characteristics. Lööf (2004) indicated that there are large and

unexpected cross-country differences in the determinants of the optimal capital

structure.

To better understand the background and empirical results of this study,

we present a brief discussion on the institutional settings in China pertaining to

firms‟ capital structure choice.

Financial institutions: Although Chinese listed firms have been reformed

into the joint-stock system for recent years, the government still controls the

majority shares through state-owned institutions such as state investment

companies, state holding companies, and state asset management agencies.

These firms are usually guaranteed by the Chinese government while

financing, and thus favored by the Chinese banks.

Most highly developed debt markets are associated with high private

sector debt ratios, which is not the case in China. The equity market in China

is more developed than the debt market, which provides more options for

corporate financing. Although the stock market is relatively developed, the

overall Chinese financial sector is still under the strong grip of the state. The

state monopoly of the financial sector has hindered the development of the

Chinese capital market and the growth of non-state financial institutions, in

particular the bond market. The access of Chinese firms to long-term debt

provided by state-owned banks has also been strictly controlled by the

government and thus the default risk of the loans is high. In such situations,

bankruptcy, even if enforced, may not be very efficient.

Capital market: There are three unique features of the Chinese capital

market that affect the capital structure choice for Chinese listed companies.

First, the capital market is not multi-layered. A complete capital market

consists of at least three components: the stock market, the bond market, and

the long-term mortgage market. Until now, the stock market is still

developing, while the long-term mortgage market is absent, and the bond

market is very small due to strict restrictive regulations. Such limitation forces

the listed companies to use bank loans instead of issuing corporate bonds.

Second, the ownership structure is complicated. The transformation of state-

owned shares and corporate shares into tradable shares has been carried out

since the release of “The Announcement on Reform of The Shareholder


Structure of Listed Companies” by the China Securities Regulatory

Commission (CSRC) in 2005. However, tradable shares only account for

32.23% of the capital market till April 2008 due to the time-lag effect. Hence,

the current ownership structure is still centralized, moreover in a transient

state. Third, information disclosure of the listed firms is irregular. Activities

such as providing false corporate statements, concealing important information

for insider trading, and non-timely information disclosure are very common.

Tax policy: In China, the central government controls local governments‟

tax policies. According to the Acting Regulations on Corporate Income Tax

launched on December 13, 1993, Chinese corporate tax rate is 33%. However,

to stimulate the development of the Chinese economy, the government

provides preferential tax rate for some regions. For example, the favorable tax

rate is 15% in five special economic zones, 32 economic and technology

development zones, 13 free trade zones, and 52 high-tech development zones.

On October 11, 2000, the Ministry of Finance announced the cancellation

of tax rebates to the listed firms after December 31, 2001. The new rule

expressly subjects the listed companies to the 33% corporate income tax rate.

Consequently, the actual tax rate for firms that had received tax rebates

increased from 15% to 33%, which greatly increases the tax advantage of debt.

Based on the special background, we expect that the tax policy has some effect

on capital structure choice for Chinese listed firms.

Law system: Chinese legal and institutional framework is still immature

and incomplete. For example, the company law is ambiguous about the debt

holders‟ rights. It is seriously flawed in granting shareholders and government

agencies too much power in bankruptcy procedures, while giving no control

rights in liquidation to debt holders.

3. THE FIRM-SPECIFIC FACTORS

Considering the above unique institutional settings in China and the extant

literatures (Harris and Raviv, 1991), we focus on the following attributes in

this study: assets structure, non-debt tax shields, growth, size, profitability,

assets liquidity, ownership structure, uniqueness, operation risk, signal, and tax

shields. The proxies (indicators) which represent the attributes from different

aspects will also be discussed accordingly.


A. Capital Structure Measurements

The choice of the corporate capital structure measures is controversial.

Due to the lack of a uniform definition of capital structure, there are so many

debt ratios can be adopted. As Rajan and Zingales (1995) pointed out, it

depends on the objective of the analysis to choose the definition of capital

structure. Some authors (e.g., Rajan and Zingales, 1995; Titman and Wessels,

1988) use the total debt, an inclusive measure of debt. Others (e.g., Chen,

2004; Huang and Song, 2006) only consider long-term debt. In this study, we

define the capital structure via the following three measures. The first measure

is simply the ratio of total debt over total assets (TDR). It is the broadest

definition of leverage and includes both long-term and short-term debt. The

second one is the ratio of long-term debt over total assets (LDR). The last one

is the ratio of short-term debt over total assets (SDR). According to Bevan and

Danbolt (2002), focusing on long-term debt when analyzing firms that

incorporate a large percentage of short-term debt will yield limited explanatory

power. In China, short-term debt plays an important role in corporate

financing. Therefore, the application of the broader measures of capital

structure is reasonable and necessary.

In addition, it is not clear whether leverage should be computed as the

ratio of book or market values of debt and equity. According to Graham and

Harvey (2001), application of book values is reasonable because financial

managers use mainly book values in decision making. Fama and French

(2002) argued that most theoretical predictions apply to book values.

Additionally, In China, there is still small proportion of circulated shares and

their market values are often unavailable. Therefore, in this chapter, the main

results are based on book values.

B. Assets Structure

Assets of a firm include tangible and intangible assets, both of which are

important factors relevant to the capital structure. However, the effects of

tangible and intangible assets on the capital structure are diverse. A firm with

more tangible assets is expected to possess a larger collateral value, and is

capable to access more debt. This relation is not obvious for intangible assets.


Previous empirical studies argued that the ratio of fixed to total assets is

an essential determinant of leverage (Titman and Wessels, 1988; Rajan and

Zingales, 1995; Fama and French, 2002). The trade-off theory argues that the

revaluation of assets is less when a company goes bankrupt, which reveals a

positive relationship between tangible assets and leverage. The pecking order

theory (Myers and Majluf, 1984) demonstrates that firms with assets to be

collateral may tend to issue more debt to take advantage of the information

asymmetry. Indicators of assets structure include the ratio of intangible assets

to total assets (INTG) and the ratio of inventory plus fixed assets to total assets

(TANG).

C. Non-Debt Tax Shields

Although some evidence (Homaifar, et al., 1994; Kim and Berger, 2008)

showed that non-debt tax shields are weakly related to firms‟ capital structure

choice, many works support the importance of the non-debt tax shields.

DeAngelo and Masulis (1980) presented that non-debt tax shields are

substitutes for the tax benefits of debt financing. Following studies (e.g., Wald,

1999; Ozkan, 2001; Korajczyk and Levy, 2003; Sogorb-Mira, 2005) argued

that a firm possessing larger non-debt tax shields is expected to use less debt.

Delcoure (2007) further found strong relations between the total, long-term,

and short-term debt ratio and non-debt tax shields. Indicators of non-debt tax

shields include the depreciation over total assets (DEPR) and a direct estimate

of non-debt tax shields over total assets (NDTS).

D. Growth Opportunity

The trade-off hypothesis of capital structure suggests that high-growth

firms tend to have larger bankruptcy costs and will use less debt. According to

the pecking order hypothesis (Myers and Majluf, 1984), information

asymmetry that outside investors have minor information about the quality of

the firms‟ investment projects demands extra premium for debt. However,

information asymmetry also motivates firms to turn to debt as a positive signal

of growth opportunities. The agency cost theories (Jensen and Meckling,

1976) argue that firms with high growth opportunities are more likely to have

high agency costs of debt due to the higher debt prices. Indicators of growth


opportunity include the percentage change in total assets (TAPC) and the

percentage change in prime operating revenue (PORPC).

E. Size

Based on the trade-off model, large firms are expected to have a high debt

capacity and be able to reduce transaction costs associated with long-term debt

issuance. Rajan and Zingales (1995) further argued that larger firms tend to

disclose more information than smaller ones, which makes it easier for larger

firms to have access to loans. However, pecking-order hypothesis suggests that

the complexity of the large firms increases the cost of information asymmetry,

and thus makes their debt financing more difficult. Indicators of size include

the natural logarithm of total assets (LTA) and the natural logarithm of prime

operating revenue (LPOR).

F. Profitability

From the pecking order theory (Myers and Majluf, 1984), internal

financing is favored by firms compared with debt financing. It can be expected

that profitable firms tend to have more retained earnings, and thus will use less

debt financing. Alternatively, tax effects predicted by the trade-off model

suggest that profitable firms should borrow more, given that they have greater

needs to shield income from corporate tax. On the other hand, the agency cost

theory describes that an increase in the debt ratio of profitable companies

signals the quality of financial management. Therefore, managers will attempt

to reduce the agency cost of the equity by increasing the company‟s debt ratio.

We use return on equity (ROE), return on assets (ROA), earnings per share

(EPS), sales gross profit rate (NGR), and net profit margin on sales (NSR) as

the indicators of profitability.

G. Liquidity

The pecking order theory pointed out that managers can manipulate liquid

assets in favor of shareholders against the interest of debt holders, which

increases the agency costs of debt. However, if firms generate substantial free

cash flows, shareholders would be motivated to cooperate with banks or


lending institutions for monitoring management spending by undertaking more

debt. Consequently, debt functions as an instrument that reduces the agency

cost caused by managers. In this study, liquidity ratio (LR) and quick ratio

(QR) are selected to measure the liquidity.

H. Ownership Structure

The shares structure of Chinese listed firms is officially classified into

state-owned shares, legal-person shares, and tradable shares. The role of state

ownership in the Chinese reform process is still controversial.

State-owned shares: A firm with substantial state ownership is more likely

to have a higher debt ratio than other firms for three reasons. First, the

financial sector in China is characterized by a bank-based system, where state-

owned commercial banks (SOCBs) play an important role. It is evident that

SOCBs‟ policies favor the state business sector much, as compared to the

private business sector, notably in terms of interest rates, banking procedures,

and collateral requirements. Second, the segregated voting and cash flow

rights related to high level of state ownership leads to acute agency problems

between owners and managers (Berkman et al., 2002). As mentioned before,

those agency problems can be alleviated by the high level of indebtedness.

Third, representatives of state ownership may discourage issuing seasoned

equity for fear of diluting state control as government shareholders often

cannot afford to subscribe new rights offerings (Xu and Wang, 1999).

Corporate shares: The agency cost theory argues that large institutional

shareholders should have enhanced incentives and capabilities to monitor

managerial behavior closely and therefore should have less demand for the

disciplinary function of debt.

Tradable shares: Tradable shares are direct substitution for debt. It is self-

evident that tradable shares ratio should be negatively correlated with

leverage.

In this study, we examine the effect of the ratios of state-owned shares,

corporate shares, and circulated shares on corporate financing decisions.

Indicators of ownership structure include the ratios of state-owned shares

(STATEP), corporate shares (CORPP), and circulated shares (CIRA) over

issued and outstanding shares.


I. Uniqueness

Titman (1984) suggested that a firm with unique products might require

its customers, suppliers and workers to undertake investments that lose value if

the firm goes into liquidation. Under this setting, lower leverage commits the

firm to a liquidation policy that takes into account the effects on its customers,

suppliers and workers. Further, customers, suppliers and workers may not be

willing to deal with a highly levered firm. For these reasons, uniqueness is

expected to be negatively associated with debt ratios. Considering the

limitation of the data, we select operating expenses ratio (OER) as the

indicator of uniqueness.

J. Operational Risk

Business risk is a proxy for the probability of financial and bankruptcy

distress and is generally expected to be negatively related to leverage (Bradley

et al., 1984; MacKie-Mason, 1990; Fama and French, 2002). According to the

trade-off theory, higher earnings volatility increases the probability of

financial distress, and consequently decreases firms‟ debt ratio conditional on

the high bankruptcy cost. According to the pecking order theory, firms with

very unstable cash flows would find that debt financing is too risky, and thus

prefer equity financing. The standard deviation of prime operating revenue

ratio (SDPOR) is used as an indicator of operational risk.

K. Signal

Dividend payout ratio (DIV) is selected as the indicator of signal. Mazur

(2007) found it is a useful discriminator in their analysis. Dividend payments

decrease the amount of internal funds and increase the need for external

financing. However, the agency cost theory argues that both debt and dividend

payouts can be used to control managerial perquisite consumption that arises

from excessive free cash flow. From this perspective, debt and dividend

payouts function as substitutions, and a negative relationship between them

can be expected.


Table 1. Measurement of Variables (Firm-specific Factors and Indicators)

Firm-

specific

Factors

Accounting Indicator Measurement

Capital

Structure

( )

Total Debt Ratio (TDR) Ratio of Total Debt to Total

Assets

Long-term Debt Ratio

(LDR)

Ratio of Long-term Debt to Total

Assets

Short-term Debt Ratio

(SDR)

Ratio of Short-term Debt to Total

Assets

Assets

Structure

( )

Tangible Assets (TANG) (Fixed Assets + Inventory)/ Total

assets

Intangible Assets (INTG) Intangible Assets/Total Assets

Non-debt

Tax Shields

( )

Depreciation (DEPR) Depreciation/Total Assets

Non-debt Tax Shields

(NDTS)

(Prime Operating Revenue -

Interest Payments - Income

Tax)/Total Assets

Growth ( ) Percentage Change in

Total Asset (TAPC)

(Ending Balance of Total Asset –

Beginning Balance of Total

Asset)/ Beginning Balance of

Total Asset

Percentage Change in

Prime Operating Revenue

(PORPC)

(Ending Balance of Prime

Operating Revenue – Beginning

Balance of Prime Operating

Revenue)/Beginning Balance of


Size ( ) LTA Natural logarithm of Total Assets

LPOR Natural logarithm of Prime

Operating Revenue

Profitability

( )

ROE Net Profit/(Total Assets - Total

Debt)

ROA Net Profit/Total Assets

Net Profit Margin on

Sales (NSR）

Net Profit/Prime Operating

Revenue

Sales Gross Profit

Rate(NGR)

Sale Revenue/(Sale Revenue –

Cost of Sales)

EPS Net Profit/Issued and Outstanding

Shares

1

2

3

4

5


Firm-

specific

Factors

Accounting Indicator Measurement

Liquidity

( )

Liquidity Ratio (LR) Current Assets / Current

Liabilities

Quick Ratio(QR) (Current Assets - Inventory)/

Current Liabilities

Ownership

Structure

( )

STATEP State Shares/ Issued and

Outstanding Shares

CORPP Legal Persons Shares/ Issued and

Outstanding Shares

CIRA Circulation Shares/ Issued and

Outstanding Shares

Uniqueness

( )

Operating Expenses Ratio

(OER)

Operating Expenses/Prime

Operating Revenue

Operation

Risk ( )


Ratio (SDPOR)

Standard Deviation of Prime

Operating Revenue / Average of


Signal ( ) Dividend Payout Ratio

(DPR)

Dividends Per Share/Earnings Per

Share

Tax Shields

( )

Income Tax Rate (ITR) Income Tax/Total Profit

L. Tax Shields

Modigliani and Miller (1963) indicated that the optimal leverage ratio of

firms is determined by the trade-off between the tax shield benefit of debt and

the higher bankruptcy costs implied by the higher degree of corporate

indebtedness. Some following studies fail to find plausible or significant tax

effects on financing behaviors, which is partially explained by the theory that

the debt-equity ratio is the cumulative result of the separate decisions during a

period (Mackie-Mason, 1990). The indicator of tax shields is the income tax

rate (ITR). The above-mentioned indicators and variables are listed in Table 1.

6

7

1x

2x

3x

4x


4. DATA AND METHODOLOGY

4.1. Data

In this study, the data were collected from the annual reports of Chinese

public-listed companies on Shanghai and Shenzhen stock exchanges over the

period 1998-2006, provided by the Wind Database. The year 2006 was chosen

because a new accounting system for business enterprises became operational

on January 1, 2007. To avoid some confounding effects, all listed companies

were required to date back to the period before 2006. In addition, we imposed

several criteria to obtain our data set: (1) all firms were set up before 1998 and

not in the financial industry. (2) We excluded the listed firms that were

suspended or withdrawn from the stock market. (3) We disregarded firms that

did not have a complete record on the variables required in our analysis. In

total, we procured 852 firms for the analysis, of which 489 were in

manufacturing and the rest in non-manufacturing that reflected the current

situation of industry of China. The distribution of industrial sectors is listed in

Table 2.

4.2. Methodology

In Section 3, we discussed a number of attributes and their indicators that

may theoretically affect firms‟ capital structure choice. Unfortunately, the

developed theories do not specify the functional forms describing how the

attributes relate to the indicators and debt ratios. The ordinary least squares

(OLS) method, a basic approach taken in previous empirical studies, has been

used to analyze regression equations with proxies for unobservable theoretical

attributes. However, the application of the OLS method in these studies may

encounter problems, such as multicollinearity among the explanatory variables

and the measurement errors in measuring the latent attributes. In addition, as

highlighted by Titman and Wessels (1988), additional problems exist in the

regression analysis with proxies for latent attributes. First, the lack of unique

representation of attributes may lead researchers to select variables based on

statistical goodness-of-fit criteria, and thus biasing the economic

interpretation. Second, the regression analysis introduces an errors-in-variables

problem due to the imperfect representation of proxy variables for attributes of

interest.


In this study, the structural equation modeling (SEM) technique is used to

overcome the abovementioned problems. Though the relevant theoretical

attributes are not directly observable, we can observe a number of proxies

(indicators) that measure the unobservable attributes in different aspects. In the

first stage, a measurement model relates observed indicators to latent

attributes. The second stage assumes a structural model to explore the effects

of latent attributes (firm-specific factors) on three debt ratios. Specifically, the

measurement model is defined as follows:

， (1)

where y is a q × 1 vector of observable indicators; is an m × 1 vector of

latent attributes, which is assumed to follow a multivariate normal distribution

N(0, ); is a q × m matrix of factor loadings; and is a q × 1 vector of

measurement errors, which is independent of , and distributed as N(0, )

with a diagonal covariance matrix . As shown in Section 3 and Table 1, we

have identified seven firm-specific factors (attributes) with 18 proxies

(indicators). Thus, y is an 18 × 1 vector of indicators, is a 7 × 1 vector of

latent factors, and is an 18 × 7 matrix of factor loadings. Through the

measurement model (1), SEM simultaneously accommodates highly correlated

explanatory variables (proxies) without encountering multicollinearity, and

measures latent attributes through proxies with different weights (factor

loadings), reflecting different contributions of proxies in measuring the latent

attributes. Compared to the use of individual indicators or simple arithmetic

mean of multiple indicators, this data-driven-based weighted average

procedure incorporates different characteristics of latent attributes, along with

their importance, thereby reflecting the attributes more accurately and

completely.

The structural model is defined as follows:

, (2)

where is a p × 1 vector of endogenous variables; X is a n × 1 vector of

covariates; B and Γ are p × n and p × m matrices of regression coefficients,

respectively; and is a p × 1 vector of error terms, which is independent of ,

and distributed as N(0, ) with a diagonal covariance matrix . As shown

y

X


in Section 3 and Table 1, we have identified three leverage measures of capital

structure (Total Debt Ratio, Long-term Debt Ratio, and Short-term Debt

Ratio), and four covariates (Uniqueness, Operation risk, Signal, and Tax

shields). Thus, in our study, is a 3 × 1 vector of observed response variables,

X is a 4 × 1 vector of covariates, and B and Γ are 3 × 4 and 3 × 7 matrices of

regression coefficients, respectively.

It is well known that the SEM defined by (1) and (2) is not identified

without imposing identification conditions. To achieve the identifiability, we

follow the common practice in SEM literature (e.g., Lu et al., 2012; Song and

Lee, 2012; Song et al., 2014) to fix appropriate elements of at pre-assigned

values, which can be decided on the basis of substantive theories. As discussed

in Section 3, each latent factor (attribute) is clearly measured by several

indicators (proxies); for example, „Assets Structure ( )‟ is measured by two

indicators: „Tangible Assets (TANG)‟ and „Intangible Assets (INTG)‟ (see

Table 1). Therefore, a non-overlapping structure of (see Table 3) is used for

achieving a clear interpretation of firm-specific factors. As shown in Table 3,

for Assets Structure , the first loading factor corresponds to TANG is fixed

at 1.0 to specify the scale of , and the second loading factor corresponds to

INTG, , remains free and needs to be estimated. The rest of elements in

the first and second rows are fixed at zero, indicating that TANG and INTG

are irrelevant to other attributes. The factor loadings of other latent attributes

have similar patterns.

In contrast to the measurement model, the structural model is totally

unrestricted. It estimates the impact of latent attributes on each of the different

financial leverages associated with capital structure. Furthermore, some

covariates (see x1 to x4 in Table 1) are incorporated in the structural model to

account for their direct effects on the interesting financial leverages. This is

different from the theory of Titman and Wessels (1988) that estimated the

effects of these covariates in measurement model.

In this study, we use the Bayesian method to perform the analysis. A

sampling-based Bayesian method is proposed for the following reasons. First,

it enables the use of authentic prior information to achieve better results.

Second, the Bayesian method does not rely on large-sample asymptotic theory,

thereby producing more reliable results even with small sample sizes. Finally,

with the rapid development of modern statistical computing techniques, the

Bayesian method is highly efficient and feasible for the analysis with latent

variables.

1

1

1

1,2


Table 2. The Industry Distribution Characteristic of Chinese listed Firms

Industry Classification No. of

Firms

The Proportion

of the

Total Sample

Communication and Cultural 6 0.70

Mining and Quarrying 8 0.94

Construction 11 1.29

Transportation and Warehousing 16 1.88

Agriculture/Forestry/Husbandry/Fishing 16 1.88

Social Services 25 2.93

Production of Electric,Coal,Gas and Water

Supplying 29 3.40

Information Technology 54 6.34

Real Estate 59 6.92

Comprehensive 64 7.51

Wholesale and Retail 75 8.80

Manufacturing 489 57.39

Total 852 100

Table 3. The Structure of the Measurement Equation

TANG 1 0 0 0 0 0 0

INTG 0 0 0 0 0 0

DEPR 0 1 0 0 0 0 0

NDTS 0 0 0 0 0 0

PORPC 0 0 1 0 0 0 0

TAPC 0 0 0 0 0 0

LTA 0 0 0 1 0 0 0

LPOR 0 0 0 0 0 0

ROA 0 0 0 0 1 0 0

ROE 0 0 0 0 0 0

NSR 0 0 0 0 0 0

NGR 0 0 0 0 0 0

EPS 0 0 0 0 0 0

1 2 3 4 5 6 7

1,2

2,4

3,6

4,8

5,10

5,11

5,12

5,13


Table 3. (Continued)

LR 0 0 0 0 0 1 0

QR 0 0 0 0 0

STATEP 0 0 0 0 0 0 1

CORPP 0 0 0 0 0 0

CIRA 0 0 0 0 0 0

5. EMPIRICAL ANALYSIS

The descriptive statistics of the debt ratios and indicators are reported in

Table 4. The debt ratios and indicators were averaged from 1998 through 2006

to reduce the noise. As it takes time for firms to move toward the target level,

we adopted the average of debt levels to reduce the effect of this adjusting

process.

Table 4. Descriptive Statistics of the Data

Variable Mean Median Maximum Minimum Std Dev

TDR 0.504 0.481 5.526 0.086 0.301

LDR 0.068 0.045 2.769 -0.011 0.118

SDR 0.436 0.412 3.350 0.039 0.243

TANG 0.542 0.506 15.210 0.122 0.611

INTG 0.038 0.025 0.753 0.000 0.046

DEPR 0.141 0.114 1.455 0.001 0.114

NDTS 0.544 0.433 4.955 -0.647 0.432

PORPC 0.357 0.182 44.418 -0.360 1.714

TAPC 0.180 0.152 5.291 -0.194 0.237

LTA 11.801 11.720 15.082 9.140 0.835

LPOR 11.010 11.006 14.769 4.101 1.222

ROA 0.050 0.052 0.202 -0.211 0.046

ROE -0.029 0.057 0.635 -19.515 0.775

NSR -0.162 0.048 5.510 -96.103 3.521

NGR 0.248 0.220 0.922 -0.393 0.131

EPS 0.142 0.145 1.263 -1.452 0.228

LR 1.642 1.413 13.523 0.304 1.060

QR 1.204 0.979 12.337 0.167 0.949

1 2 3 4 5 6 7

6,15

7,17

7,18


Variable Mean Median Maximum Minimum Std Dev

STATEP 0.350 0.372 0.814 0.000 0.219

CORPP 0.192 0.135 0.750 0.000 0.193

CIRA 0.394 0.379 1.000 0.092 0.122

OER 0.063 0.044 0.557 -0.019 0.067

SDPOR 0.513 0.472 1.758 0.063 0.266

DPR 0.249 0.205 2.067 -0.199 0.236

ITR 0.193 0.169 13.477 -1.731 0.489

5.1. The Characteristics of Capital Structure of Chinese Listed

Companies

Table 5 shows the characteristics of leverages for Chinese companies.

Different measures of leverage for Chinese listed companies have two notable

characteristics. First, the debt ratios exhibit increasing trend, but still lower

than the level of developed countries. For example, in 1998-2001, the mean of

total debt ratios was about 45%. Since 2002, the total debt ratios continued to

increase, in 2005 and 2006 to a value of nearly 60%, while the same ratio in

Japan is around 70%. Second, the short-term debt ratios were high and

increasing in China, which demonstrates that Chinese firms mainly use short-

term debt. It is evident that in 1998-2002 the long-term debt ratios were only

about 7% in China compared with the mean of 41% in the G-7 countries and

22% in other developing countries. Numerous studies in China argue that this

low amount of long-term debt reflects the fact that Chinese listed companies

are raising long-term capital via equity instead of debt. It is noticeable from

the last column of Table 5 that the ratio of equity financing is high, though it

had been decreasing. Besides the constraints of the debt financing, capital

gains resulting from secondary shares trading are substantially high, usually

about six to eight times the IPO prices1, which also makes the equity financing

favorable.

Another possible reason for the low debt level in China is the difference

between the management incentives of Chinese firm managers and those of

firm managers in developed countries. Managers in developed countries care

more about the upside profits of firms because they are rewarded substantially

by the profits. In China, however, managers, especially those serving in state-

owned companies, tend to be more concerned about downside risks because

they might receive administrative sanctions if the firms are not managed well.

1 China Securities Regulatory Commission.


This makes the Chinese managers of state-owned firms more risk-averse than

their western counterparts, which may partially explain the low debt level.

To verify this finding, we performed a formal test described as follows.

Given that the data in developed countries are difficult to obtain and the

management incentive system in Chinese private firms is relatively similar to

that in the developed world, we divided our samples into two groups: state-

owned firms and private firms. A two-sample t-test was performed to

determine whether the average debt levels of the two types of firms differ. We

find (i) no significant difference between the long-term debt ratios (LDR) in

the two types of firms; and (ii) the total and short-term debt ratios in private

firms are significantly (at 1% level) higher than those in state-owned firms.

This result may be explained by the industry difference between state-owned

and private firms as well as provide evidence for the statement that the

managers of state-owned firms are more risk-averse.

5.2. Empirical Estimation Results

In this section, we present the empirical results. The parameter estimates

of the measurement model are reported in Table 6. Most of the factor loadings

are highly significant, suggesting that the identified indicators are highly

associated with the corresponding theoretical attributes. The estimates of

regression coefficients in the structural model are reported in Table 7,

reflecting the impacts of the theoretical attributes and covariates on the

financial leverages. The results are summarized as follows.

Table 5. The Characteristic of Leverage for Chinese Listed Firms

Year TDR (%) SDR (%) LDR (%) STER (%)

1998 43.82 37.44 6.39 54.70

1999 44.56 38.38 6.17 54.15

2000 43.81 37.97 5.84 54.17

2001 45.80 39.94 5.84 51.88

2002 49.64 42.32 7.31 48.49

2003 51.62 44.13 7.48 45.83

2004 54.64 47.27 7.33 41.96

2005 58.97 51.66 7.29 37.81

2006 59.95 52.43 7.48 36.90

Note: STER stands for the ratio of equity to total assets


Firstly, the most significant determinants of the capital structure choice

are the growth opportunity ( ), profitability ( ), and liquidity ( ). The

growth opportunity ( ) has a positive effect on debt levels at the 1% level,

regardless whether it is measured by short-term, long-term or total debt ratios.

The result is in line with the signaling model which predicts that the firms with

the best earnings and growth prospects will employ the highest leverage. This

is because the high growth opportunity represents the firm can produce more

valuable goods, which makes companies less likely to fall into bankruptcy

(Ross, 1977).

The effects of profitability ( ) on debt measures (TDR, LDR, and SDR)

are all negative, indicating higher profitability of a firm leads to more benign

internal financing. This finding agrees with the traditional pecking order

theory.

Liquidity ( ) has negative effects on total and short-term debt ratios at

the 1% significant level and on long-term debt ratio at the 5% significant level,

which confirms the pecking order theory. Managers of listed companies can

finance their investment projects by manipulating the liquid assets at the

expense of their creditors‟ benefits, which increases the agency costs of debt.

Assets structure ( ) has a positive effect on long-term debt ratio at the

1% significant level. Therefore, the ratio of tangible assets is positively related

to the long-term debt ratio, which is consistent with both the pecking order

hypothesis and the trade-off theory. This result confirms the situation in China

that most long-term loans must be guaranteed by the long-term fixed assets.

Non-debt tax shields ( ) positively influence at the 1% level both total

debt and short-term debt ratios. This result agrees with the findings by Bradley

et al. (1984).

Size ( ) has negative effects on total and short-term debt ratios at the

1% level, but has no significant effect on long-term debt ratio. This result

confirms the pecking order theory, based on which the information asymmetry

is expected to be lower for larger firms. Thus, larger firms should be more

capable of issuing informational sensitive securities. In China, however, there

might be another reasonable explanation for this relation, which is the fact that

larger firms have better access to the capital market for equity finance because

of their reputation in the market.

3 5 6

3

5

6

1

2

4

Table 6. Measurement Equation: Factor Loading for Independent Variables

Variable

TANG INTG DEPR NDTS TAPC PORPC LTA LPOR ROE

1

-

1.654

(4.504)

***

1

-

2.434

(7.356)

***

1

-

-0.110

(-1.142)

1

-

1.155

(39.406)

***

1

-

ROA NSR NGR EPS LR QR STATEP CORPP CIRA

3.883

(9.062)

***

1.073

(5.726)

***

1.031

(5.585)

***

3.749

(9.040)

***

1

-

1.003

(75.131)

***

1

-

-0.962

(-22.542)

***

-0.148

(-2.792)

***

Residual Variance

TANG INTG DEPR NDTS TAPC PORPC LTA LPOR ROE

0.976

(20.423)

***

0.882

(16.032)

***

0.977

(20.247)

***

0.603

(11.884)

***

0.817

(19.599)

***

0.989

(20.769)

***

0.286

(17.177)

***

0.075

(9.653)

***

0.937

(20.723)

***

ROA NSR NGR EPS LR QR STATEP CORPP CIRA

0.143

(10.007)

***

0.926

(20.594)

***

0.930

(20.662)

***

0.201

(12.656)

***

0.066

(12.548)

***

0.065

(12.440)

***

0.165

(5.626)

***

0.238

(7.869)

***

0.972

(20.309)

***

Note: t statistics are listed in parenthesis. *** stand for statistically significant at the 1%.

Table 7. Estimates of Structural Coefficients

Debt

Measures

Attributes

TDR 0.540

(0.701)

2.684

(4.397)

***

3.251

(9.096)

***

-1.003

(-4.197)

***

-4.419

(-6.826)

***

-0.937

(-6.771)

***

0.348

(2.499)

**

0.019

(0.675)

0.147

(5.416)

***

0.027

(0.912)

-0.011

(-0.411)

LDR 3.491

(4.173)

***

-1.002

(-

1.161)

3.410

(7.283)

***

0.154

(0.495)

-3.397

(-5.072)

***

-0.528

(-2.602)

**

0.465

(2.850)

***

-0.039

(-1.166)

0.107

(3.156)

***

0.081

(2.287)

**

-0.007

(-0.201)

SDR -1.023

(-1.370)

3.804

(6.593)

***

2.368

(6.576)

***

-1.315

(-6.516)

***

-3.818

(-6.646)

***

-0.903

(-6.624)

***

0.205

(2.638)

***

0.043

(1.627)

0.130

(5.076)

***

-0.006

(-

0.204)

-0.010

(-0.424)

Note: t statistics are in parenthesis. ***, **, * stand for statistically significant at the 1%, 5%, 10%.

1 2 3 4 5 6 7 1x 2x 3x 4x


Ownership structure ( ) has significantly positive effects on all

leverages. Specifically, the state-owned shares ratio positively influences all

leverages, whereas corporation shares and tradable shares ratios negatively

influence all leverages. This result is consistent with our preceding argument

and the actual situation in China.

Uniqueness ( ) is not a significant variable. This result contradicts the

trade-off theory and the explanation by Titman and Wessels (1988). The

insignificant association between the amount of selling expenses and the debt

ratio can be explained by the fact that Chinese firms are less likely to take into

account their stakeholders‟ interests.

For operational risk ( ), the estimated effects of earnings volatility on

leverages are not negative and close to zero. Firms may ignore the volatility of

earnings if the risk and costs of entering liquidation are low. This may occur if

ownership is concentrated (Deesomsak et al., 2004), as is the case in China. In

such a situation, coupled with the particular micro policy characteristics in

China, the bankruptcy judgment, even if enforced, may not be very effective.

In fact, many companies, especially state-owned ones, often obtain substantial

help from the government. For example, during the recent financial crisis

(2008-2012), the Chinese government made effort to rejuvenate the economy

and launched a huge stimulus package. This indicates that the effect of

financial distress on Chinese firms is minor compared with firms in developed

countries. Therefore, it is rational for Chinese firms to assign less weight to

risks of bankruptcy when they make their decisions on the capital structure.

For signal ( ), we find that dividend payout ratio has no significant

effects on short-term and total debt ratios but has a positive effect on the long-

term debt ratio. The above findings support the asymmetric information theory

which argues that a high dividend payout ratio implies that the firm is in a

good financial situation, making debt financing more accessible. The dividend

payout ratio in China is only 25% on average (see Table 4), and the

information passed by the low ratio would be displayed in a long period.

Therefore, dividend payout ratio does not significantly influence short-term

leverage.

Tax shields ( ) are weakly related to the leverages. This result does not

confirm extant capital structure theories, but is consistent with the analysis of

the Chinese tax system in previous studies. Income tax rate is generally low

because of the tax preferential policies, which leads to the insignificance of tax

shields effect.

7

1x

2x

3x

4x

The Determinants of Capital Structure Choice for Chinese Listed … 25

5.3. Model Assessment and Robustness

Under the Bayesian framework, a commonly used statistic for model

assessment is the partial posterior predictive (PPP) p-value (Bayarri and

Berger, 2000). The model fitting is good if the PPP p-value is close to 0.5. In

this analysis, the PPP p-value of the proposed SEM is 0.456, indicating the

good fitting.

To check the robustness of the proposed SEM and the obtained results, we

conducted a regression analysis using proxies directly with the OLS method.

The parameter estimates are produced with SAS and presented in Table 8. We

find that except for firm size and ownership structure, the relationships

between debt ratios and the proxies are consistent with those between the debt

ratios and the relevant attributes identified by the SEM. The inconsistent

results in firm size and ownership structure may be caused by the

multicollinearity problem. To verify this, the CI2 value is 54.90 for LTA and

LPOR, and 19.56 for STATEP, CORPP, and CIRA, implying high

multicollinearity. The empty cells in Table 8 indicate that the corresponding

proxies are deleted by automatic variable selection procedure in SAS, which

shows that some useful proxies of the latent attributes are not included in the

OLS regression model due to multicollinearity or statistical insignificance.

5.4. Multisample Analysis

In China, state-owned firms and private firms have very different

characteristics in terms of accessing capital and industry distribution.

Therefore, determining whether the result based on the whole samples still

holds for the two types of firms is of interest. Therefore, we divided the data

set into two groups: state-owned firms (SF) and private firms (PF). The SF

group is made up of 538 firms, whereas the PF group comprises 314 firms. For

each group, we re-conducted the empirical analysis using the proposed SEM.

The following results were obtained. The factor loadings are almost identical

in the two groups, implying similar weighting systems in measuring the latent

attributes. To save space, the estimates of factor loadings are not presented.

However, the effects of the determinants on capital structure choice exhibit

diverse patterns in the two groups (see Table 9). The relationships between

2 The condition index (CI) is used to measure multicollinearity. If CI is larger than 15, there is a

possible problem with collinearity between variables. For details please refer to Belsley et al.

(1980).


three debt ratios and growth ( ), profitability ( ), and liquidity ( ) are

irrelevant to the types of companies. The ownership structure ( ) has no

significant impact because the corresponding information has been

incorporated in the type of companies. The relationships that are relevant to

the type of companies are given below.

Assets structure ( ) has no significant impact on the total and short-term

debt ratios in both groups. This is similar to the result obtained in Section 5.2.

However, its impact on long-term debt ratio is positive in the SF group but

negative in the PF group. This difference may stem from the large

manufacturing industry structure in China. The majority of manufacturing

enterprises are state-owned, and they have higher proportions of tangible

assets with longer operating lives. Therefore, they have more collateral assets

for long-term debt financing.

Table 8. Estimates of Regression Coefficients

Variables TDR LDR SDR

TANG

INTG 0.265 (10.37)**

DEPR 0.248 (11.52)*** 0.201 (35.95)***

NDTS 0.052 (5.26)** -0.035 (15.00)*** 0.061 (6.18)**

PORPC

TAPC 0.165 (20.81)*** 0.093 (29.93)*** 0.073 (7.17)***

LTA 0.021 (16.53)* -0.063 (11.12)***

LPOR 0.023 (5.76)** 0.065 (15.00)***

ROA -2.149 (41.40)*** -2.133 (64.21)***

ROE

NSR -0.005 (4.65)** -0.006 (10.96)***

NGR 0.136 (5.84)**

EPS -0.352 (26.30)*** -0.128 (45.31)*** -0.272 (27.84)***

LR -0.078 (96.98)*** -0.007 (3.14)* -0.040 (4.74)**

QR -0.035 (2.88)*

STATEP -0.210 (11.51)*** -0.195 (17.88)***

CORPP -0.171 (5.71)** -0.151 (7.92)***

CIRA

OER

SDPOR 0.174 (29.42)*** 0.052 (12.65)*** 0.142 (32.50)***

DPR

ITR

Note: F statistics are in parenthesis. ***, **, * stand for statistically significant at the

1%, 5%, 10%.

3 5 6

7

1

Table 9. Estimates of Structural Coefficients for State-owned and Private Firms

Types of

Company Leverages

Attributes

State-

owned

TDR 0.024

(0.046)

1.800

(5.458)

***

1.509

(5.224)

***

-0.484

(-3.663)

***

-2.113

(-6.395)

***

-0.595

(-4.876)

***

0.073

(0.937)

0.062

(3.023)

***

0.118

(5.599)

***

-0.033

(-1.686)

*

0.074

(1.180)

LDR 0.771

(1.700)

*

-1.359

(-3.769)

***

0.987

(4.366)

***

0.333

(2.219)

**

-0.783

(-2.220)

**

-0.280

(-2.229)

**

0.068

(0.820)

-0.051

(-2.003)

**

0.107

(2.759)

***

0.035

(1.483)

**

0.095

(1.260)

SDR -0.344

(-0.469)

2.883

(6.846)

***

0.831

(3.538)

***

-0.760

(-5.279)

***

-2.233

(-5.985)

***

-0.601

(-3.950)

***

0.057

(0.670)

0.102

(4.357)

***

0.112

(4.724)

***

-0.057

(-2.522)

**

0.045

(0.627)

Private TDR 0.046

(1.334)

1.766

(1.693)

*

5.226

(9.291)

***

-0.599

(-1.563)

-6.567

(-6.120)

***

-0.589

(-2.516)

**

0.164

(0.339)

0.047

(0.806)

0.086

(1.378)

0.134

(1.626)

*

-0.024

(-

0.652)

LDR -2.911 (-1.734)

*

5.703 (5.558)

***

4.713 (4.216)

***

-0.619 (-1.030)

-2.773 (-2.114)

**

-0.058 (-0.166)

-0.834 (-1.361)

0.044 (0.592)

0.113 (1.362)

0.138 (1.311)

-0.023 (-

0.458)

SDR 2.122 (1.597)

-0.581 (-0.593)

4.177 (7.342)

***

-0.440 (-1.368)

-6.774 (-7.500)

***

-0.700 (-3.696)

***

0.607 (1.504)

0.036 (0.724)

0.052 (0.958)

0.099 (1.364)

-0.019 (-

0.596)

Note: t statistics are in parenthesis. ***, **, * stand for statistically significant at the 1%, 5%, 10%.

1 2 3 4 5 6 7 1x 2x 3x 4x


The effects of non-debt tax shields ( ) on long-term debt ratio of the two

groups have reverse directions. The negative impact on state-owned firms is in

accordance with the findings in many studies (e.g., Wald, 1999; Ozkan, 2001;

Korajczyk and Levy, 2003; Sogorb-Mira, 2005). The positive impact on

private firms is also reasonable and confirmed (e.g., Bradley et al., 1984;

Delcoure, 2007). We also noticed that the relationship between non-debt tax

shield and short-term debt ratio is positively significant in SF but insignificant

in PF.

Firm size ( ) has no significant impact on debt ratios for private firms,

which can be attributed to their small size. The relationships between firm size

and total and short-term debt ratios for state-owned firms are the same as those

in Section 5.2. Notably, firm size has a positive impact on long-term debt ratio

for the state-owned firms, which supports the asymmetric information theory.

While in the previous analysis, this positive impact is insignificant.

Government policies that support large state-owned companies to use debt also

contribute to the positive relationship.

The effects of uniqueness(x1) on debt ratios are significant in the SF group

but insignificant in the PF group because most of the companies in

monopolistic industries are state-owned. Furthermore, uniqueness in state-

owned firms is positively related to total and short-term debt ratios, and

negatively related to long-term debt ratio. This result confirms Titman and

Wessels (1988), which states that monopolistic companies can easily obtain

short-term loans.

The effects of operational risk (x2) on debt ratios are positively significant

in the SF group but insignificant in the PF group. This confirms our

explanation in Section 5.2, where we stated that the size of state-owned

companies is usually very large. If these companies go bankrupt, huge social

costs will be generated. Thus, the government usually implements effective

measures to help these companies survive bankruptcy. Consequently, state-

owned banks are more willing to provide loans to state-owned companies even

though the operational risk of these companies may be high.

Dividend payout ratio (x3) has negative effects on total and short-term

debt ratios in state-owned firms. This means that if the state-owned firms are

in good financial standing, they will have enough retained surplus to sustain

their operations. For the private firms, however, dividend payout ratio is

positively related to total debt ratio, indicating that dividends are used as a

signal of good financial condition to investors. With positive signals, private

firms can easily access debt financing.

2

4


Finally, similar to the preceding result, tax shields (x4) have no impact on

debt ratios in both groups. In conclusion, with the effects of unique Chinese

institutional settings and industry difference between Chinese private and

state-owned firms, some similarities and differences in the influential patterns

of the determinants on capital structure choice are identified.

CONCLUSION

China has special characteristics including the imperfect capital market

and banking system, poorly specified property rights and laws, and

institutional uncertainty. In this chapter, we employed a structural equation

model with Bayesian approach to analyze the determinants of capital structure

choice for Chinese listed companies. We find a remarkable difference between

the capital structure of firms in China and developed countries, which lies in

the Chinese firms‟ low overall debt levels with a small portion of the long-

term debt. Three possible reasons have been identified. First, constraints of

debt financing in China, especially for the long-term loans, are rather

restrictive. Second, Chinese listed companies use equity more frequently than

debt in raising long-term investment capital. Third, compare with their

counterparts in developed countries, Chinese firm managers are more risk-

averse, and tend to take less debt.

The results of this empirical study show that some insights from modern

finance theories are applicable to China, especially the pecking order theory.

However, there are also some unique patterns of the determinants of capital

structure choice for Chinese listed firms. First, the most influential attributes

of the capital structure choice are growth opportunities, profitability, and

liquidity. Profitability and liquidity of the firms have negative impacts on the

three debt ratios. This supports the pecking order theory. Growth has positive

impacts on debt ratios, which verifies the agency theory. Second, asset

structure has no significant effect on total debt ratio, but has significantly

positive effect on long-term debt ratio, which agrees with the pecking order

theory. Third, leverages, as measured by short-term debt and total debt ratios,

increase with non-debt tax shield. Fourth, firm size has negative impacts on

short-term debt and total debt ratios, which agrees with the pecking order

theory. Fifth, the ownership structure significantly affects the firms‟ capital

structure choice. This unique pattern is consistent with the special

characteristics of China. Sixth, leverages increase with operational risk, which

is inconsistent with the pecking order theory. This unique feature is related to


the institutional settings and government policies in China. Seventh, the

association between firms‟ signal and leverage confirms the asymmetric

information theory. Finally, the insignificant effects of uniqueness and tax

shields on the capital structure choice contradict the extant theories, but are

consistent with the Chinese special reality.

The same analysis was also conducted for Chinese state-owned and

private firms, respectively. Some different patterns for the two groups of firms

have been identified. First, the effects of assets structure on the long-term debt

are opposite for the two groups mainly because of the industry difference.

Second, the impacts of non-debt tax shield on long-term debt for the two

groups are diverse, both of which are meaningful. Third, firm size is not a

significant determinant of debt ratios for private firms, but an important factor

that affects debt ratios of state-owned firms. Forth, uniqueness and operational

risk are also only related to the capital structure choice for state-owned firms.

Finally, dividend acts as a positive signal for the private firms to better access

the external loans, while the state-owned firms that have profitable projects do

not favor outside debts.

In summary, although western capital structure theories do not completely

explain the determinants of capital structure choice for Chinese listed

companies, the attributes identified by those theories are of highly useful.

ACKNOWLEDGMENT

The research was supported by the NSFC 11471277, and 4053087 from

the Direct Grant of the Chinese University of Hong Kong.

REFERENCES

Agarwal, S. & Mohtadi, H. (2004). Financial Markets and the Financing

Choice of Firms: Evidence from Developing Countries. Global Finance

Journal, 15(1), 57-70.

Bayarri, M. J. & Berger, J. O. (2000). P Values for Composite Null Models.

Journal of the American Statistical Association, 95, 1127-1142.

Belsley, D. A., Kuh, E. & Welsch, R. E. (1980). Regression Diagnostics. New

York: Wiley.


Berkman, H., Cole, R. & Fu, J. (2002). From State to State: Improving

Corporate Governance When the Government is a Large Block Holder.

University of Auckland, New Zealand, Working Paper.

Bevan, A. A. & Danbolt, J. (2002). Capital Structure and Its Determinants in

the United Kingdom–a Decompositional Analysis. Applied Financial

Economics, 12(3), 159-170.

Booth, L., Aivazian, V., Demirguc-Kunt, A. & Maksimovic, V. (2001).

Capital Structures in Developing countries. Journal of Finance, 56(1), 87-

130.

Bradley, M., Jarrell, G. A. & Kim, E. H. (1984). On the Existence of an

Optimal Capital Structure: Theory and Evidence. Journal of Finance,

39(3), 857-878.

Chang, C. F., Lee, A. C. & Lee, C. F. (2009). Determinants of Capital

Structure Choice: a Structural Equation Modeling Approach. The

Quarterly Review of Economics and Finance, 49(2), 197-213.

Chen, J. J. (2004). Determinants of Capital Structure of Chinese Listed

Companies. Journal of Business Research, 57(12), 1341-1351.

DeAngelo, H. & Masulis, R. (1980). Optimal Capital Structure under

Corporate and Personal Taxation. Journal of Financial Economics, 8(1),

3-29.

Deesomsak, R., Paudyal, K. & Pescetto, G. (2004). The Determinants of

Capital Structure: Evidence from the Asia Pacific Region. Journal of

Multinational Financial Management, 14(4, 5), 387-405.

Delcoure, N. (2007). The Determinants of Capital Structure in Transitional

Economies. International Review of Economics and Finance, 16(3), 400-

415.

De Haas, R. & Peeters, M. (2006). The Dynamic Adjustment towards Target

Capital Structures of Firms in Transition Economies. Economics of

Transition, 14(1), 133-169.

Drobetz, W. & Wanzenried, G. (2006). What Determines the Speed of

Adjustment to the Target Capital Structure?. Applied Financial

Economics, 16(13), 941-958.

Fama, E. & French, K. (2002). Testing Trade-off and Pecking Order

Predictions about Dividends and Debt. Review of Financial Studies, 15(1),

1-33.

Fan, J. P., Titman, S. & Twite, G. (2012). An International Comparison of

Capital Structure and Debt Maturity Choices. Journal of Financial and

Quantitative Analysis, 47(1), 23-56.


Fattouh, B., Scaramozzino, P. & Harris, L. (2005). Capital Structure in South

Korea: a Quantile Regression Approach. Journal of Development

Economics, 76(1), 231-250.

Feidakis, A. & Rovolis, A. (2007). Capital Structure Choice in European

Union: Evidence from the Construction Industry. Applied Financial

Economics, 17(12), 989-1002.

Graham, J. R. & Harvey, C. R. (2001). The Theory and Practice of Corporate

Finance: Evidence from the Field. Journal of Financial Economics,

60(2, 3), 187-243.

Harris, M. & Raviv, A. (1991). The Theory of the Capital Structure. Journal of

Finance, 46(1), 297-355.

Homaifar, G., Zietz, J. & Benkato, O. (1994). An Empirical Model of Capital

Structure: Some New Evidence. Journal of Business Finance &

Accounting, 21(1), 1-14.

Huang, G. H. & Song, F. M. (2006). The Determinants of Capital Structure:

Evidence from China. China Economic Review, 17(1), 14-36.

Jensen, M. & Mecking, W. (1976). Theory of the Firm: Managerial Behavior,

Agency Costs, and Capital Structure. Journal of Financial Economies,

3(4), 305-360.

Kayo, E. K. & Kimura, H. (2011). Hierarchical Determinants of Capital

Structure. Journal of Banking & Finance, 35(2), 358-371.

Kim, H. & Berger, P. D. (2008). A Comparison of Capital Structure

Determinants: the United States and the Republic of Korea. The

Multinational Business Review, 16(1), 79-100.

Korajczyk, R. A. & Levy, A. (2003). Capital Structure Choice:

Macroeconomic Conditions and Financial Constraints. Journal of

Financial Economics, 68(1), 75-109.

Kraus, A. & Litzenberger, R. H. (1973). A State-Preference Model of Optimal

Financial Leverage. Journal of Finance, 28(3), 911 – 922.

Lööf, H. (2004). Dynamic Optimal Capital Structure and Technical Change.

Structural Change and Economic Dynamics, 15(14), 449-468.

Lu, B., Song, X. Y. & Li, X. D. (2012). Bayesian Analysis of Multi-group

Nonlinear Structural Equation Models with Application to Behavioral

Finance. Quantitative Finance, 12(3), 477-488.

Mackie-Mason, J. K. (1990). Do Taxes Affect Corporate Financing

Decisions?. Journal of Finance, 45(5), 1471-93.

Manos, R., Murinde, V. & Green, C. J. (2007). Leverage and Business

Groups: Evidence from Indian Firms. Journal of Economics and Business,

59(5), 443-465.


Mazur, K. (2007). The Determinants of Capital Structure Choice: Evidence

from Polish Companies. International Atlantic Economic Society, 13(4),

495-514.

Modigliani, F. & Miller, M. H. (1963). Corporate Income Taxes and the Cost

of Capital: a Correction. American Economic Review, 53(3), 433-443.

Modigliani, F. & Miller, M. H. (1958). The Cost of Capital, Corporation

Finance and the Theory of Investment. American Economic Review, 48(3),

267-297.

Myers, S. C. & Majluf, N. S. (1984). Corporate Financing and Investment

Decision When Firms Have Information that Investor Do Not Have.

Journal of Financial Economic, 13(2), 187-221.

Nguyen, T. D. K. & Ramachandran, N. (2006). Capital Structure in Small and

Medium-sized Enterprises, the Case of Vietnam. Asean Economic

Bulletin, 23(2), 192-211.

Nivorozhkin, E. (2005). Financing Choices of Firms in EU Accession

Countries. Emerging Markets Review, 6(2), 138-169.

Ozkan, A. (2001). Determinants of Capital Structure and Adjustment to Long

Run Target: Evidence from UK Company Panel Data. Journal of Business

Finance & Accounting, 28(1-2), 175-198.

Öztekin, Ö. (2014). Capital Structure Decisions Around the World: Which

Factors are Reliably Important?. Journal of Financial and Quantitative

Analysis, Forthcoming.

Öztekin, Ö. & Flannery, M. J. (2012). Institutional Determinants of Capital

Structure Adjustment Speeds. Journal of Financial Economics, 103(1),

88-112.

Rajan, R. G. & Zingales, L. (1995). What Do We Know about Capital

Structure? Some Evidence From International Data. Journal of Finance,

50(5), 1421-1460.

Ross, S. (1977). The Determinants of Financial Structure: the Incentive-

signaling Approach. Bell Journal of Economics, 8(1), 147-175.

Sogorb-Mira, F. (2005). How SME Uniqueness Affects Capital Structure:

Evidence from a 1994-1998 Spanish Data Panel. Small Business

Economics, 25(5), 447-457.

Song, X. Y. & Lee, S. Y. (2012). Basic and Advanced Bayesian Structural

Equation Modeling: With Applications in the Medical and Behavioral

Sciences. UK: Wiley.

Song, X. Y., Lu, Z. H. & Feng, X. N. (2014). Latent Variable Models with

Nonparametric Interaction Effects of Latent Variables. Statistics in

medicine, 33(10), 1723-1737.


Titman, S. (1984). The Effect of Capital Structure on the Firm‟s Liquidation

Decision. Journal of Financial Economics, 13(1), 137-152.

Titman, S. & Wessels, R. (1988). The Determinants of Capital Structure

Choice. The Journal of Finance, 43(1), 1-19.

Wald, J. K. (1999). How Firm Characteristics Affect Capital Structure: an

International Comparison. Journal of Financial Research, 22(2), 161-187.

Xu, X. N. & Wang, Y. (1999). Ownership Structure and Corporate

Performance in Chinese Stock Companies. China Economic Review,

10(1), 75-98.



Chapter 2

AN EXAMINATION OF PREDICTORS

AND OUTCOMES RELATED TO SCHOOL

CLIMATE USING LATENT CLASS ANALYSIS

Christine DiStefano, Elizabeth Leighton,

Mihaela Ene and Diane M. Monrad University of South Carolina, US

ABSTRACT

A favorable school climate provides the structure within which students,

teachers, administrators, and parents function cooperatively and

constructively. Measures of school climate, however, have received only

passing interest from policy makers as critical elements in accountability

reporting. This study used a state-wide dataset of climate ratings from

610 elementary schools and considered multidimensional information

from both teachers and students to produce latent classes of school

climate. Two variables, school size and a school’s poverty index, were

used as covariates when creating latent classes. In addition, two measures

of school performance were examined as distal outcomes. The study

identified four classes, where classes were distinguished based upon

school climate scores. Differences in outcome variables and covariates

were observed across the classes. The information may be used by school

Correspondence: 138 Wardlaw Hall, Columbia, SC 29208, [email protected], 803-777-

4362

Christine DiStefano, Elizabeth Leighton, Mihaela Ene et al. 36

personnel in examinations of malleable factors related to school

performance.

INTRODUCTION

Declining aptitude test results during the 1970s and 1980s, combined with

unflattering international comparisons, led legislators across the United States

to enact far reaching educational accountability measures for public schools.

Perhaps the most controversial, No Child Left Behind (NCLB) Act of 2001,

set a goal for all children to demonstrate achievement at least equal to their

grade level by the year 2014. While the merits of NCLB and other high stakes

accountability systems have been heatedly argued, there is no doubt that in

every state there has been an intense focus on academic performance of both

the students and the school.

Along with a focus on academic performance, the “school report card” or

“school profile” has become ubiquitous in the accountability movement.

Report cards, required under the provisions of NCLB, most typically include

mandated information on student achievement, teacher qualifications,

attendance, and other variables that provide descriptive information about the

school and its programs.

All the states have some form of a school-level reporting system

accessible over the world-wide web, providing a dizzying array of information

and data. Student attendance (and dropout data for secondary schools), student

behavior indices (e.g., incidents of tardiness, cutting class, and disruptive or

criminal acts), and teacher qualifications are customarily reported. Measures

of school climate, however, have received only passing interest from policy

makers as critical elements in accountability reporting.

Too often, the importance of school climate as a critical contextual factor

in which teachers teach and students learn has tended to fade into the

background, a casualty of other "priorities." And yet, among the preconditions

for school success, few rival the motivation to teach and the motivation to

learn: Do students wish to attend school and engage in learning activities? Do

teachers want to return to the same school next year? Do parents desire to

become involved with their child's school programs? The answers to these

questions very often hinge on the levels of support, challenge, collaboration,

and partnership provided to them by the school: in short, its climate.

An Examination of Predictors and Outcomes … 37

Defining School Climate

Every school may be thought of as having a distinct personality or

climate. A favorable school climate provides the structure within which

students, teachers, administrators, and parents function cooperatively and

constructively. Hoy and Miskel (1982) defined school climate as a school‟s

personality, and its importance has intrigued researchers for approximately 50

years (Anderson, 1982). Edmunds (1982) and Lezotte (1990) were prominent

in linking climate directly to school effectiveness more than 35 years ago.

According to Perkins (2006), school climate is the learning environment

created through the interaction of human relationships, physical setting, and

psychological atmosphere.

School climate is typically thought to involve four distinct parts (Allen,

Thompson, Hoadley, Engelking, & Drapeaux, 1997; Sackney, 1988): ecology,

milieu, social system, and culture. Ecology comprises physical and material

features of schools, such as age of the building and cleanliness. The milieu

involves the personnel (e.g., administrators, teachers, parents, staff, students,

etc.) involved with a school. A social system is described as the “rules” which

a school uses to interact with members. Finally, school culture consists of

shared norms, values, and beliefs of the members. The two related topics of

climate and culture are delineated by Allen et al. (1997) where …“culture

establishes normative behavior for the members of organizations, and climate

is the perceptions of those norms” (p.1). Together, students, teachers,

administration, parents, and the broader community all contribute to the school

climate (National School Climate Center [NSCC], Center for Social and

Emotional Education [CSEE], & National Center for Learning and Citizenship

at Education Commission of the States, 2008).

The construct of school climate is generally characterized as

multidimensional and representative of shared perceptions of behavior

including customs, goals, values, relationships, teaching practices, and

structures within the school (Ashforth, 1985; Cohen, 2009; CSEE, 2010; Hoy,

1990; Van Houtte, 2005). Most studies include four primary components when

measuring school climate: (1) safety of students and staff, (2) school culture

and relationships, (3) elements of teaching and learning, and (4) the

institutional environment (Cohen, 2009; CSEE, 2010; Tagiuri, 1968).

Research Involving School Climate. Demographic variables such as

ethnicity and socioeconomic status of students are other components that

influence school climate and achievement (Chen & Weikart, 2008). Several

researchers have identified a relationship between school climate and school


effectiveness. Specifically, positive school climate has been found to correlate

with higher rates of academic achievement including standardized test scores,

as well as increased classroom engagement, student participation, and

motivation to learn (CSEE, 2010; Chen & Weikart, 2008; DiStefano, Monrad,

May, McGuinness, & Dickenson, 2007; Edmunds, 1982; Greenberg, 2004;

Lee & Burkham, 1996; Lezotte, 1990; NSCC et al., 2008; Roney, Coleman, &

Schlictin, 2007; Sebring, Allensworth, Bryk, Easton, & Luppescu, 2006;

Stewart, 2007). Positive school climate has also been linked to indicators of

school success reported for accountability purposes including academic

achievement, annual yearly progress (AYP) measures, and school report card

information (Greenberg, 2004; Lee & Burkham, 1996; Macneil, Prater, &

Busch, 2009; DiStefano et al., 2007; Monrad, May, DiStefano, Smith, Gay,

Mindrila, Gareau, & Rawls, 2008; Tubbs & Garner, 2008).

In addition to impacting learning outcomes, a supportive school climate

has been associated with positive psychological and behavioral student

outcomes. Supportive school climate has been linked to reductions in

behavioral conduct problems, instances of bullying, rates of depression and

substance use, self-esteem, absenteeism, and dropout rates (Brand, Felner,

Shim, Seitsinger, & Dumas, 2003; Bryk & Thum, 1989; Christle, Jolivette, &

Nelson, 2007; CSEE, 2010; Gottfredson, Gottfredson, Payne, & Gottfredson,

2005; Loukas & Murphy, 2007; NCSS et al., 2008; Rumberger, 1995; Way,

Reddy, & Rhodes, 2007). Fostering a positive school climate provides students

a behavioral model for how society operates and provides examples of

appropriate conduct outside of the school walls (NSCC et al., 2008).

Patterns of climate variables have also been related to trust (Hoy, Tarter,

& Kottkamp, 1991). When trust is high, educators are more likely to

experiment with new practices and work together with parents to advance

improvements (Bryk & Schneider, 2002). Trust matters because effective

school leadership depends upon the competence and cooperation of a school

team; important school goals cannot be developed and accomplished by a

single person (Tschannen-Moran, 2004). Trust levels between students and

staff also influence student behavior and educational outcomes (Virtanen,

Kivimaki, Luopa, Vahtera, Elovainio, Jokela, & Pietikainen, 2009). Research

indicates that, for teachers, some of the most important aspects of the school

climate include the freedom to disclose stress to administrators, student

behavior, and collaborative relationships with parents (Grayson & Alvarez,

2008). Teacher benefits of a positive working environment include increased

job satisfaction (Grayson & Alvarez, 2008; Ma & MacMillan, 1999; Tubbs &

Garner, 2008), increased retention and attendance, and better home-school


relationships (Brown & Medway, 2007). Teacher and staff perceptions were

pivotal in measuring school climate in early research, however, there has been

an increasing interest in examining students‟ perceptions of school climate

(e.g., Koth, Bradshaw, & Leaf, 2008; Way, Reddy, & Rhodes, 2007).

Previous research suggested the existence of a relationship between school

climate and school poverty level, with higher levels of poverty being

associated with a less positive school climate (Bernstein, 1992; “School

Climate, Discipline, and Safety”, 2013). In addition, many researchers have

noted the negative impact of poverty on educational outcomes such as

academic achievement (Sirin, 2005; Malecki & Demaray, 2006; Monrad et al.,

2008; Hopson & Lee, 2011), behavior problems (Hopson & Lee, 2011),

dropout rates (Cataldi, Laird, & Kewal-Ramani, 2009), and graduation rates

(Monrad et al., 2008). More specifically, these studies suggested that higher

levels of poverty are associated with lower grades and overall GPA, lower

scores on standardized tests, lower graduation rates, as well as higher rates of

problem behavior and dropout rates. Furthermore, results showed a strong

negative relationship between school poverty and school absolute value which

is the basis for determining school absolute rating (Monrad et al., 2008).

Larger Scale Studies of School Climate. Most studies involving school

climate administer a survey to participants at a smaller level, such as one

school or one district, and identify factors of climate that are important to

consider. See Sackney (1998) for a comprehensive review of factors identified

with previous school climate studies. School climate, as a factor to increase

student achievement, has been receiving increased attention in the school

improvement literature. The Consortium on Chicago School Research (CCSR)

used information from principals, teachers, and students across over 200

schools to identify “five essential supports for school improvement” (Sebring

et al., 2006). The CCSR found the important factors to be: leadership,

professional capacity (e.g., knowledge, skills, and disposition of faculty),

parent-community ties, climate, and instruction. To gain a greater

understanding of the impact of the five supports on school achievement, the

CCSR examined the relationship between the five factors and student

achievement as measured by a standardized test. The findings showed that

schools strong in most (e.g., 3 to 5) essential areas were up to 10 times likely

to make gains in both reading and mathematics standardized test scores on the

Iowa Test of Basic Skills (grades 3-8). Sebring et al. (2006) also found that

improvements in the essential supports also led to improved achievement.

An earlier study of elementary schools in Chicago also highlighted the

importance of positive school climate characterized by mutual trust and


respect. According to Bryk and Schneider (2002), schools with a high degree

of “relational trust” between administrators, teachers, and parents are far more

likely to make the kinds of changes needed to improve student achievement

than schools where relationships are poor. Bryk and Schneider compared 100

schools that made the greatest improvement on achievement tests (reading and

mathematics) between 1991 and 1996 with 100 schools that made little or no

improvement. They discovered that schools with high levels of trust at the

beginning of reform efforts had a 1 in 2 chance of making significant

improvements in reading and math achievement, while schools with low levels

of trust had a 1 in 7 chance of making achievement gains. Among the schools

with initially low levels of trust, only those schools where trust was

strengthened over the course of reform efforts showed achievement gains. No

school that continued to have low levels of relational trust improved student

achievement levels to any appreciable degree.

There have been few studies that have investigated school climate on a

state or national level. Using exploratory factor analysis (EFA) techniques

with the California School Climate and Safety Survey and 7,524 students in

grades 6 to 12, Furlong et al., (2005) identified two broad categories of

climate: school climate (support from teachers, enforcement of school rules)

and school safety (perceptions of safety and gang activity). However, EFA

techniques were conducted within each dimension separately rather than

across the survey.

Also, Greenberg (2004) used a national dataset for 4th

, 8th

, and 12th

graders

to determine how NAEP mathematics scores were affected by school climate.

Using EFA, three components of school climate were identified: student

behavior, parental involvement, and school morale. Further, regression

analyses showed that NAEP mathematics scores were increased as climate

scores increased, even when school characteristics (e.g., poverty, urbanicity,

type of school, school size) were controlled.. While this study provided an

investigation of the relationship between student achievement and school

climate, only mathematics was studied.

The studies by Greenberg (2004), Furlong et al. (2005), and Sebring et al.

(2006) recognize the dimensional nature of school climate and its relationship

to achievement. Compared with other barriers which cannot be controlled by

schools, such as high child poverty, previous work has supported the notion

that school climate is not a fixed school condition and that climate can be

changed (Greenberg, 2004), potentially affecting accountability ratings.

Considering the benefits, both socially and academically, of a positive school

climate, it would be of interest for states or districts to group schools based


upon the level of school climate and to intervene for those schools suffering

from negative climate.

South Carolina is currently one of only a few states in the country that

includes climate data from surveys of students, teachers, and/or parents on

their school report cards. South Carolina‟s report card was developed in

response to requirements of the state‟s Education Accountability Act of 1998

(SC Code of Laws, Section 2, Chapter 18, Title 59). The specific variables and

data elements were selected by the General Assembly‟s Education Oversight

Committee working in collaboration with the State Department of Education

and the State Board of Education. The inclusion of school climate data from

“evaluations of the school by parents, teachers, and students” in the school,

district, and state report cards is a specific requirement of the state‟s

accountability legislation (SC Code of Law, 59-18-900 (D)). School climate

data in South Carolina is collected annually from questionnaires administered

to parents, teachers, and students.

Using a state-wide database of both teacher and student responses, cluster

analysis was used to identify groups of schools related to climate (DiStefano et

al., 2007). These groups were replicated over a two-year period. The authors

identified four categories of schools, where schools differed in the degree to

which they had positive school climate. In addition, schools were differentially

related to report card outcomes, where schools with the most positive average

climate ratings also showed the most positive report card factors, such as

higher standardized test scores, lower teacher turnover, higher student

attendance, and higher AYP scores. Also, schools within each cluster had

varying levels of poverty, showing that low-poverty schools do not necessarily

have poor school climate.

Other cluster analysis studies supported some of these findings and also,

showed that the relationship between school climate cluster membership and

student achievement is consistent across core content areas such as English,

mathematics, history, and science (Bergren, 2014; Smith, 2005). In addition to

supporting the relationship between school climate group membership and

school performance, researchers also used cluster analysis to examine the

moderating effects of school climate on school interventions. Specifically,

findings showed differential effects of a violence prevention intervention by

school climate type, with the intervention having a more positive effect on

student behavior in schools with conducive climate than in schools with

average or distressed climate (Dymnicki & the Multisite Violence Prevention

Project, 2013). Whereas these findings were able to provide evidence of

climate groups and relations among school climate and report card outcomes


or school-level interventions, an older grouping method was used. Instead of

cluster analysis, a newer methodology, mixture modeling, may be employed to

create groups of cases, while controlling for covariates and examining

relations with outcome variables in the same analysis.

Mindrila, DiStefano, Monrad, and Ene (2014) also examined the relation

between four latent classes of climate and school absolute ratings while

controlling for poverty. The study found that classes with more positive

climate generally showed lower levels of poverty. In addition, more positive

school climate was positively related to a school‟s absolute rating. While this

study is similar to the current study, only teacher ratings were used; no data

concerning students‟ view of school climate were included in the analyses.

Further, only one covariate (poverty level) and one outcome (school absolute

rating) were used.

Therefore, the purpose of this study is to identify latent classes of school

climate at the elementary school level, based on teachers‟ and students‟

perception of school climate. Elementary level was chosen because there were

the most schools at this level, and it was thought the higher number of schools

this would produce the most stable typology. Covariates such as school size

and a school‟s poverty level were included to recognize their influence when

creating classes of school climate; outcome variables from school report cards

were used to examine relations between classes.

Latent Class Analysis

Cluster analysis refers to a set of classification procedures used to uncover

homogeneous groups underlying a data set where the number of groups is not

known a priori (Aldenderfer & Blashfield, 1984; Blashfield & Aldenderfer,

1988; Everitt, 1980). This method has been very popular in the social sciences

as a methodology to create groups; however, it is not without criticism. For

example, few fit indices are available with cluster analysis to help researchers

identify an optimal solution (e.g., DiStefano & Kamphaus, 2006). In addition,

cluster solutions may be sensitive not only to the clustering algorithm used to

group cases, but also to the ordering of the cases within the dataset (Blashfield

& Aldenderfer, 1988).

An alternative approach to grouping cases via cluster analysis is to use

latent class clustering methods (e.g., Bacher, 2000; Bensmail, Celeux, Raftery,

& Robert, 1997; Clogg, 1995; Collins & Lanza, 2010; Everitt, 1993; Heinen,

1996; Muthén & Muthén, 2000, 2007; Vermunt & Magidson, 2002). Latent


class cluster analysis has been called other names in the literature, such as

finite mixture modeling (Pastor & Gagne, 2013; McLachlan & Peel, 2000),

model-based clustering (e.g., Banfield & Raftery, 1993), and mixture

likelihood approach to clustering (Everitt, 1993). Latent class clustering

encompasses a broad family of methods that use the same general model,

including latent class analysis, latent profile analysis, mixed-mode clustering,

and latent transition analysis. Although the latent class methods have been

available for many years (e.g., Gibson, 1959; Lazarsfeld & Henry, 1968), the

techniques are enjoying increased popularity through improved computer

capabilities and available software (Vermunt & Magidson, 2002). While there

are similarities between latent class cluster analysis and cluster analysis, there

are also distinctions between the two methods.

As with cluster analysis, latent class cluster (LCC) analysis has a similar

overarching goal: to classify cases into groups where members within a group

are similar to each other and different from individuals in other groups

(Vermunt & Magidson, 2002). Similarly, cases are thought to belong to one of

K groups underlying the dataset where the number of groups is unknown a

priori. The goal is to uncover the total number of classes (termed K)

underlying the dataset, where each class (noted as k) may be thought of as a

sub-population which is discrete and mutually exclusive (Clogg, 1995;

Heinen, 1996).

Given that k different classes underlie a population, individuals within a

certain class have the same probability distribution with respect to the

categorical latent variable. However, while each case is grouped into only one

class, k, the LCC model recognizes that there may be uncertainty in the

classification. Therefore, each case is given a probability value of belonging to

each of the K groups. Values for the weights range between 0 and 1 per class

and sum to 1 across the set of classes.

The use of latent class clustering models has additional advantages over

traditional clustering methods. One advantage is that LCC models use fit

indices which help researchers select a model and report how well the model

performs to fit the data. Another advantage is that LCC allows researchers to

select parameters of specific interest to be included in the model estimation or

to restrict parameters that are not of interest to estimate. The former is referred

to as freeing parameters, the latter as fixing parameters. Model parameters

may be fixed or freed in line with assumptions, model characteristics, or

relationships among variables. Third, covariates and outcomes may be

included in the same model. This allows LCC to incorporate important

covariates which may affect the creation of the latent classes and to examine


classes on important outcomes to see if the groups display distinct

relationships.

LCC subsumes both latent class analysis and latent profile analysis. Both

methods have the same objective, but differ in the level of the observed

variables used to group cases into classes. Measured variables are categorical

within latent class analysis and on a continuous metric with latent profile

analysis (Muthén & Muthén, 2000). As the analyses to be conducted will use

latent profile analysis, discussion is restricted to this technique.

With latent profile analysis, the distribution for each of the k groups can

be defined by the class mean vector and variance-covariance matrix (μk, Σk).

The μk vector represents the class-specific centroid and the Σk matrix

represents the homogeneity of the observed variables within class (i.e.,

indicator variances – represented on the diagonal of Σk); the relationships

between variables beyond what is accounted for by the latent variable are

represented on the off-diagonal elements of Σk (i.e., indicator covariances).

Besides the probability of class membership, these means and variance-

covariance elements are the parameters which will be estimated in an LCC

analysis. With LCC, researchers can relax restrictions to determine which

parameters should be estimated (i.e., freed) or constrained (i.e., fixed).

Imposing different model restrictions allows the evaluation of different

solutions to evaluate which model provides optimal fit to the data in terms of

parsimony, fit indices, and interpretability (Collins & Lanza, 2010). To select

an appropriate model, it is recommended that researchers test different models

by imposing different restrictions on the variance-covariance matrix.

When fitting LCCs, researchers need to know how many latent classes are

optimal. With latent class clustering, the most common way to uncover the

number of underlying groups is to test the fit of various models with

increasing numbers of latent classes (Nylund, Asparouhov, & Muthén, 2007).

In other words, within a certain model specification (e.g., conditional

independence model), a range of class solutions are run and interpreted (e.g., 2

– 6 classes). As classes are added, model fit tends to improve, and one may

define the optimal solution by looking for the most parsimonious model. For

each model, the number of classes extracted may be increased until a solution

does not converge (DiStefano & Kamphaus, 2006; Vermunt & Magidson,

2002). Researchers can evaluate and compare converged solutions to

determine which model best fits the data.

LCC uses model fit criteria to help choose the optimal solution (Clogg,

1995; Vermunt & Magidson, 2002). It is noted, however, that the fit criteria

are largely heuristic in nature. The model log likelihood value may be thought


of as a global fit index, providing information about the overall fit of the

model to the data. Relative fit indices are useful to use when comparing

alternative models to determine which model illustrates a better fit to the data.

Indices in this class may be used to compare models which differ in the

number of groups requested and/or the model specifications. When comparing

models, there is a need to balance information from fit indices and also the

principle of parsimony. Parsimony suggests selecting the solution with the

minimum number of classes possible while achieving an acceptable model fit.

Additional fit indices may be used to compare competing models to select a

model.

The Akaike Information Criteria (AIC) and the Bayesian Information

Criteria (BIC) offer comparative evidence to evaluate different solutions

(Muthén, 2001; Vermunt & Magidson, 2002). The AIC and BIC are

"parsimony criteria" used to compare different model solutions (i.e., different

numbers of groups underlying the data) in order to determine which model fits

the data best. For these indices, the more parameters that are estimated, the

higher the value of AIC/BIC. In addition for adjusting for the number of

parameters, BIC adjusts for the sample size (N), yielding larger values as

sample size increases, all other factors held constant. With AIC and BIC,

comparatively lower values indicate better fitting models (Pastor et al., 2013).

Other LCC-based fit indices measure the degree of uncertainness in the

classifications. These methods require that individuals are grouped to

determine how well the model works to classify cases. Posterior probabilities

denote the probability of class membership and are computed using both the

model characteristics across the set of K classes and a case‟s pattern of

observed scores (see Vermunt & Magidson, 2002 for more details). Under

mixture modeling, cases may associate with more than one group through the

mixing weight, and can have fractional group membership across all groups.

To judge model uncertainty, cases are typically assigned to the one group with

which it has the highest posterior probability of association. This type of group

classification is termed modal assignment (e.g., Pastor & Gagné, 2013). At the

individual level, the higher probability value, the greater confidence one may

have concerning a case's class assignment. A better fitting solution will have

higher classification rates (i.e., fewer cases which are difficult to classify) for

each class, interpreted as a greater certainty of the classification. Perfect

classification would be illustrated by probability values of 1 for each class.

Entropy is a measure of uncertainness or randomness in the classification

procedure, and provides a summary of the information presented in a

classification table with one index (Pastor et al., 2007). Within LCC analyses,


entropy values indicate how well the model predicts class memberships

(Akaike, 1977). Values range from 0 to 1, where entropy values closer to 1

illustrate better prediction (Vermunt & Magdison, 2002). Entropy is computed

as the maximum of the probability density distribution underlying the latent

class cluster model (Akaike, 1977) and is calculated using the posterior

probability information, as well as the number of classes K modeled and

sample size N.

As suggested by latent class researchers (Collins & Lanza, 2010; Muthén,

2001; Muthén & Muthén, 2000, 2007; Vermunt & Magdison, 2002), each

class solution can be evaluated using multiple criteria. Finally, an optimal

solution can be “named” through interpreting patterns of high and low

parameter estimates within each class to ensure that each latent class has

substantive meaning (Muthén & Muthén, 2000) and matches to theory. It is

noted that while interpretability relies on the judgment of the researcher rather

than fit indices, it is still a very important component. This is similar to factor

analysis, where a researcher evaluates the sensibility of the solutions when

deciding between different numbers of factors (Crocker & Algina, 1986).

When both covariates and outcomes are available, a three-step procedure

is recommended (Asparouhov & Muthén, 2014; Vermunt, 2010). Generally,

identifying the model is the first, and most important, step in the mixture

modeling process. This involves identifying the optimal number of classes

using the procedures described above. First, the models are estimated where

no covariate or outcomes are included. These are called unconditional models,

as they are not ”conditional” or dependent upon additional auxiliary variables.

This information is used to estimate different class models. Based on this

information, the preferred model may be chosen. Second, using modal class

assignment, cases are assigned to latent classes. Modal assignment information

provides classification information to include with the auxiliary variables (i.e.,

covariates and outcomes). Third, fixed values are obtained from the latent

classification procedures in step 1 (i.e., log odds of the classification

probabilities; Asparouhov & Muthén, 2014). Fixing measurement

relationships between the latent class variable and the most likely class

variable helps to account for the imprecision in the classification and produces

correct estimates and standard errors for the relationships of class membership

with other variables (Asparouhov & Muthén, 2014). Upon completion of these

three steps, auxiliary information may be included without influencing the

measurement of the latent classes. After performing these three steps,

covariates and distal outcomes may be incorporated. Covariates may help to

predict class membership and may be added into the model using the set


number of classes, k, found in the first step. Finally, the distal outcome

variables related to the latent class variable and/or indicators may be included

in the last step. Outcome variables (e.g., standardized test scores) may be

assessed to determine if latent groups provide differences with regard to

performance and accountability measures.

METHODS

Schools and Participants

As mentioned, a unique feature of the current study was the availability of

a statewide data set with a large number of responses. The current study used

data collected from the 2013 climate survey administration and contained

approximately 54,750 responses from students in elementary grades 3-6 and

18,370 responses from teachers working in elementary schools. A total of 660

elementary schools from across the state of South Carolina were included in

the database. Survey responses from students and teachers were included for

analysis to provide average information about ratings of a school‟s climate.

Instrumentation

Students and parents at selected grades (typically grades 5, 8 and 11) as

well as all teachers at each school are asked to complete a survey at the end of

each academic year to assess characteristics about a school‟s learning

environment, parent-school relationships, and social and physical factors

related to the school. Three items from each survey (one from each main

section noted below) are included on the report card. However, the surveys

consist of many items, and relationships among these items may illuminate

differences between schools with differing climate perspectives. Two forms

were used to create climate groups: student and teacher forms. The student

survey consisted of 43-items and includes questions from three areas:

Learning Environment, measuring students‟ perceptions about the learning

context (18 items); Social and Physical Environment measuring students‟

thoughts about building cleanliness, appearance of the grounds, classroom

management/ behavior, school safety, and relationships with other

teachers/students (17 items); and Home and School Relations measuring the

relationship between schools and parents (8 items). Students respond to each


item using a 4-point Likert scale ranging from 1=Disagree; 2=Mostly

Disagree; 3=Mostly Agree; to 4=Agree.

There are 53 items included on the teacher climate survey. While the

items differ, the three scales hypothesized for the students are also

hypothesized for teachers. There are 26 items included on the Learning

Environment scale, 16 items on the Social and Physical Environment scale,

and 11 items on the Home and School Relations scale. Teachers responded to

each item using the same 4-point Likert scale: 1=Disagree; 2=Mostly

Disagree; 3=Mostly Agree; 4=Agree.

Before analyses, each dataset was examined. Duplicate cases were

removed from each dataset, as well as cases having more than 25% of the

responses missing within each scale. For cases with 25% or less missing data

on each section of the survey, missing item responses were imputed. Missing

item data were replaced with the average of the individual‟s responses for

other items on the same scale, thereby maximizing sample sizes for analyses.

PRELIMINARY ANALYSES

Statistical analyses of the imputed teacher and student data sets began

with confirmatory factor analysis (CFA). This multivariate statistical

procedure aims to determine how well the survey items measure the climate

constructs. This procedure is appropriate to use when researchers hold prior

knowledge of the underlying latent structure of an instrument (Benson, 1998;

Byrne, 1998; Hoyle & Panter, 1993). CFA was appropriate as it was preceded

by exploratory factor analysis (EFA), which yielded the same factor structure

across two consecutive years (2006 and 2007) and CFA for teachers, students,

and parents (Monrad et al., 2008) with additional independent samples (survey

data collected in 2008, 2009, and 2010).

For each data set, the factor structure derived from exploratory procedures

was used to specify the measurement model in CFA. The confirmatory factor

analyses conducted in this study included only the teacher items present in the

optimal exploratory factor solutions (53 teacher items and 43 student items).

Subsequent item analysis showed that CFA results of the current study

replicated closely the prior EFA solutions. CFAs of the teacher, student, and

parent data sets were conducted using the CALIS procedure provided in the

SAS 9.2 statistical software package. Parameters and model fit indices were

estimated using the Maximum Likelihood procedure. This estimator is

frequently used in CFA studies with (distributionally) normally distributed


categorical data that represents underlying continuous constructs and with at

least 4 ordered categories (Finney & DiStefano, 2013). Results from multiple

ad-hoc fit indices were used to arrive at the optimal final solution. The series

of analyses identified six factors underlying the teacher dataset and four

factors underlying the student dataset.

For the teacher survey, a six-factor solution was determined to be the most

interpretable. This six factor solution was identified within each of the three

organizational levels (elementary, middle, and high school; Monrad et al.,

2008). For teachers, the first factor, Working Conditions/Leadership, describes

the administrative leadership, perceptions of inclusion of teachers, and

enforcement of work-related policies. This factor included items such as: “The

school administration provides effective instructional leadership” and “The

school administration communicates clear instructional goals for the school.”

Home-School Relationship describes the relationship between parents and their

involvement with school activities. Example items include: “I am satisfied

with the home-school relations” and “Parents attend school meetings and other

school events.” The third factor, Instructional Focus, measures an

understanding of instructional standards and high expectations for students to

meet those standards. The Resources factor assesses teachers‟ views of the

availability of textbooks and classroom materials needed for teaching. Sample

items on this fourth factor included: “Our school has sufficient computers for

instructional use.” and “I have sufficient space in my classroom to meet the

educational needs of my students.” The Physical Environment factor measures

teachers‟ views of the physical environment of the schools and were closely

associated with building cleanliness and maintenance. Finally, the sixth factor,

Safety, expressed teachers‟ perceived safety during the school day and while

going to and coming from school.

A four-factor solution was thought to be optimal for the student survey;

again, this structure was identified within each organizational level (Monrad et

al., 2008). These factors have been named: Learning Environment, Social-

Physical Environment, Home-School Relationship, and Safety. The Learning

Environment factor is defined by items such as: “My classes are interesting

and fun”, “My teachers spend enough time helping me learn”, and “I am

satisfied with the learning environment in my school.” Positive student

responses to these items suggest the existence of a nurturing learning

environment in which the student feels supported by teachers and engaged in

learning. The second factor, Social-Physical Environment, is similar to the

Physical Environment factor for teachers, with items relating to building

cleanliness and maintenance. The third dimension, Home-School Relationship,


is primarily associated with parent involvement with the school and student

learning. The Safety factor for students is comparable to that outlined for

teachers: the perception of security both at school and coming to and going

from school.

CFA Factor Scores. Each CFA run was followed by the computation of

factor scores using a least squares regression approach (Thurstone, 1935).

Regression factor scores predict the location of each survey participant on the

distribution of each of the climate factors, and may be used for subsequent

statistical analysis. They are standardized scale scores developed from the

factor structure and based upon the weights assigned to individual items.

Values generally range from a low of -3 to a high of 3, representing three

standard deviations from the mean, where values near zero represent an

average performance. With respect to climate, positive factor scores depict

above average ratings whereas negative scores describe a climate rating that is

below average. To identify climate characteristics within each school, as well

as to compare these characteristics across schools, factor scores were

aggregated at the school level for the latent class clustering analysis. Resulting

analyses, thus, included students‟ and teachers‟ perceptions of their school‟s

climate across multiple domains. This procedure provided a school average

estimate for each climate dimension, and it allowed researchers to determine

where each school is located on every teacher and student factor. These

10factor scores across student and teacher solutions were used to create latent

classes of school climate. Thus, we recognize latent profile analysis was used

as the variables used to create the classes were continuous in nature.

COVARIATES AND OUTCOMES

As noted, LCC can incorporate covariates and outcomes using a three-step

procedure (Asparouhov & Muthén, 2014). The covariates and outcomes used

are described below.

Covariates. Two variables, school size and school poverty index, were

used as covariates. These variables were included because they may have an

effect on a school‟s overall climate, and could impact the number of classes if

the variables are not included in the estimation of latent classes. A poverty

composite score was used to estimate the effect of school poverty levels on the

school climate classification. This index is provided by the South Carolina

State Department of Education and is based on the proportion of students


receiving Medicaid and/or reduced meal plans. Its values range from 0 to 100,

where higher values indicate higher levels of poverty.

Outcomes. School climate data was compared to the school report card

information. Specifically, the school Absolute index and the school‟s

composite score or “grade” used for federal accountability purposes were

included as outcomes. For elementary schools, the Absolute index provides a

measure of a school‟s performance on standardized tests and attendance (e.g.,

http://www.eoc.sc.gov/Information%20for%20Educators/Accountability%20

Manuals/2012/Ratings%20for%20School%20Districts%2011-12.pdf).

Beginning in 2012, South Carolina applied for, and received, a waiver

for several of the requirements for the Elementary and Secondary Education

Act (ESEA). This “ESEA waiver” allowed for more flexibility in reporting for

federal accountability purposes. As such, schools now receive a composite

score calculated based on student performance on state standardized tests. This

composite score takes into account both students who meet the pre-defined

proficiency goal, as well as students who do not meet the proficiency goal but

show improvement in test scores from the previous school year (SC

Department of Education, 2013). Schools are then assigned a letter grade

based on this composite score where scores 90-100 = “A”, 80-89.9 = “B”, 70-

79.9= “C”, 60-69.9 = “D”, and below 60 = “F” (SC Department of Education,

2013).

Statistical Methods

To conduct analyses, the software package Mplus (v. 7.3) was used

(Muthén & Muthén, 1998-2014). Within the current software version, Mplus

includes automated procedures for the three-step approach when mixture

models consist of a single latent class variable and auxiliary information

consisting of either covariates or outcomes. In this study, however, the model

included both covariates and distal outcomes, so it was necessary to perform

the three-step method manually. In addition, cases which were missing

covariates, outcomes, or school-level climate scores for more than four scales

were eliminated from the analyses.

First, a series of unconditional models, that is, models which did not

include covariates or outcomes, were fit. Here, it is typical practice to start

with a one-class model and then successively increase the number of classes

by one to find the optimal number of classes. Fit statistics were collected for

each model and compared to the previous model (i.e., the model with one less


class) to identify the optimal number of groups underlying the dataset.

Because there is no one single index which can be used to assess model fit, we

considered multiple fit indices.

As a measure of global fit, the Log Likelihood value of the model was

examined. To examine relative fit, both the AIC and the BIC values were

compared across models, where the model with the lowest BIC was preferred

(e.g., Nylund et al., 2007; Pastor & Gagné, 2013). In addition, entropy values

and classification probabilities were examined, where higher values show

greater ease in classifying schools to a given class.

After the preferred model was chosen, modal assignment was used to

assign schools to latent classes of climate. Third, we included fixed values

accounting for measurement error in class assignment. All values were

included in the output from the unconditional model. Upon completion of the

three-step procedures, auxiliary information (i.e., covariates and outcomes)

was then included without influencing the measurement of the latent classes.

Finally, the covariates and outcomes were examined for statistically significant

differences across pairwise class comparisons.

RESULTS

After examination of the preliminary dataset, 40 cases were deleted due to

missing data. Overall, the analyzed sample included 52,436 student responses

and 17,843 teacher responses from 610 elementary schools. These schools had

complete profiles of cases among the set of 10 factor scores as well as

information on covariates and outcomes. Across schools, the average Absolute

index was 3.20, average school size was 513 students, the average ESEA

waiver composite score was 83.13, and the average poverty index was 75.56.

Latent Class Results

As the first step of the procedures, the latent profile analyses ignored

covariate and outcome information and concentrated on finding the optimal

class solution. To begin, two latent classes were extracted and successive

classes were added to the solution until no additional solutions could be

extracted due to nonconvergence problems (Nylund et al., 2007; Vermunt &

Magidson, 2002). Based on this process, two to a maximum of five class

solutions were extracted. Fit indices were examined to help identify the


optimal number of classes. Table 1 presents fit information for the two through

five class solutions examined.

Table 1. Model Fit Indices for Latent Profile Analyses

Classes Free

Parameters

LL AIC BIC Entropy LMR

p-

value

Unconditional Models

2 31 67 -74 62 .899 .0003

3 42 565 -1046 -861 .865 .0451

4 53 829 -1552 -1315 .909 .1052

5 64 996 -1864 -1581 .881 .7312

Four Class Solutions*

4A: All σ2

equal;

σ = 0

53 829 -1552 -1315 .909

4B: Free σ2

within class, σ

= 0**

83 1345 -2524 -2157 .922

4C: Free Σk

within class 104 2737 -5265 -4806 .759

Notes: LL = Log-likelihood value, LMR = Lo-Mendell-Rubin hypothesis test; * = μk

freely estimated within each class. **= Model 4B was chosen for the three-step

procedure.

As shown in Table 1, the four-class solution had the highest Entropy

value, along with low AIC and BIC values. The five-class solution did have

lower BIC and AIC values; however, it had a lower entropy value, showing

higher amounts of classification error. In addition, Mplus offers the Lo-

Mendell-Rubin (LMR) hypothesis test to help decide if the tested solution (k

classes) fits acceptably or if an additional class (k + 1) is necessary. Analysts

can use the accompanying p-value to help decide if the additional class is

needed (via a small p-value). From the information across the solutions, the

LMR test suggested that four classes were adequate to describe the elementary

school climate.

After determining that four classes were acceptable, different models were

tested. The baseline model, Model 4A, constrained variances of the variables

used to group cases (i.e., factor scores) to be equal across all four classes and


set covariance terms to zero. Model 4B allowed variance terms to vary across

classes, and covariance terms to be estimated, but constrained to be equal

across the set of classes. The last model tested, Model 4C, allowed all variance

and covariance terms to vary freely within class. For all models, vectors of

mean scores (i.e., centroid values) were uniquely estimated within each class.

Fit indices, as well as the interpretability of the class solution, were examined

for each model.

As shown at the bottom of Table 1, Model 4B produced the highest

entropy level and the lowest AIC value. Model 4C yielded lower information

fit criteria, but showed low entropy and was hard to interpret. The difficulty in

interpreting solutions with correlated variables has been recognized previously

(Muthén, 2002). Based on the information, the baseline model, Model 4B was

used for the three-step procedure as it illustrated acceptable fit, interpretability,

and match to prior research with state-wide datasets using both cluster analysis

(DiStefano et al., 2007) and latent profile analysis (Mindrila et al., 2010).

Model 4B was used for the remaining steps in the procedure. Probability

information was used to assign schools to classes. Subsequently, class

probabilities were constrained while the influence of covariates and distal

outcomes were included. Estimated parameter values for the classes are

provided in Table 2 and a representation of the class centroids is provided in

Graph 1.

Class one (n=129, 21% of elementary school sample) was named “Poor

Climate” based on the mean profile of the student and teacher factor scores.

There were relatively few classification problems, with an average prior

probability of .974. The Poor Climate group had the lowest scores of the set of

classes, with negative teacher and student scores. Here, teacher scores were

lower than student scores, showing greater levels of dissatisfaction with the

school environment. Average teacher scores also reported higher levels of

variability than average student scores.

Class two (n=186, 30% of elementary schools) was termed “Average

Climate”. Again, class average prior probability values were high, .949. All of

the mean values for this class were positive, but close to zero. This group also

had low variability, with small variances reported for both teacher and student

parameters.

Table 2. Average Latent Profiles of School Climate Variables, by Class (N = 610 Elementary Schools)

Class 1 Class 2 Class 3 Class 4

Poor

Climate

Average

Climate

Average Teacher/

Positive

Student Climate

Positive

Climate

n (%) 129 (21.1%) 186 (30.3%) 155 (25.3%) 140 (23.2%)

Classification Probability 0.968 0.958 0.943 0.962

Student Factor Scores

Learning Environment 0.136 (0.030) 0.182 (0.011) 0.383 (0.007) 0.403 (0.022)

Social-Physical Environment -0.108 0.052) 0.018 (0.017) 0.308 (0.013) 0.392 (0.037)

Home-School Relations 0.089 (0.022) 0.141 (0.007) 0.328 (0.005) 0.368 (0.015)

Safety -0.983 (0.054) 0.011 (0.011) 0.264 (0.010) 0.325 (0.023)

Teacher Factor Scores

Working Conditions/ Leadership -0.404 (0.187) 0.133 (0.044) 0.088 (0.071) 0.447 (0.015)

Home-School Relations -0.418 (0.106) 0.103 (0.058) 0.158 (0.068) 0.536 (0.026)

Instructional Focus -0.292 (0.124) 0.117 (0.016) 0.117 (0.023) 0.346 (0.008)

Resources -0.338 (0.115) 0.099 (0.035) 0.040 (0.057) 0.362 (0.017)

Physical Environment -0.333 (0.221) 0.080 (0.074) 0.022 (0.120) 0.370 (0.021)

Safety -0.408 (0.158) 0.109 (0.028) 0.092 (0.032) 0.322 (0.008)

Note: Variances shown in parentheses.


Figure 1. Profile Plots of Class Centroids, Latent Profile Analysis.

Class three (n=155, 25% of sample) also appeared to be a group of schools

with average climate scores. However, what distinguished this group was that

students were more positive than the teachers. Average scores here were

positive and above zero, while teacher factor scores were positive and close to

zero. Additionally, students showed low variability in their responses. This

class was named “Average Teacher/High Student”. Regarding classification,

the average prior probability value was .952, showing relatively few

classification problems.

The final class, class four (n=140, 23% of sample), was named “Positive

Climate”. This group reported the highest teacher and student average factor

scores. Again, the classification probability was high at .959. Variability was

also low across the set of parameters, showing little discrepancy in average

scores.

Influence of Covariates and Outcome Variables

Covariates. Descriptive statistics for the covariates and outcome variables

included in the study by latent class are presented in Table 3. Relationships

show a differential pattern related to school climate.

Table 3. Descriptive Statistics by Latent Class

Covariates Outcome variables

Latent class

School size Poverty index Absolute index ESEA Composite

n M SD M SD M SD M SD

Poor Climate 129 454 202.84 88.57 12.97 2.81 0.38 69.36 20.59

Average Climate 186 521 216.09 78.92 17.23 3.12 0.36 82.06 16.03

Average Teacher/

Positive Student

155 521 195.47 73.15 19.82 3.32 0.35 87.56 12.87

Positive Climate 140 548 220.86 61.80 20.15 3.51 0.32 92.34 7.87

Note: ESEA = this score, used for federal accountability purposes, represents a composite of student performance on SC state

standardized tests; this score is based on students who meet a pre-defined proficiency goal or who show growth in test scores from

the previous school year.

Table 4. ESEA Waiver School Grade Distribution between Latent Classes

School grade

Latent class

A B C D F

n

% within

class

n

% within

class

n

% within

class

n

% within

class

n

% within

class

Poor Climate 23 17.8% 25 19.4% 24 18.6% 16 12.4% 41 31.8%

Average Climate 71 38.2% 60 32.3% 19 10.2% 19 10.2% 17 9.1%

Average Teacher/

Positive Student

81 52.3% 48 31.0% 13 8.4% 2 1.3% 11 7.1%

Positive Climate 98 70.0% 33 23.6% 6 4.3% 3 2.1% 0 0.0%

Note. The ESEA composite score or index can be converted into a letter grade as follows: scores 90-100 = “A”, scores 80-89.9 = “B”,

scores of 70-79.9= “C”, scores 60-69.9 = “D”, and scores below 60 = “F”.


For example, schools belonging to the Poor Climate class (class 1) were

associated with the highest poverty index, smallest school size, and lowest

Absolute index and ESEA composite score. Conversely, schools belonging to

the Positive Climate group (class 4) have the lowest poverty index, largest

school size, and highest Absolute index and ESEA composite score.

Considering the influence of the covariates on the latent classes, analyses

were conducted, where classes are compared to a referent class; here, the

Positive Climate (class 4) was used as the comparison. Compared to the most

Positive Climate class (class 4), there was a significant effect of poverty on

Poor Climate (class 1) and Average Climate (class 2) groups, where these

classes were more likely to have higher poverty. School size was also

significant; however, coefficients were close to zero and odds ratios

(comparing each class to the referent group) were 1, showing no great impact

of school size on school climate. The Average Teacher/High Student Climate

class did not report significantly different effects on poverty or school size as

compared to the Positive Climate class.

Outcomes. Table 3 provides information about the relationship between

class membership and performance indicators. Results indicated that schools

with the poorest climate, as defined by negative, below average factor scores,

did worst on achievement outcomes; schools in successively more favorable

climate groups showed progressively higher achievement outcomes. One

feature of the ESEA waiver composite score is that it can be converted into a

letter grade for each school.

Table 4 highlights the school “grade” distribution between the identified

latent classes. As climate profiles become more positive, the proportion of

schools receiving a school grade of an “A” increases, while the proportion of

schools receiving an “F” decreases. For example, approximately 32% of

schools belonging to the Poor Climate class received a school grade of an “F”,

while 70% of schools belonging to the Positive Climate class received an “A.”

DISCUSSION

The current study used a state-wide sample of climate ratings from 610

elementary schools. Survey results from teachers and students were aggregated

to the school level to identify the number of underlying latent classes and

estimate the impact of a categorical school climate variable on school absolute

ratings and composite scores based on student achievement ESEA, while

taking into account the effect of school size and poverty on the classification


process. Results showed a strong relationship between school poverty and

school climate latent profile memberships. Specifically, schools with lower

poverty were assigned to latent profiles with more positive school climate.

This was also observed with previous research that only included teacher

ratings of school climate (Mindrila et al., 2014). The previous findings can be

extended to models where both student and teacher ratings are used to create

latent classes.

Latent profiles were also described using poverty, school size, and school

performance information. Results showed that schools assigned to profiles

with poor school climate generally have higher poverty indices and smaller

school sizes. The average poverty index gradually decreases for latent profiles

with more positive school climate. Alternatively, school size is positively

related to climate. While the direct cause is unknown, one hypothesis is that

the recent tendency within the state is to construct new, larger schools when

possible. Thus, more affluent areas of the state may have larger elementary

schools instead of conducting repairs to older, smaller buildings. In contrast,

the average absolute rating value and ESEA index showed the lowest value

with the Poor Climate class. Values gradually increased for profiles with more

positive climate.

Using the poverty index and the school size as covariates allowed for

controlling the effects of these variables on the classification of schools

according to student and teacher climate ratings, and to estimate the impact of

the resulting school climate classification on school performance. Results

showed that school climate latent profile membership has a significant impact

on performance. Specifically, as schools are assigned to groups with more

positive school climate the probability of having higher performance measures

is significantly higher.

Our work with the school climate surveys and other non-survey report

card indicators over the past several years has led to a better understanding of

their relationship to both school achievement and to poverty. We have begun

to think of poverty, not only as an indicator of parental income, but also as: a)

the attitudes of parents, students, and teachers about schooling, b) the

perceived and real levels of support for and focus on the learning environment,

and c) the attendance rates and other indicators of time-on-task afforded to

students. Schools with large concentrations of poor students often have fewer

highly qualified teachers and administrators, higher teacher turnover, lower

student attendance, higher student suspensions, and parents less likely to be

actively participating in and supportive of the school and its learners. The

clarification of this constellation of relationships is an essential step in


developing the goals, strategies, and programs necessary to effectively address

educational improvement. It is for these reasons that we have focused upon

variables that could be addressed by school communities.

SCHOLARLY SIGNIFICANCE

Wang, Haertel, and Walberg (1997) conducted a meta-analysis using a

database consisting of 11,000 statistical findings and determined that

instruction and climate affect learning as much as student characteristics. Their

work supports “the idea that climate is a real factor in the lives of learners and

that it is measurable, malleable and material to those that work and learn in

schools” (Freiberg, 1999, p. 17). There is a compelling body of literature

providing support for the importance of school climate. Compared to other

barriers which are not within the locus of control of schools, such as high child

poverty and low state funding, negative school climate factors can be

improved. Although there is a growing literature dealing with the assessment

of school climate, efforts to systematically improve it have been limited.

Changing school climate “requires explicit, targeted, and aligned change

efforts at the leverage points” (McGuigan, 2008, p. 112). Results from this

study may be used to foster such efforts by providing greater insight about

how climate may impact selected accountability outcomes.

While some obstacles, such as poverty, are not easily surmountable,

school climate can be improved with limited expenses. Therefore, training on

changing the school climate should be provided at the school level for teachers

and administrators. These efforts should be stronger in high poverty areas, as

poverty has a negative impact on both school climate and school performance.

Therefore, with an increased focus on accountability and academic

improvement nationwide, the current research provides support for an

increased attention to school climate as a critical dimension for school leaders

to focus school improvement efforts. By evaluating the practices at the school-

level to determine which are promoting positive school climate, schools may

also see improvement in achievement outcomes.

The current work provides a framework for evaluating school climate data

as well as providing direction for the potential application of school climate

data for use in school improvement. For example, an extension of the current

work includes utilizing the climate data to develop multi-year school climate

profiles that could provide low-performing schools with a practical tool to use

in identifying critical areas for school improvement. Assessment and


evaluation efforts could be tailored to identify school climate needs and

measure implementation of targeted strategies to improve climate and

achievement outcomes. The current school climate research provides a starting

point to begin narrowing the gap between research, policy, and the practice of

implementing and evaluating approaches that includes school climate as one

important facet of school improvement.

LIMITATIONS OF THE STUDY AND CONCLUSION

This study represents an analysis of relationships among climate factors

and measures of performance, while controlling for poverty and school size. A

large data sample was used; however, the outcome measures are specific to

South Carolina‟s curriculum and accountability standards. Thus, the findings

may or may not generalize to educational systems in other locations.

Furthermore, this was an associative study of archival cross-section data, not

an experimental study designed to measure the impact of an intervention. The

large statewide sample is a unique characteristic of this study: most

investigations do not have access to such a large sample across organizational

levels.

Understanding school climate and its relation to school performance can

benefit school-community leaders and policy makers as they seek to improve

student learning. For teachers, a better school climate can help foster a positive

working environment by reducing absenteeism and stress, lowering teacher

turnover rates, and increasing job satisfaction. For students and parents, the

crucial importance of attendance and engagement in a supportive learning

environment is validated. For researchers, the analyses can point the way

toward structuring future studies into the relationship among student learning

and the concerns of teachers, parents, administrators and other stakeholders in

the community.

In summary, school climate provides a critical backdrop for efforts to

improve schools. Within the context of a poor school environment, even the

most well-documented reform strategy is unlikely to succeed. The current

school climate research provides a starting point to begin narrowing the gap

between research, policy, and the practice of implementing and evaluating

approaches that includes school climate as one important facet of school

improvement.


REFERENCES

Akaike, H. (1977). On the entropy maximization principle. In P. R. Krishnaiah

(Ed.), Applications of statistics (pp. 27-41). Amsterdam: North-Holland.

Aldenderfer, M. S., & Blashfield, R. K. (1984). Cluster analysis. Beverly

Hills, CA: Sage.

Allen, S.M., Thompson, R.H., Hoadley, M., Engelking, J., & Drapeaux, J.

(1997, March). Improving school climate: Creating a circle of

communication between educators and families. Paper presented at the

meeting of the American Educational Research Association, Chicago, IL.

Anderson, C.S. (1982). The search for school climate: A review of the

research. Review of Educational Research, 52(3), 368-420.

Ashforth, S.J. (1985). Climate formations: Issues and extensions. Academy of

Management Review, 25, 837-847.

Asparouhov, T., & Muthén, B. (2014). Auxiliary variables in mixture

modeling: Three-Step approaches using Mplus. Structural Equation

Modeling: A Multidisciplinary Journal, 21(3), 329-341.

Bacher, J. (2000). A probabilistic clustering model for variables of mixed type.

Quality & Quantity, 34, 223-235.

Banfield, J.D., & Raftery, A.E. (1993). Model-based Gaussian and non-

Gaussian clustering. Biometrics, 49, 803-821.

Bensmail, H., Celeux, G., Raftery, A. E., & Robert, C. P. (1997). Inference in

model based clustering. Statistics and Computing, 7, 1-10.

Benson, J. (1998). Developing a strong program of construct validation: A test

anxiety example. Educational Measurement: Issues and Practice, 17, 10-

22.

Bergren, D. A. (2014). The impact of school climate on student achievement in

the middle schools of the commonwealth of Virginia: A quantitative

analysis of existing data (Doctoral dissertation). Retrieved from ProQuest.

(Publication Number 3615518)

Bernstein, L. (1992). Where is reform taking place? An analysis of policy

changes and school climate. Educational Evaluation and Policy Analysis,

14(3), 297-302.

Blashfield, R. K., & Aldenderfer, M. S. (1988). The methods and problems of

cluster analysis. In J. R. Nesselroade & R. B. Cattell (Eds.), International

handbook of multivariate experimental psychology (pp. 311-359). New

York: Plenum.

Brand, S., Felner, R., Shim, M., Seitsinger, A., & Dumas, T. (2003). Assessing

school climate middle school improvement and reform: Development and


validation of a school-level assessment of climate, cultural pluralism and

school safety. Journal of Educational Psychology, 95, 570–588.

Brown, K.E., & Medway, F.J. (2007). School climate and teacher beliefs in a

school effectively serving poor South Carolina (USA) African-American

students: A case study. Teaching and Teacher Education, 23, 529-540.

Bryk, A.S. & Schneider, B. (2002). Trust in schools: A core resource for

improvement. NY: Russell Sage Foundation.

Bryk, A.S. & Thum, Y.M. (1989). The effects of high school organization on

dropping out: An exploratory investigation. American Educational

Research Journal, 26, 353-383.

Byrne, B. A. (1998). Structural equation modeling with LISREL, PRELIS, and

SIMPLIS: Basic concepts, applications, and programming. Mahwah, NJ:

Erlbaum.

Cataldi, E.F., Laird, J., & KewalRamani, A. (2009, September 23). High

school dropout and completion rates in the United States: 2007. National

Center for Education Statistics, Institute of Education Sciences, U.S.

Department of Education. Washington, DC. Retrieved July 16, 2013, from

http://nces.ed.gov/pubsearch/pubsinfo.asp?pubid=2009064.

Center for Social and Emotional Education (2010). School Climate Research

Summary—January 2010, School Climate Brief Vol. 1 No. 1. Retrieved

February 23, 2011 from: http://www.schoolclimate.org/climate/

documents/SCBrief_v1n1_Jan2010.pdf

Chen, G. & Wiekart, A. (2008). Student background, school climate, school

disorder, and student achievement: An empirical study of New York City's

middle schools, Journal of School Violence, 7(4), 3‐20.

Christle, C., Jolivette, K., & Nelson, M. (2007). School characteristics related

to high school dropout rates. Remedial and Special Education, 28(6), 325-

339.

Clogg, C. C. (1995). Latent class models. In G. Arminger, C. C. Clogg, &M.

E. Sobel (Eds.), Handbook of statistical modeling for the social and

behavioral sciences (pp. 313-314). New York: Springer.

Cohen, J. (2009). Transforming school climate: Educational and

psychoanalytic perspectives: Introduction. Schools: Studies in Education,

6(1), 99-103.

Collins, L. M., & Lanza, S. T. (2010). Latent class and latent transition

analysis: With applications in the social, behavioral, and health sciences.

New York: Wiley.

Crocker, L., & Algina, J. (1986). Introduction to classical and modern test

theory. Philadelphia: Harcourt Brace Jovanovich College Publishers.


DiStefano, C., & Kamphaus, R.W. (2006). Investigating subtypes of child

development: A comparison of cluster analysis and latent class cluster

analysis in typology creation. Educational and Psychological

Measurement, 66(5), 778-794.

DiStefano, C., Monrad, D.M., May, R.J., McGuiness, P., & Dickenson, T.

(2007, April). Using school climate surveys to categorize schools and

examine relationships with school achievement. Paper presented at the

meeting of the American Educational Research Association, Chicago, IL.

Dymnicki, A. B. & the Multisite Violence Prevention Project (2013).

Moderating effects of school climate on outcomes for the multisite

violence prevention project universal program. Journal of Research on

Adolescence, 24(2), 383-398.

Edmunds, R.R. (1982). Programs of school improvement: An overview.

Educational Leadership, 40(3), 4-11.

Everitt, B. S. (1993). Cluster analysis. London: Edward Arnold.

Finney, S. J., & DiStefano, C. (2013). Nonnormal and categorical data in

structural equation models. In G.R. Hancock & R.O. Mueller (Eds.),

Structural equation modeling: A second course (2nd

ed., pp. 439-492).

Greenwich, CT: Information Age.

Freiberg, H. J. (1999). School climate: Measuring, improving, and sustaining

healthy learning environments. New York, NY: Routledge.

Furlong, M. J., Greif, J. L., Bates, M. P., Whipple, A. D., Jimenez, T. C., &

Morrison, R. (2005). Development of the California school climate and

safety survey‐short form. Psychology in the Schools, 42(2), 137-149.

Gibson, W. A. (1959). Three multivariate models: Factor analysis, latent

structure analysis, and latent profile analysis. Psychometrika, 24, 229-252.

Gottfredson, G.D., Gottfredson, D.C., Payne, A.A., & Gottfredson, N.C.

(2005). School climate predictors of school disorder: Results from a

national study of delinquency prevention in schools. Journal of Research

in Crime and Delinquency, 42, 412-444.

Grayson, J. & Alvarez, H. (2008). School climate factors relating to teacher

burnout: A mediator model. Teaching and Teacher Education, 24, 1349-

1363.

Greenberg, E. (2004). Climates for learning. Paper presented at the annual

meeting of the American Educational Research Association, San Diego,

California.

Heinen, T. (1996). Latent class and discrete latent trait models: Similarities

and differences. Thousand Oaks, CA: Sage.


Hopson, L.M. & Lee, E. (2011). Mitigate the effect of family poverty on

academic and behavioral outcomes: The role of school climate in middle

and high school. Children and Youth Services Review, 33, 2221-2229.

Hoy, W.K. & Miskel, C.G. (1982). Educational administration: Theory,

research, and practice. (2nd ed., pp. 185-219). New York: Random

House.

Hoy, W.K. (1990). Organizational climate and culture: A conceptual analysis

of the school workplace. Journal of Educational and Psychological

Consultation, 1(2), 149-168.

Hoy, W.K., Tarter, C.J., & Kottkamp, R.B. (1991). Open schools, healthy

schools: Measuring organizational climate. Newbury Park, CA: Sage.

Hoyle, R. H., & Panter, A. T. (1993). Writing about structural equation

models. In R. H. Hoyle (Ed.), Structural equation modeling: Concepts,

issues, and applications (pp. 158–176). Newbury Park, CA: Sage.

Koth, C.W., Bradshaw, C.P., & Leaf, P.J. (2008). A multilevel study of the

predictors of student perceptions of school climate: The effect of

classroom-level factors. Journal of Educational Psychology, 100(1), 96-

104.

Lazarsfeld, P. F., & Henry, N. W. (1968). Latent structure analysis. Boston:

Houghton Mifflin.

Lee, V.E. & Burkham, D.T. (1996, February). Gender differences in middle-

grade science achievement: Subject, domain, ability level, and course

emphasis. Science Edition, 80(6), 613–650.

Lezotte, L.W. (1990). A guide to the school improvement process based on

effective schools research. Okemos, MI: Effective Schools Products.

Loukas, A. & Murphy, J.L. (2007). Middle school perceptions of school

climate: Examining protective functions on subsequent adjustment

problems. Journal of School Psychology, 45, 293-309.

Ma, X. & MacMillan, R.B. (1999). Influences of workplace conditions on

teachers‟ job satisfaction. The Journal of Educational Research, 93(1), 39-

47.

Macneil, A.J., Prater, D.L., & Busch, S. (2009). The effects of school culture

and climate on student achievement. International Journal of Leadership

in Education, 12(1), 73-84.

Malecki, C.K. & Demaray, M.K. (2006). Social support as a buffer in the

relationship between socioeconomic status and academic performance.

School Psychology Quarterly, 21(4), 375-395.

McGuigan, L. (2008). Systems thinking and culture change in urban school

districts. In Hoy, W.K., & DiPaola, M. (Eds.), Improving schools: Studies


in leadership and culture (pp.99-116). Charlotte, NC: Information Age

Publishing, Inc.

McLachlan, G.L., & Peel, D. (2000). Finite mixture models. New York: Wiley

Mindrila, D., DiStefano, C., Monrad, D., & Ene, M. (2014, April). School

climate latent profiles: Relationships with poverty and school report card

data. Paper presented at the meeting of the American Educational

Research Association, Philadelphia, PA.

Monrad, D.M., May, R.J., DiStefano, C., Smith, J., Gay, J., Mindrila, D.,

Gareau, S., & Rawls, A. (2008, April). Parent, student, and teacher

perceptions of school climate: Investigations across organizational levels.

Paper presented at the meeting of the American Educational Research

Association, New York, NY.

Muthén, B. O. (2001, December 11). LCA and cluster analysis. Message

posted to MPLUS discussion list, archived at http://www.statmodel.com/

discussion/messages/13/155.html?1077296160

Muthén, B. O. (2002). Beyond SEM: General latent variable modeling.

Behaviormetrika, 29(1), 81-117.

Muthén, B. O. (2002, March 8). Latent variable mixture modeling: Latent

profile analysis. Message posted to MPLUS discussion list, archived at

http://www.statmodel.com/discussion/messages/13/115.html?1114649917

Muthén, B., & Muthén, L. K. (2000). Integrating person-centered and

variable-centered analyses: Growth mixture modeling with latent

trajectory classes. Alcoholism: Clinical and Experimental Research, 24,

882-891.

Muthén, L. K., & Muthén, B. O. (1998-2014). Mplus users's guide (7th

ed.).

Los Angeles, CA: Muthén & Muthén.

National School Climate Center, Center for Social and Emotional Education,

& National Center for Learning and Citizenship at Education Commission

of the States (2008). The School Climate Challenge: Narrowing the Gap

Between School Climate Research and School Climate Policy, Practice

Guidelines and Teacher Education Policy. Retrieved February 23, 2011

from: http://www.ecs.org/html/projectsPartners/nclc/docs/school-climate-

challenge-web.pdf

Nylund, K.L., Asparouhov, T., & Muthen, B.O. (2007). Deciding on the

number of classes in latent class analysis and growth mixture modeling: A

Monte Carlo simulation study. Structural Equation Modeling: A

Multidisciplinary Journal, 14(4), 535-569.


Pastor, D. A., & Gagné, P. (2013). Mean and covariance structure mixture

models. In G. R. Hancock & R. O. Mueller (Eds.), Structural equation

modeling: A second course (2nd

Ed.). Greenwich, CT: Information Age.

Perkins, B.K. (2006). Where we learn: The CUBE survey of urban school

climate. Retrieved June 4, 2009 from http://www.nsba.org/

SecondaryMenu/CUBE/Publications/CUBEResearchReports/WhereWeLe

arnReport/WhereWeLearnFullReport.aspx

Roney, K., Coleman, H., & Schlichtin, K.A. (2007). Linking the

developmental health of middle grades schools to student achievement.

NASSP Bulletin, 91(4), 289-321.

Rumberger, R.W. (1995). Dropping out of middle school: A multilevel

analysis of students and schools. American Educational Research Journal,

32(3), 583-625.

Sackney, L. (1988). Enhancing school learning climate: Theory, research and

practice. In D. O. E. Administration (Ed.) (Vol. 180, pp. 146): SSTA

Research Centre.

School climate, discipline, and safety: Gauging educator attitudes. (2013).

Retrieved July 16, 2013, from http://www.edweek.org/ew/qc/2013/school-

climate-gauging-attitudes.html

Sebring, P.B., Allensworth, E., Bryk, A.S., Easton, J.Q., & Luppescu S.

(2006). The essential supports for school improvement. Retrieved March

9, 2007, from http://ccsr.uchicago.edu/content/publications.php?

pub_id=86.

Sirin, S.R. (2005). Socioeconomic status and academic achievement: A meta-

analytic review of research. Review of Educational Research, 75(3), 417-

453.

Smith, J. J. (2005). The relationship between school division climate and

student achievement of school divisions in the commonwealth of Virginia

(Unpublished doctoral dissertation). Virginia Polytechnic Institute and

State University, Blacksburg, VA.

South Carolina Department of Education. (2013). ESEA federal accountability

system: Calculating the grades for districts and schools: Technical

manual. Retrieved from http://ed.sc.gov/data/esea/2012/

documents/ESEA_Technical_Documentation-Draft.pdf

Stewart, E.B. (2007, June). School structural characteristics, student effort,

peer associations, and parental involvement: The influence of school- and

individual-level factors on academic achievement. Education and Urban

Society 2008, 40, 179-204.


Tagiuri, R. (1968). The concept of organizational climate. In R. Tagiuri &

G.H. Litwin (Eds.), Organizational climate: Exploration of a concept.

Boston: Harvard University.

Thurstone, L. L. (1935). The vectors of mind. Chicago: University of Chicago

Press.

Tschannen-Moran, M. (2004). Trust matters: Leadership for successful

schools. San Francisco, CA: Jossey-Bass.

Tubbs, J. & Garner, M. (2008). The impact of school climate on school

outcomes. Journal of College Teaching and Learning, 5(9), 17-26.

Van Houtte, M. (2005). Climate or culture? A plea for conceptual clarity in

school effectiveness research. School Effectiveness and School

Improvement, 16(1), 71-89.

Vermunt, J. K. (2010). Latent class modeling with covariates: Two improved

three step approaches. Political Analysis, 18, 450–469.

Vermunt, J. K., & Magidson, J. (2002). Latent class cluster analysis. In J. A.

Hagenaars & A. L. McCutcheon (Eds.), Applied latent class analysis (pp.

89-106). Cambridge, UK: Cambridge University Press.

Virtanen, M., Kivimaki, M., Luopa, P., Vahtera, J., Elovainio, M., Jokela, J.,

& Pietikainen, M. (2009). Staff reports of psychosocial climate at school

and adolescents‟ health, truancy and health education in Finland.

European Journal of Public Health, 19(5), 554-560.

Wang, M. C., Haertel, G. D., & Walberg, H. J. (1997). Learning influences. In

H. J.Walberg & G. D. Haertel (Eds.), Psychology and educational

practice (pp. 199–211). Berkeley, CA: McCutchan

Way, N., Reddy, R., & Rhodes, J. (2007). Students‟ perception of school

climate during the middle school years: Associations with trajectories of

psychological and behavioral development. American Journal of

Community Psychology, 40, 194-213.



Chapter 3

ASSESSING MEDIATION IN SIMPLE

AND COMPLEX MODELS

Thomas Ledermann1,* and Siegfried Macho2

1University of Basel 2University of Fribourg, Switzerland

ABSTRACT

This chapter addresses the testing of specific effects and contrasts in three

types of mediation models: models with up to four simultaneous

(parallel) mediators, models with two sequential mediators, and single-

mediator models with two initial variables. We use the delta method and

provide equations to calculate standard errors for simple and total indirect

effects, total effects, and specific contrasts in each type of model. We also

demonstrate how bootstrap interval estimates of specific effects and

contrasts can be obtained using phantom models and how indirect effects

involving different initial variables can be compared in a scale-free

fashion. Testing contrasts, we show how common requirements for

complete mediation can be made stronger. Limitations of both, statistics

using standard errors based on normal theory and bootstrapping to test

mediation, along with new methods are discussed. The methods are

illustrated using publicly available datasets. Supplementary material

* Corresponding author: Thomas Ledermann, [email protected].

Thomas Ledermann and Siegfried Macho 70

available online includes Amos, OpenMx, and Mplus files to estimate the

models and an Excel spreadsheet to calculate the effects.

Keywords: mediation, specific effects, contrasts, delta method, bootstrapping,

phantom models

The assessment of mediational processes is of great importance in the

social and behavioral sciences if researchers are to understand the mechanism

through which an effect unfolds. Mediation is said to occur when the effect of

an initial variable on an outcome variable is transmitted through one or more

third variables, called mediator or intervening variables (Baron & Kenny,

1986; James & Brett, 1984; MacKinnon, 2008).

The analysis of models involving multiple mediators along with the

comparison of effects in mediation models has attracted researchers over the

last two decades. The testing of specific effects, such as an indirect effect, and

contrasts in such models can be accomplished by means of the z-statistic or the

bootstrap method (Efron, 1979), which both have their own strengths and

weaknesses.

The purpose of this chapter is to addresses the testing of specific effects

and contrasts in simple and complex mediation models. Using the delta

method, we provide equations to calculate standard errors (SE) for specific

effects and contrasts in mediation models with up to four simultaneous

(parallel) mediators, models with two sequential mediators, and models with

two initial variables and demonstrate how bootstrap interval estimates can be

obtained using phantom models (Macho & Ledermann, 2011). We also show

how contrasts among mediation effects involving different initial variables can

be assessed in a scale-free fashion. To ease the estimation of the models and

testing of effects Amos (Arbuckle, 1995–2012), OpenMx (Boker et al., 2011),

and Mplus (Muthén & Muthén, 1998-2012) files and an Excel spreadsheet

calculating SEs, z and p values, and normal confidence intervals for all types

of effects as well as specific contrasts are available at

http://thomasledermann.com/mscm/.

In what follows, we discuss common types of mediation models and

different types of effects that can be assessed in each. We then address the

assessment of mediation and focus on the distinction between partial and

complete mediation. Finally, we review the use of the z-statistic and the

bootstrap method to test mediating effects and contrasts. To illustrate the

assessment of mediation in complex models, publicly available data sets are

used.

Assessing Mediation … 71

COMMON TYPES OF MEDIATION MODELS

The most basic form of a mediation model is shown in Figure 1A. This

model consists of three random variables: the independent variable X, the

mediator M, and the outcome Y. In this simple mediation model, there are

three direct effects, a, b, and c′. The effect of X on Y can be apportioned into a

direct effect represented by c′ and an indirect effect through M, which is the

product ab. The sum of these two effects is the total effect. This simple

mediation model can be extended in various ways. Here, we focus on three

common extensions.

The first extension is a mediation model with multiple intervening

variables that simultaneously mediate the effect of the initial variable on the

outcome variable. The coaction of two simultaneous mediators has been

studied, for example, by Fosco and Grych (2008) who demonstrated that the

effect of parental conflicts on children‟s internalizing problems is mediated

simultaneously by children‟s appraisals of threat and self-blame. A model with

three simultaneous mediators is presented in Figure 1B. In this model, there

are three simple indirect effects, a1b1, a2b2, and a3b3, sometimes called specific

indirect effects. The sum of them is the total indirect effect (i.e., a1b1 + a2b2 +

a3b3). The total effect that X exerts on Y is the total indirect effect plus the

direct effect c′ (i.e., a1b1 + a2b2 + a3b3 + c′). We echo Preacher and Hayes

(2008) and suggest the inclusion of covariances between the residuals of the

mediators, because constraining a substantial residual covariance to zero may

result in serious model misspecification and in biased estimates of the SEs of

the b paths.

Another extension of the simple mediation model is a model with multiple

mediators that act in series. In this type of models, the effect of an initial

variable on an outcome variable is mediated through a chain of two or more

sequential mediators. For example, Perkinson-Gloor, Lemola, and Grob (2013)

found that sleep duration influences daytime tiredness, which, in turn, affects

behavioral persistence, which, ultimately, predicts academic achievement. A

mediation model with two sequential mediators is given in Figure 1C. In this

model, X and Y are linked by two simple indirect effects, a1b1 and a2b2, a

three-path indirect effect, a1b12b2, a term used by Taylor, MacKinnon, and

Tein (2008), and the direct effect c′. The total indirect effect of X on Y that

goes through both mediators is the sum of all indirect effects (i.e., a1b12b2 +

a1b1 + a2b2). The total effect of X on Y is again the total indirect effect plus the

direct effect c′ (i.e., a1b12b2 + a1b1 + a2b2 + c′).


Figure 1. Model A: Simple mediation model. Model B: Mediation model with three

simultaneous mediators. Model C: Mediation model with two sequential mediators.

Model D: Mediation model with two initial variables.

A third extension is a simple mediation models with multiple initial

variables that affect a mediator, which, in turn, affects an outcome. For

example, Klainin (2009) hypothesized that both family stress and occupational

stress affect physical health, which in turn affects psychological stress. A

mediation model with two initial variables is presented in Figure 1D. In this

Figures

Model A

Model B

Model C

Model D

1 r1

M

1 r2

Y X

b a

c′

1 r1 1 r2

M1

X

b a1

c′

1 r3

Y

M2

a2

b12

b1

b2

1 r4

Y

1 r1

M1 b1 a1

a2

c′

b2

1 r2

M2

1 r3

M3

X

a3

b3

1 r1

M

1 r2

Y b

a1

c′2

X1

a2

c′1

X2


model, there are two simple indirect effects, a1b and a2b, sharing the direct

effect b, two direct effects c′, c′1 and c′2, and two total effects, a1b + c′1 and a2b

+ c′2. We next address the assessment of mediation and the testing of specific

effects and contrasts in these types of mediation models.

ASSESSING MEDIATION

For a good understanding of the mediation process all direct effects and

indirect effects in a model should be estimated and tested (e.g., Ledermann &

Macho, 2009).

Direct and Indirect Effects

There are two reasons to test the direct effects separately (Judd & Kenny,

2010). First, for mediation to occur all direct effects that constitute an indirect

effect have to be substantial because it makes no sense to speak of mediation if

not all direct effects that make up an indirect effect are substantial. Second,

mediation can be inconsistent (Maassen & Bakker, 2001; MacKinnon, Krull,

& Lockwood, 2000). Inconsistent mediation, also called suppression, occurs

when an indirect effect and the respective direct effect c′ have opposite signs.

To assess whether consistent mediation (i.e., the indirect effect and the

respective direct effect c′ are of the same sign) or inconsistent mediation has

occurred one need to know the sign of the direct effects.

Contrasts

In addition to the information whether consistent or inconsistent mediation

occurs, the knowledge of the relative importance of a specific mediator can

further refine the understanding of the pathways through which an initial

variable exerts an effect on an outcome. In any mediation model with a direct

effect c′, each simple indirect effect and total indirect effect can be compared

with the respective direct effect c′.3 In a model with multiple mediators, we

3 In mediation models with a direct effect c′, a quantity commonly reported is the relative

proportion of the indirect effect in the total effect. For the simple mediation model this is

ab/(ab + c′). However, it has been found that the accuracy of this ratio is poor unless the

sample size is at least 500 (MacKinnon, Warsi, & Dwyer, 1995). Additionally, this ratio can


may wish to know whether two indirect effects differ in magnitude. For

example, a researcher might find that the effect of sleep problems on school

achievement is stronger through tiredness than through negative mood.

In mediation models, very often the effects being compared have the same

initial and outcome variable and so the same metric (e.g., contrasting an

indirect effect with the respective direct effect c′ or two indirect effects in a

model with multiple sequential mediators). In this case, the effects are

quantified by the same units of measurement (e.g., Cheung, 2007; MacKinnon,

2000; Preacher & Hayes, 2008) and, so, the comparison of the effects is

independent whether the variables are standardized or unstandardized.

Sometimes the effects being compared do not have the same initial or

outcome variable. In such situations, one can use either unstandardized or

standardized estimates that are likely to produce different results. Consider a

simple model where an outcome Y is regressed on X1 and X2. If we want to

know whether an increase in X1 by one unit has the same effect on Y than an

increase by one unit in X2 we use unstandardized estimates. This practice

presupposes that the unit of measurement of both predictors is meaningful in

itself and in comparison to each other.4 For instance, in a study on the

influence of time spent with family and close friends on people‟s satisfaction,

we might be interested in whether satisfaction is more influenced by hours

spent with family vs. hours spent with close friends and so use the

unstandardized predictors with hours as unit of measurement.

When the predictors have different scales the unit of measurement for

each predictor has to be chosen in such a way that the comparison makes most

sense on a practical level. For example, a researcher may want to know

whether weight loss is more influenced by the reduction of calories consumed

or sport work and find that reducing the consumption of food energy by one

kilocalorie a week has a bigger effect on weight loss than increasing sport

work by one hour. Measuring food energy in calories, it is most likely that

increasing sport work by one hour has a bigger influence on weight loss than

reducing food energy by one calorie.

If in models with multiple initial (or outcome) variables the units of

measurement of the variables have no definitive meaning, a comparison of

effects makes most sense if done in an independent fashion of the units of

measurement (e.g., Raykov, Brennan, Reinhardt, & Horowitz, 2008). This can

be large when the total effect is very small and can be greater than one when inconsistent

mediation occurs. 4 In models with one outcome, the outcome‟s unit of measurement has no influence on the

comparison of the effects.


be achieved by dividing each predictor by the respective standard deviation.

This practice allows then to determine whether a change of one standard

deviation in one predictor produces the same change in the outcome than a

change of one standard deviation in another predictor.

This strategy of standardizing the variables, however, is inappropriate in

those cases where the parameter estimates are based on data other than the

data used to standardize the variables because the standardization would have

to be done on the data used by the program to obtain correct parameter

estimates. This problem concerns all popular resampling methods, including

bootstrapping that we discuss below. Using resampling methods, a proper

solution to standardize predictors is to implement a latent variable for each

predictor with the variance set to 1 and a direct path from the latent variable to

the predictor (e.g., Cheung, 2009; Jöreskog & Sörbom, Du Toit, & Du Toit,

1996). Figure 2 shows how a mediation model with two initial variables, one

mediator, and one outcome variable looks like after implementing the latent

variables for the initial variables. In this model, the unstandardized coefficients

of l1 and l2 represent the standard deviation of the predictors X1 and X2.5

Partial and Complete Mediation

A distinction often made in mediation analysis, is the one between

complete and partial mediation (e.g., Kenny, Kashy, & Bolger, 1998). Partial

mediation is said to occur when the indirect effect and the respective direct

effect c′ are nonzero and both effects are of the same sign. Complete (full or

perfect) mediation is said to occur when the indirect effect is nonzero and the

direct effect c′ is zero. Although this distinction has been found to be useful in

theory testing, it can be misleading when the decision about the type of

mediation relies upon significance tests (Hayes & Preacher, 2014; Preacher &

Kelley, 2011; Rucker, Preacher, Tormala, & Pitty, 2011; Wood, Goodman,

Beckmann, & Cook, 2008). For example, Kenny and Judd (2014) could

demonstrate that the statistical power of an indirect effect is often greater than

the power of the direct effect between an initial variable and an outcome. A

common misunderstanding pointed out by Rucker et al. (2011) is that

5 Lau and Cheung (2012) and Raykov et al. (2008) provide details to estimate and test

standardized effects using a SEM software program, such as OpenMx, LISREL (Jöreskog &

Sörbom, 2006), or RAMONA (Browne & Mels, 2005), that allow researchers to specify the

models on the level of the matrices. For users of Mplus or lavaan (Rosseel, 2012), Cheung

(2009) showed how standardized effects can be estimated in simple mediation models.


complete mediation suggests that the process by which an initial variable

affects an outcome is completely explained and, so, there is no need to test for

other mediating variables.

Stronger assumptions about complete mediation can be made by

contrasting the indirect effect with the respective direct effect c′ and requiring

that the indirect is bigger in size than the direct effect c′. If these two effects do

not differ one could set them equal, which results in that both effects are either

not statistically significant (i.e., no mediation), or significant (i.e., partial

mediation). Another important criterion for complete mediation often used by

researcher is the assumption that the absolute value of the standardized direct

effect c′ is smaller than .10. The case where the indirect effect and the direct

effect c′ do not differ statistically and the absolute value of the standardized

direct effect c′ is smaller than .10 and not statistically significant, may be

considered as weak complete mediation until further research will shed more

light into the mechanism. However, it is noteworthy that the comparison of

effects relies upon statistical tests that are sensitive to sample size. Next, we

describe how indirect effects, total effects, and specific contrasts can be

probed by using the z-statistic or bootstrapping.

TESTING METHODS

Indirect effects are most often tested by either the z-statistic or Efron‟s

(1979) bootstrap method. We describe both methods that can also be used to

test direct effects, total effects, and contrasts among effects.

Figure 2. Mediation model with standardized initial variables.

l2 s

2 = 1

l1 s

2 = 1

1 r1

M

1 r2

Y b

a1

c′2

X1

a2

c′1

X2

Xl1

Xl2


The Z Statistic

The z-statistic is a prominent method to test effects in mediation models.

A z-score can be obtained by dividing the effect being tested through its SE.

That is, for the indirect effect ab in Model 1A, z equals . This

statistical test is known as the product-of-coefficients approach or Sobel test.

For complex functions (effects) involving multiple direct effects, SEs can be

obtained by applying the good old multivariate delta method based on the

Taylor series (e.g., Bishop, Fienberg, & Holland, 1975; MacKinnon, 2000;

Rao, 1973; Raykov & Marcoulides, 2004). Using the first-order Taylor series

expansion, the delta method provides a general approach for computing

asymptotic variances for functions of estimated model parameters, such as

indirect effects (see Bollen, 1987; Sobel, 1982, 1986). The square roots of

these variances represent approximate standard errors of the effects. The first-

order version of the delta method has been used by MacKinnon (2000),

Preacher and Hayes (2008), and Taylor et al. (2008) to test indirect and total

effects and some contrasts in models with multiple mediators. In Appendix A,

we provide details on how the variance of a complex effect can be derived

using the first-order delta method and present equations to test all indirect and

total effects and the aforementioned contrasts in models with up to four

simultaneous or two sequential mediators and mediation models with two

initial variables.

Bootstrapping

The bootstrap method has been widely advocated for testing indirect

effects (e.g., Bollen & Stine, 1990; Preacher & Hayes, 2008; Shrout & Bolger,

2002) as well as contrasts among effects (Williams & MacKinnon, 2008). In

several popular SEM software packages, including Amos and EQS (Bentler,

2000-2008), the built-in bootstrap procedure for testing effects is limited to

direct effects, total indirect effects, and total effects. As we have seen, in

complex mediation models, simple indirect effects are often part of a total

indirect.

To test specific effects and contrasts using a SEM software program with

limited capabilities to do so the phantom model approach can be used (Macho

& Ledermann, 2011). This method provides a flexible means that stands out

due to its ease of use in obtaining both point and bootstrap interval estimates

for specific effects and contrasts in structural equation models and multiple


group analysis. It can be used with all software packages that report point

estimates of total effects, have bootstrap functions, and allow for latent

variables and parameter constraints. Phantom models are set up along and

estimated simultaneously with the main model. For each effect that cannot be

classified as direct, total indirect, and total effect, a phantom model is to build

whose structure represents the effect being estimated. Further details are given

in Appendix B. For more applications see Ledermann, Macho, and Kenny

(2011) and Perera (2013).

Choice of the Method

The decision of which method to use should be based on the data and

variables being analyzed. The z-statistic for testing contrasts and indirect

effects works well in large samples (i.e., in the high hundreds) and if the data

are normally distributed. Using maximum likelihood (ML) estimation the z-

statistic can be applied if data are available in raw form or in form of the

variances-covariance matrix and if they are incomplete. Though bootstrapping

dominates these days the testing of mediation effects, it is not without

limitations. The ordinary nonparametric bootstrap method requires access to

raw data whose empirical distribution function is assumed to be a good

representation of the population distribution function. Although the

nonparametric bootstrap method makes no assumption about the distribution

in the population it is likely to fail to provide consistent estimates when the

original sample has an extreme distribution and outliers and when there are

many missing values. Dichotomous variables in combination with a small

sample are also likely to lead to estimation problems or inconsistent results.

Depending on the distribution of the data, the missing values, and the types of

variables, bootstrapping can require fairly large sample sizes with extreme

distributions, many missing values, and dichotomous variables all require

larger samples (see Chernick, 2008). Indeed, in small samples of 20 to 80

cases, bootstrap confidence intervals can be inconsistent (Efron & Tibshirani,

1993). Koopman, Howe, Hollenbeck, and Sin (2015) evaluated the bootstrap

method for small samples and note that even for moderate effect sizes samples

of 100 cases were required. The use of bootstrapping can further be

cumbersome when standardized effects should be analyzed because the

standardization would need to be done separately for each bootstrap sample. In

addition, some programs, including Amos, require complete data to perform

bootstrap analysis but parametric bootstrapping may be used in lieu of the


ordinary bootstrapping with incomplete data. Finally, as all resampling

techniques, the bootstrap method violates Gleser‟s “first law of applied

statistics” that two people using the same data and the same method should

always obtain the same results.

In conclusion, the z-statistic may be preferred when the bootstrap method

fails or leads to inconsistent estimates or when there are many missing values.

If the sample size is large either the bootstrap or the z-statistic may be chosen

because the conclusion would be almost identical (e.g., Cheung, 2007;

Williams & MacKinnon, 2008). If the sample size is medium the bootstrap

method has more statistical power than methods based on normal theory (e.g.,

MacKinnon, Lockwood, & Williams, 2004) and, so, is often preferred to test

both indirect effects and contrasts.

There are alternative methods to test mediation effects. One is Bayesian

analysis (see Kline, 2013, and Kruschke, Aguinis, & Joo, 2012, for a brief

introduction), which has been strongly recommended for small samples (e.g.,

less than 100 cases) and multilevel data (Enders, Fairchild, & MacKinnon,

2013; Koopman et al., 2015; Yuan & MacKinnon, 2009). Another method is

robust analysis based on median regression, which has most recently been

demonstrated to be useful when data are non-normal, including heavy-tailed

and skewed data (Yuan & MacKinnon, 2014).

ILLUSTRATIONS

We illustrate the assessment of mediation for a model with three

simultaneous mediators, a model with two sequential mediators, and a single-

mediator model with two initial variables using publicly available datasets. For

illustrative purpose, we used both the z-statistic and the bootstrap method to

test specific effects and contrasts. The bootstrap estimates presented here are

based on 5,000 bootstrap samples. To determine whether an effect is

statistically significant, we followed Cheung‟s recommendation (2007) and

reported the bias-corrected (BC) bootstrap confidence intervals for the

unstandardized effects. Cases with missing data were excluded prior to the

analysis. We analyzed the data using the Amos software program.


Mediation Model with Three Simultaneous Mediators

The data for the model with three simultaneous mediators (Figure 1B)

were taken from the study Quality of American Life conducted by Campbell

and Converse (1978). The purpose of this study was to investigate the

perceived quality of life of Americans 18 years of age and older. Here, we

used interview data from 1350 persons who provided complete data on the

selected variables. We used satisfaction with oneself as a person (X) to predict

satisfaction with the life (Y) through the simultaneous mediators satisfaction

with the job (M1), satisfaction with marriage (M2), and degree to which

respondents enjoy their life (M3). The satisfaction measures could range from

1 (completely dissatisfied) to 7 (completely satisfied) and the enjoyment of life

measure could range from 1 (rarely) to 4 (all the time). Due to the large sample

size we report 99 percent confidence intervals.

The effect estimates, the standard errors (estimated by Amos and

Equations A2, A4, A11, A15, and A17 for the variances), and the normal and

BC bootstrap confidence intervals of the effects are presented in Table 1

(phantom models were set up to obtain point and interval estimates for a1b1,

a2b2, a3b3, a1b1 – c′, a2b2 – c′, a3b3 – c′, a1b1 + a2b2 + a3b3 – c′, a1b1 – a2b2, a1b1

– a3b3, and a2b2 – a3b3). We found that all six direct effects that make up an

indirect effect were positive and statistically significant. The direct effect c′

was positive and significant too, which suggests that self-satisfaction had an

effect on life satisfaction over and above the effects of the three mediators. For

the indirect effects, all three simple indirect effects and the total indirect effect

were statistically significant indicating that the effect from satisfaction with

oneself on satisfaction with the life was simultaneously mediated through job

satisfaction, marriage satisfaction, and enjoyment of life. The total effect was

also positive and significant. The finding that all indirect effects as well as the

direct effect c′ had the same sign and were statistically significant indicates

that partial mediation occurred. Contrasting the three simple indirect effects

and the total indirect effect with the direct effect c′, we found that c′ was

significantly stronger than any indirect effect. Among the three indirect

effects, those through enjoyment of life and marital satisfaction were

significantly stronger than the one through job satisfaction.

Table 1. Testing mediation of satisfaction with oneself on satisfaction with the life through the simultaneous

mediators satisfaction with job, satisfaction with marriage, and enjoyment of life

Effect Estimate z-statistic Bootstrapping

SE z p 99% CI BC 99% CI

Direct effects

self sat job sat (a1) 0.362 0.034 10.795 <.001 [0.275, 0.448] [0.264, 0.461]

self sat mar sat (a2) 0.318 0.028 11.342 <.001 [0.246, 0.390] [0.233, 0.408]

self sat enjoy (a3) 0.201 0.013 15.182 <.001 [0.167, 0.235] [0.163, 0.237]

job sat life sat (b1) 0.110 0.017 6.288 <.001 [0.065, 0.155] [0.056, 0.169]

mar sat life sat (b2) 0.206 0.021 9.698 <.001 [0.151, 0.261] [0.126, 0.287]

enjoy life sat (b3) 0.395 0.045 8.736 <.001 [0.279, 0.511] [0.268, 0.521]

self sat life sat (c′) 0.375 0.024 15.763 <.001 [0.314, 0.436] [0.286, 0.458]

Indirect effects and total effect

a1b1 0.040 0.007 5.433 <.001 [0.021, 0.058] [0.020, 0.067]

a2b2 0.065 0.009 7.371 <.001 [0.043, 0.088] [0.038, 0.099]

a3b3 0.079 0.010 7.572 <.001 [0.052, 0.106] [0.052, 0.112]

a1b1 + a2b2 + a3b3 0.184 0.015 12.501 <.001 [0.146, 0.222] [0.137, 0.236]

a1b1 + a2b2 + a3b3 + c′ 0.559 0.023 23.922 <.001 [0.499, 0.619] [0.482, 0.625]

Contrasts

a1b1 – c′ -0.335 0.026 12.885 <.001 [-0.402, -0.268] [-0.428, -0.239]

a2b2 – c′ -0.309 0.026 11.689 <.001 [-0.377, -0.241] [-0.406, -0.204]

a3b3 – c′ -0.295 0.028 10.464 <.001 [-0.368, -0.223] [-0.392, -0.195]

Table 1. (Continued)



a1b1 + a2b2 + a3b3 – c′ -0.190 0.032 5.963 <.001 [-0.273, -0.108] [-0.310, -0.071]

a1b1 – a2b2 -0.026 0.012 2.224 .026 [-0.056, 0.004] [-0.065, 0.013]

a1b1 – a3b3 -0.040 0.013 3.024 .002 [-0.073, -0.006] [-0.078, -0.003]

a2b2 – a3b3 -0.014 0.014 0.970 .332 [-0.050 0.023] [-0.057, 0.030]

Note: SE = standard error; BC CI = bias-corrected confidence interval. The formula used to compute normal 99% CI is estimate

2.58 SE.

Table 2. Testing mediation of satisfaction with payment on satisfaction with the life through the sequential

mediators satisfaction with job and satisfaction with family life


SE z P 95% CI BC 95% CI

Direct effects

pay sat job sat (a1) 0.244 0.028 8.658 <.001 [0.189, 0.299] [0.181, 0.308]

pay sat fam sat (a2) 0.057 0.030 1.908 .056 [-0.002, 0.116] [-0.004, 0.118]

job sat fam sat (b12) 0.194 0.040 4.858 <.001 [0.116, 0.272] [0.109, 0.282]

job sat life sat (b1) 0.403 0.050 8.062 <.001 [0.305, 0.501] [0.305, 0.511]

fam sat life sat (b2) 0.437 0.049 8.989 <.001 [0.342, 0.533] [0.342, 0.532]

pay sat life sat (c′) 0.104 0.037 2.808 .005 [0.031, 0.177] [0.026, 0.183]

Indirect effects and total effect

a1b1 0.098 0.017 5.900 <.001 [0.066, 0.131] [0.064, 0.139]

a2b2 0.025 0.013 1.867 .062 [-0.001, 0.051] [-0.001, 0.055]

a1b12b2 0.021 0.005 3.832 <.001 [0.010, 0.031] [0.011, 0.035]

a1b12b2 + a1b1 + a2b2 0.144 0.022 6.478 <.001 [0.101, 0.188] [0.099, 0.193]

a1b12b2 + a1b1 + a2b2 + c′ 0.248 0.040 6.271 <.001 [0.171, 0.326] [0.156, 0.338]

Contrasts

a1b1 – c′ -0.006 0.044 0.131 .896 [-0.092, .080] [-0.096, .085]

a2b2 – c′ -0.079 0.040 1.994 .046 [-0.157, -.001] [-0.161, .005]

a1b12b2 – c′ -0.083 0.038 2.216 .027 [-0.157, -.010] [-0.164, .004]

a1b12b2 + a1b1 + a2b2 – c′ 0.040 0.047 0.859 .390 [-0.051, .131] [-0.051, .135]

Note: SE = standard error; BC CI = bias-corrected confidence interval. The formula used to compute normal 95% CI is estimate

1.96 SE.


In sum, the effect of satisfaction with oneself on satisfaction with the life

seemed partially transmitted through the simultaneous mediators job

satisfaction, marriage satisfaction, and enjoyment of life. In addition, evidence

indicated that the direct effect of self-satisfaction on life satisfaction was

stronger than the indirect effects through the three mediators and that the

mediators enjoyment of life and martial satisfaction seemed to be more

important than the mediator job satisfaction for the association between

satisfaction with oneself and satisfaction with the life.

Mediation Model with Two Sequential Mediators

The data for the model with two sequential mediators (Figure 1C) are part

of the Quality of Employment Survey conducted by Quinn and Graham

(1977). The aim of this survey was to investigate the working conditions in

American labor force by workers aged 16 or older. There were 640 people

who provided complete data for the variables used in this example. We used

the variable satisfaction with payment (X) to predict satisfaction with the life

(Y) through the mediators satisfaction with the job (M1) and satisfaction with

family life (M2) that act in turn. The answers for payment, job, and family

satisfaction could range from 1 to 4, the answers for life satisfaction from 1 to

3, with higher scores indicating greater satisfaction. Table 2 presents the effect

estimates, the standard errors (estimated by Amos and Equations A2, A6, A7,

A13, A15, A19 and A20 for the variances), and the normal and BC bootstrap

confidence limits (phantom models were set up to obtain point and interval

estimates for a1b1, a2b2, a1b12b2, a1b1 – c′, a2b2 – c′, a1b12b2 – c′, and a1b12b2 +

a1b1 + a2b2 – c′). All direct effects constituting the indirect effects were

positive and statistically significant with the exception of the effect from

satisfaction with payment on family life satisfaction (a2), which was not

significant. These direct effects make up two positive simple indirect effects

(a1b1 and a2b2) and a positive three-path indirect effect (a1b12b2). Because a2

was trivial the focus here is on a1b1 and a1b12b2. These two indirect effects

were both statistically significant, which indicates that the effect from

satisfaction with payment on life satisfaction was mediated by both the effect

through the sequential mediators job and family satisfaction (a1b12b2) and the

effect through job satisfaction (a1b1). Also, the total indirect effect and the

total effect were significant. The direct effect c′ was positive and statistically

significant, which means that the mediators accounted partially for the effect

of satisfaction with payment on life satisfaction. For the comparisons of the

Assessing Mediation 85

indirect effects with the direct effect c′, we found that the total indirect effect

(a1b12b2 + a1b1 + a2b2) and the simple indirect effect through job satisfaction

(a1b1) were equally strong as c′. However, c′ was significantly stronger than

the simple indirect effect through family satisfaction (a2b2) and the three-path

indirect effect a1b12b2. In sum, these results revealed that the effect of

satisfaction with payment on satisfaction with the life was partially mediated

by job satisfaction and satisfaction with family life. The simple mediating

effect through job satisfaction and the total mediating effect were as important

as the direct effect from satisfaction with payment on satisfaction with the life.

Mediation Model with Two Initial Variables

For the model with two initial variables (Figure 1D), we used cross-

sectional data collected in 2000 that are part of the study Marital Instability

Over the Life Course conducted by Booth, Johnson, Amato, and Rogers

(2010). For the variables used in this example, 711 people provided complete

data. As X1 and X2 we used respondent‟s income and partner‟s income,

respectively, to predict the degree of happiness (Y) through self-esteem (M).

Income can take the values: 0, 2.5, 7.5, 12.5, 17.5, 22.5, 27.5, 35.0, 45.0, 55.0,

65.0, 75.0, 85.0, 95.0, and 105.0 (unit of measurement is $1,000). Self-esteem

was measured by six items (1= strongly agree, 4 = strongly disagree;

Cronbach‟s alpha = .806) that were combined to a composite score with higher

scores indicating higher self-esteem. The answers for degree of happiness

could range from 1 to 3 with higher scores reflecting greater happiness. Here,

we used the unstandardized effects because the initial variables, respondent‟s

and partner‟s income, were measured by the same scale and the unit of

measurement of this scale was meaningful. This is a common practice in

dyadic research where dyad members typically provide information on the

same variables (Kenny & Ledermann, 2010). The effect estimates, the

standard errors (estimated by Amos and Equations A2, A8, A14, A15, A22,

and A23 for the variances), and the normal and BC bootstrap confidence

intervals of the effects are given in Table 3 (phantom models were set up to

obtain point and interval estimates for a1b + a2b, a1b + c′1 + a2b + c′2, a1b – c′1,

a2b – c′2, a1b – a2b, and a1b + c′1 – (a2b + c′2). We found that a1, a2, and b were

positive and statistically significant. These direct effects constitute two

positive simple indirect effects (a1b and a2b) that were statistically significant.

Table 3. Testing mediation of respondent’s and partner’s income on degree of happiness through the

mediator self esteem



Direct effects

R‟s income self-esteem (a1) 0.0182 0.0040 4.5834 <.0001 [0.0104, 0.0260] [0.0110, 0.0260]

P‟s income self-esteem (a2) 0.0148 0.0040 3.6765 .0002 [0.0069, 0.0227] [0.0071, 0.0228]

self-esteem happiness (b) 0.0909 0.0081 11.2685 <.0001 [0.0751, 0.1067] [0.0741, 0.1076]

R‟s income happiness (c'1) 0.0016 0.0009 1.8154 .0695 [-0.0001, 0.0033] [-0.0002, 0.0033]

P‟s income happiness (c'2) 0.0016 0.0009 1.7874 .0739 [-0.0002, 0.0033] [-0.0002, 0.0034]

Indirect and total effects

a1b 0.0017 0.0004 4.2456 .0001 [0.0009, 0.0024] [0.0010, 0.0025]

a2b 0.0013 0.0004 3.4952 .0005 [0.0006, 0.0021] [0.0007, 0.0022]

a1b + c'1 0.0032 0.0009 3.4818 .0005 [0.0014, 0.0050] [0.0014, 0.0050]

a2b + c'2 0.0029 0.0009 3.0938 .0020 [0.0011, 0.0048] [0.0010, 0.0048]

a1b + a2b 0.0030 0.0007 4.5547 <.0001 [0.0017, 0.0043] [0.0019, 0.0045]

a1b + c'1 + a2b + c'2 0.0061 0.0018 3.3604 .0008 [0.0026, 0.0097] [0.0029, 0.0092]

Contrasts

a1b – c'1 0.0001 0.0010 0.0845 .9327 [-0.0018, 0.0020] [-0.0018, 0.0021]

a2b – c'2 -0.0002 0.0010 0.2228 .8237 [-0.0021, 0.0017] [-0.0022, 0.0018]

a1b – a2b 0.0003 0.0004 0.7575 .4487 [-0.0005, 0.0011] [-0.0005, 0.0011]

a1b + c'1 – (a2b + c'2) 0.0003 0.0010 0.3060 .7596 [-0.0017, 0.0024] [-0.0016, 0.0023]

Note: SE = standard error; BC CI = bias-corrected confidence interval. The formula used to compute normal 95% CI is estimate 1.96

SE.


The two total effects, their sum, and the sum of the two indirect effects

were all positive and statistically significant. The two direct effects c′ were not

significant. The fact that both indirect effects were not significantly stronger

than the respective direct effects c′ does not support complete mediation. The

finding that the two indirect effects and the two total effects were equal in size

indicated that a person‟s degree of happiness seemed to be affected to the

same extent by one‟s own income and that of the partner. That is, an increase

of the partner‟s income by one unit would have had the same effect on a

person‟s degree of happiness as an increase of his or her own income.

In sum, results suggested that the effect of both one‟s own income and the

partner‟s income on his or her happiness was mediated by his or her self-

esteem. Moreover, partner‟s income affected the degree of happiness to the

same extent as one‟s own income.

DISCUSSION

The assessment of specific indirect effects and contrasts of effects, as

discussed in this chapter, offers much promise for a good understanding of the

mechanism through which one or multiple initial variables affect an outcome.

Specifically, the information whether a specific indirect effect is stronger than,

weaker than, or equal in strength to its respective direct effect c′ or whether in

a model with multiple mediators one indirect effect is stronger or weaker than

another indirect effect can (a) help to foster a better understanding of the

significance of the mediators in a model and (b) provide insights that allow a

researcher to draw firm conclusions about where it is appropriate to intervene.

Moreover, comparing the indirect effect with the respective direct effect c′ and

requiring for complete mediation that the indirect effect is bigger in size than

the respective direct effect c′ adds to the requirements for complete mediation

that proponents of the distinction between partial and complete mediation may

find useful.

Mediation analysis has been criticized because mediation implies a causal

mechanism that often remains uncertain (Bullock, Green, & Ha, 2010) due to

the existence of statistically equivalent models (e.g., Lee & Hershberger, 1990;

MacCallum, Wegener, Uchiono, & Fabrigar, 1993). This problem of causal

inference is alleviated, if theoretical considerations preclude alternative

models or if experimental designs provide strong evidence for the underlying

mediational process (e.g., Spencer, Zanna, & Fong, 2005). A method widely

discussed to reduce the number of statistically equivalent models and to detect


model misspecification is the use of instrumental variables (e.g., Foster, &

McLanahan, 1996; Joffe, Small, Have, Brunelli, & Feldman, 2008;

MacKinnon & Pirlott, 2015; Pearl, 2014; Reardon, Unlu, Zhu, & Bloom,

2014; Shrout & Bolger, 2002). Another method recommended by Imai, Keele,

and Tingley (2010) is sensitivity analysis

In conclusion, mediation models are often used to shed some light into the

process through which an initial variable exerts an effect on an outcome. The

testing of specific effects and contrast can provide important insights into the

causal pathways and contribute to a refinement of the theories underlying a

model. Specific effects and contrasts can be tested by means of the z-statistic

or the phantom model method, which provides a flexible means for testing and

contrasting specific effects in recursive and non-recursive structural equation

models, including multilevel models (Preacher, Zyphur, & Zhang, 2010).

APPENDIX A. THE MULTIVARIATE

DELTA METHOD

Bollen (1987, 1989) and Sobel (1982, 1988) give details on how first

order (approximate) standard errors of indirect and total effects can be derived

using the multivariate delta method (and maximum likelihood or generalized

least squares). For a function f of parameter estimates (e.g., ) the

asymptotic variance can be obtained by:

, (A1)

where is the column vector of the partial derivatives of the function

with respect to its parameters ,6 is the variance-covariance matrix of

the estimates, and superscript T denotes the transpose. The square root of

represents the estimate of the approximate standard error of the

function . Next, we give the asymptotic variances of the indirect and total

effects and specific contrasts in models with up to four simultaneous or two

sequential mediators and models with two initial variables. For the sake of

6 The partial derivative of a function (e.g., ab – c′) with respect to one of its parameters (e.g., a) is

the ordinary derivative of the function with respect to that parameter with all other

parameters considered as constants.


simplicity, the hats above the parameter estimates are dropped in the

remainder of this section. We note that in mediation models, the a parameters

are independent from the b and c′ parameters. Consequently, the covariances

between the a parameters and the b and c′ parameters are zero.

Indirect Effects

Using the first order delta method, Sobel (1982) derived a formula to

calculate the asymptotic variance for simple indirect effects. For the effect ab

this is

2222)(Var ab sbsaab , (A2)

where a and b are estimated parameters, 2

as and 2

bs are estimated variances of

a and b, respectively. This equation can be used to test any simple indirect

effect in the models of Figure 1.

Mediation models with multiple simultaneous mediators (Figure 1B). The

asymptotic variances for the indirect effects a1b1 + a2b2, a1b1 + a2b2 + a3b3,

and a1b1 + a2b2 + a3b3 + a4b4 are (see Preacher & Hayes, 2008, p. 882)

, (A3)

323231312121

323231312121

2

3

2

3

2

2

2

2

2

1

2

1

2

3

2

3

2

2

2

2

2

1

2

1332211

222

222

)(Var

aaaaaa

bbbbbb

aaabbb

sbbsbbsbb

saasaasaa

sbsbsbsasasabababa

, (A4)

and

434342423232

41413131212143434242

3232414131312121

2

4

2

4

2

3

2

3

2

2

2

2

2

1

2

1

2

4

2

4

2

3

2

3

2

2

2

2

2

1

2

144332211

222

22222

2222

)(Var

aaaaaa

aaaaaabbbb

bbbbbbbbaa

aabbbb

sbbsbbsbbc

sbbsbbsbbsaasaa

saasaasaasaasbsb

sbsbsasasasababababa

, (A5)

where sa1a2, sa1a3, … sb3b4 are the covariance between the estimated parameters.

21212121

2

2

2

2

2

1

2

1

2

2

2

2

2

1

2

12211 22)(Var aabbaabb sbbsaasbsbsasababa


Mediation models with two sequential mediators (Figure 1C). The

asymptotic variance for the three-path indirect effect a1b12b2 (see Taylor et al.,

2008, p. 245) and the total indirect effect a1b12b2 + a1b1 + a2b2 are

2

1

2

2

2

12

2

12

2

2

2

1

2

2

2

12

2

12121 )(Var abb sbbsbasbabba (A6)

and

2121

2

12121

2

21221122

2

212112

2

1

2

1

2

2

2

12

2

12

2

2

2

1

2

2

2

12

2

1

2

2

2

2

2

1

2

1

2

2

2

2

2

1

2

122112121

22

222

)(Var

bba

bbabbab

baabb

saasbbb

sbaasbasbasbbsba

sbasbsbsasabababba

.(A7)

Mediation models with two initial variables (Figure 1D). The asymptotic

variance for the sum of the indirect effects a1b and a2b is

21

22

2

2

1

222

2121 2)(Var aaaab sbssbsaababa . (A8)

Total Effects

Simple mediation model and models with multiple mediators (Figure 1A-

1C). The asymptotic variances for the total effects ab + c′, a1b1 + a2b2 + c′,

a1b1 + a2b2 + a3b3 + c′, a1b1 + a2b2 + a3b3 + a4b4 + c′, and a1b12b2 + a1b1 + a2b2

+ c′ are

2

''

2222 2)'(Var cbcab sassbsacab , (A9)

2

''22'1121212121

2

2

2

2

2

1

2

1

2

2

2

2

2

1

2

12211

2222

)'(Var

ccbcbaabb

aabb

ssasasbbsaa

sbsbsasacbaba

, (A10)

2

''33'22'1132323131

2121323231312121

2

3

2

3

2

2

2

2

2

1

2

1

2

3

2

3

2

2

2

2

2

1

2

1332211

22222

2222

)'(Var

ccbcbcbaaaa

aabbbbbba

aabbb

ssasasasbbsbb

sbbsaasaasaasb

sbsbsasasacbababa

, (A11)


2

''44'33'22'114343

42423232414131312121

4343424232324141

31312121

2

4

2

4

2

3

2

3

2

2

2

2

2

1

2

1

2

4

2

4

2

3

2

3

2

2

2

2

2

1

2

144332211

22222

22222

2222

22

)'(Var

ccbcbcbcbaa

aaaaaaaaaa

bbbbbbbb

bbbbaaaa

bbbb

ssasasasasbb

sbbsbbsbbsbbsbb

saasaasaasaa

saasaasbsbsbsb

sasasasacbabababa

, (A12)

and

2

''2121'22

'112121

2

12121

2

21221

122

2

212112

2

1

2

1

2

2

2

12

2

12

2

2

2

1

2

2

2

12

2

1

2

2

2

2

2

1

2

1

2

2

2

2

2

1

2

122112121

22

2222

22

)'(Var

ccbcb

cbbbab

babbabb

aabb

ssbasa

sasaasbbbsbaa

sbasbasbbsbasba

sbsbsasacbababba

. (A13)

Mediation model with two initial variables (Figure 1D). The asymptotic

variance for the sum of the two total effects (i.e., a1b + c′1 + a2b + c′2) is

2

2'

2

1'2'1'2'22'11'21'1

2

2121

22

2

22

1

222

2

22

12211

22222

22)''(

ccccbcbcbcbc

baaaabb

ssssasasasa

saasbsbsbsasacbacbaVar

. (A14)

Contrasts

Simple mediation model (Figure 1A). The asymptotic variances for the

contrasts ab – c′ (see MacKinnon, 2000, p. 151) is

2

''

2222 2)'(Var cbcab sassbsacab . (A15)

Contrasting the indirect effects with c′ in models up to four simultaneous

mediators (Figure 1B). The asymptotic variances for a1b1 + a2b2 – c′, a1b1 +

a2b2 + a3b3 – c′, a1b1 + a2b2 + a3b3 + a4b4 – c′ are

2

''22'1121212121

2

2

2

2

2

1

2

1

2

2

2

2

2

1

2

12211

2222

)'(Var

ccbcbaabb

aabb

ssasasbbsaa

sbsbsasacbaba

, (A16)


2

''33'22'11

32323131

2121323231312121

2

3

2

3

2

2

2

2

2

1

2

1

2

3

2

3

2

2

2

2

2

1

2

1332211

222

22

2222

)'(Var

ccbcbcb

aaaa

aabbbbbb

aaabbb

ssasasa

sbbsbb

sbbsaasaasaa

sbsbsbsasasacbababa

, (A17)

and

2

''44'33'22'11

43434242

3232414131312121

43434242

3232414131312121

2

4

2

4

2

3

2

3

2

2

2

2

2

1

2

1

2

4

2

4

2

3

2

3

2

2

2

2

2

1

2

144332211

2222

22

2222

22

2222

)'(Var

ccbcbcbcb

aaaa

aaaaaaaa

bbbb

bbbbbbbb

aaaab

bbb

ssasasasa

sbbsbb

sbbsbbsbbsbb

saasaa

saasaasaasaa

sbsbsbsbsa

sasasacbabababa

. (A18)

Contrasting the indirect effects with c′ in models with two sequential

mediators (Figure 1C). The asymptotic variances for a1b12b2 – c′ and a1b12b2 +

a1b1 + a2b2 – c′ are

2

''2121

2

1

2

2

2

12

2

2

2

12

2

1

2

12

2

2

2

12121 2)'( ccbabb ssbasbbsbasbacbbaVar , (A19)

and

2

''2121'22'112121

2

12121

2

21221122

2

212112

2

1

2

1

2

2

2

12

2

12

2

2

2

1

2

2

2

12

2

1

2

2

2

2

2

1

2

1

2

2

2

2

2

1

2

122212121

2222

2222

'Var

ccbcbcbbb

abbabb

abb

aabb

ssbasasasaa

sbbbsbaasbasba

sbbsbasba

sbsbsasacbababba

. (A20)

Contrasting two simple indirect effects (Figure 1B and 1D). The

asymptotic variance for the contrasts a1b1 – a2b2 (see Preacher & Hayes, 2008,

p. 884) and a1b – a2b are

21212121

2

2

2

2

2

1

2

1

2

2

2

2

2

1

2

12211 22Var aabbaabb sbbsaasbsbsasababa (A21)


and

21

22

2

2

1

222

2121 2)(Var aaaab sbssbsaababa . (A22)

Contrasting two total effects in models with two initial variables (Figure

1D). The asymptotic variance for the contrast a1b + c′1 – (a2b + c′2) is

2

2'

2

1'2'1'2'211'2121

2

2

2

22

1

222

212211

2222

)'('Var

ccccbcbcaa

aab

ssssaasaasb

sbsbsaacbacba

. (A23)

APPENDIX B. PHANTOM MODELS

Figure B1 shows the phantom models for assessing the simple indirect

effect a1b1 and the three-path indirect effect a1b12b2. To identify the models

each path coefficient in the phantom model is set equal to the coefficient of the

corresponding path in the main model (e.g., a1 in Model B1B is equated to a1

in Model 1B). In addition, the variance of the initial phantom variable Pin is set

to 1 and the mean to zero while the intercepts of the other phantom variables

(e.g., P1, and Pout) are fixed to zero.

To obtain the point estimate of a specific effect represented by a phantom

model the total effect is estimated between the phantom model‟s initial

variable and its final outcome variable. In the models of Figure B1, the total

effects between Pin and Pout equal a1b1 and a1b12b2, respectively. To test a

Figure B1.Phantom models for assessing indirect effects. Model A: Simple indirect

effect a1b1. Model B: Three-path indirect effect a1b12b2.

Model A

Model B

P1 b1 a1 Pin Pout

Pin Pout P1 b12

P2 b2 a1


specific effect, the bootstrap confidence limit of the phantom model‟s total

effect is estimated. We note that in contrast to the equations for the standard

errors, phantom models do not depend whether the residual covariances in the

main model are constrained or freely estimated.

Phantom models for assessing specific contrasts are shown in Figure B2

and B3. The models of Figure B2 are designed to assess the difference

between a specific indirect effect and the direct effect c′. Model B2A contrasts

Figure B2. Phantom models for assessing contrasts involving the direct effect c′.

Model A: Contrast ab – c′. Model B: Contrast a1b1 + a2b2 + a3b3 – c′. Model C:

Contrast a1b12b2 – c′. Model D: Contrast a1b12b2 + a1b12b2 + a2b2 – c′.

Model A

Model B

Model C

Model D

P2

a1

Pout c′ Pin

b1

-1

P1

P4

a1

Pout c′

Pin

b1

-1

P1

P2

P3

b2

b3

a2

a3

b2 P1

b12 P2

a1

P3 -1 c′

Pin Pout

P5 -1 c′ Pin Pout

b2

P1 b12

P2

a1

P4

P3 a1

a2 b2

b1


the simple indirect effect a1b1 with c′. Model B2B tests a1b1 + a2b2 + a3b3 – c′,

Model B2C a1b12b2 – c′, and Model B2D a1b1b2 + a1b1 + a2b2 – c′. For

mediation models containing fewer or more simultaneous mediators the

phantom model B2B can readily be adapted: Excluding a phantom variable,

say P3, with the connecting paths a3 and b3 yields a phantom model with two

simultaneous mediators; adding a new phantom variable, say P4, connected by

the paths a4 and b4 results in a phantom model with four simultaneous

mediators. The models of Figure B3 enable the comparison of two simple

indirect effects and two total effects: Model B3A tests a1b1 – a2b2, Model B3B

b(a1 – a2), and Model B3C a1b + c′1 – (a2b + c′2).

Figure B3.Phantom model for contrasting simple indirect effects and total effects.

Model A: Contrast a1b1 – a2b2. Model B: Contrast a1b – a2b. Model C: Contrast a1b +

c′1 – (a2b + c′2).

Model A

Model B

Model C

-1

P2 b2

P3

a2 P1

b1 a1 Pin Pout

-1

P2 b

P3

a2 P1

b a1 Pin Pout

-1

P2 b

P3

a1 P1

c′1

a2 c′2

b

Pin Pout


Estimating a model with two initial variables, a researcher may also wish

to assess the sum of the two indirect effects and the sum of the two total

effects. This can be achieved by changing in Model B3B and B3C the path P3

Pout from -1 to 1. The total effects are then b(a1 + a2) and a1b + c′1 + a2b +

c′2, respectively.

AUTHOR NOTE

The research used three data sets: Quality of American Life 1978; the

Quality of Employment Survey 1977; and Marital Instability Over the Life

Course: A six-Wave Panel Study, 1980, 1983, 1988, 1992-1994, 1997, 2000.

The data collected by Angus Campbell and Philip Converse; Robert Quinn and

Staines Graham; and Alan Booth, David Johnson, Paul Amato, and Stacy

Rogers were made available through Ann Arbor, MI: Inter-university

Consortium for Political and Social Research.

Supplementary material is available at http://thomasledermann.com

/mscm/.

We thank David A. Kenny for helpful comments on an earlier version of

this chapter.

REFERENCES

Arbuckle, J. L. (1995–2012). IBM SPSS Amos 20 user's guide. Chicago, IL:

SPSS Inc.

Baron, R. M. & Kenny, D. A. (1986). The moderator-mediator variable

distinction in social psychological research: Conceptual, strategic, and

statistical considerations. Journal of Personality and Social Psychology,

51, 1173-1182.

Bentler, P. M. (2000-2008). EQS 6 structural equations program manual.

Encino, CA: Multivariate Software.

Bishop, Y. M. M., Feinberg, S. E. & Holland, P. W. (1975). Discrete

multivariate analysis: Theory and practice. Cambridge, MA: MIT Press.

Boker, S., Neale, M., Maes, H., Wilde, M., Spiegel, M., Brick, T., . . . Fox, J.

(2011). OpenMx: An open source extended structural equation modeling

framework. Psychometrika, 76, 306-317.


Bollen, K. A. (1987). Total, direct, and indirect effects in Structural Equation

Models. Sociological Methodology, 17, 37-69.

Bollen, K. A. (1989). Structural equations with latent variables. New York:

Wiley.

Bollen, K. A. & Stine, R. (1990). Direct and indirect effects: Classical and

bootstrap estimates of variability. Sociological Methodology, 20, 115-140.

Booth, A., Johnson, D., Amato, P. & Rogers, S. (2010). Marital Instability

Over the Life Course: A Six-Wave Panel Study, 1980, 1983, 1988, 1992-

1994, 1997, 2000 [Computer file]: Ann Arbor, MI: Inter-university

Consortium for Political and Social Research [distributor].

Browne, M. W. & Mels, G. (2005). Path analysis (RAMONA). In SYSTAT

11: Statistics III [Computer software and manual] (III-1–III-61).

Richmond, CA: SYSTAT Software.

Bullock, J. G., Green, D. P. & Ha, S. E. (2010). Yes, but what‟s the

mechanism? (don‟t expect an easy answer). Journal of Personality and

Social Psychology, 98, 550-558.

Campbell, A. & Converse, P. E. (1978). Quality of American Life, 1978

[Computer file]. Conducted by University of Michigan, Survey Research

Center. Ann Arbor, MI: Inter-university Consortium for Political and

Social Research [producer and distributor], 1984.

Chernick, M. R. (2008). Bootstrap methods: A guide for practitioners and

researchers (2nd ed.). New York: Wiley.

Cheung, M. W. L. (2007). Comparison of approaches to constructing

confidence intervals for mediating effects using Structural Equation

Models. Structural Equation Modeling, 14, 227-246.

Cheung, M. W. L. (2009). Comparison of methods for constructing confidence

intervals of standardized indirect effects. Behavior Research Methods, 41,

425-438.

Efron, B. (1979). Bootstrap methods: Another look at the jackknife. The

Annals of Statistics, 7, 1-26.

Efron, B. & Tibshirani, R. J. (1993). An introduction to the bootstrap. New

York: Chapman & Hall/CRC.

Enders, C. K., Fairchild, A. J. & MacKinnon, D. P. (2013). A Bayesian

approach for estimating mediation effects with missing data. Multivariate

Behavioral Research, 48, 340-369.

Fosco, G. M. & Grych, J. H. (2008). Emotional, cognitive, and family systems

mediators of children's adjustment to interparental conflict. Journal of

Family Psychology, 22, 843-854.


Foster, E. M. & McLanahan, S. (1996). An illustration of the use of

instrumental variables: Do neighborhood conditions affect a young

person's chance of finishing high school? Psychological Methods, 1, 249-

260.

Hayes, A. F. & Preacher, K. J. (2014). Statistical mediation analysis with a

multicategorical independent variable. British Journal of Mathematical

and Statistical Psychology, 67, 451-470.

Imai, K., Keele, L. & Tingley, D. (2010). A general approach to causal

mediation analysis. Psychological Methods, 15, 309-334.

James, L. R. & Brett, J. M. (1984). Mediators, moderators, and tests for

mediation. Journal of Applied Psychology, 69, 307-321.

Joffe, M. M., Small, D., Have, T. T., Brunelli, S. & Feldman, H. I. (2008).

Extended instrumental variables estimation for overall effects. The

International Journal of Biostatistics, 4, 1-20.

Jöreskog, K. G. & Sörbom, D. (2006). LISREL 8.8 for Windows. Lincolnwood,

IL: Scientific Software International, Inc.

Jöreskog, K. G., Sörbom, D., Du Toit, S. & Du Toit, M. (1999). LISREL 8:

New Statistical Features. Chicago: Scientific Software International.

Judd, C. M. & Kenny, D. A. (2010). Data analysis in social psychology:

Recent and recurring issues. In D. T. Gilbert, S. T. Fiske & G. Lindzey

(Eds.), Handbook of social psychology (5th ed., 115-139). New York:

Wiley.

Kenny, D. A. & Judd, C. M. (2014). Power anomalies in testing mediation.

Psychological Science, 25, 334-339.

Kenny, D. A., Kashy, D. A. & Bolger, N. (1998). Data analysis in social

psychology. In D. T. Gilbert, S. T. Fiske & G. Lindzey (Eds.), Handbook

of social psychology (4th ed., Vol. 1, 233-265). New York: McGraw-Hill.

Kenny, D. A. & Ledermann, T. (2010). Detecting, measuring, and testing

dyadic patterns in the Actor-Partner Interdependence Model. Journal of

Family Psychology, 24, 359-366.

Klainin, P. (2009). Stress and health outcomes: The mediating role of negative

affectivity in female health care workers. International Journal of Stress

Management, 16, 45–64.

Kline, R. B. (2013). Bayesian estimation and best practices summary Beyond

significance testing: Statistics reform in the behavioral sciences (2nd ed.,

289-312). Washington, DC US: American Psychological Association.

Koopman, J., Howe, M., Hollenbeck, J. R. & Sin, H.-P. (2015). Small sample

mediation testing: Misplaced confidence in bootstrapped confidence

intervals. Journal of Applied Psychology, 100, 194-202.


Kruschke, J. K., Aguinis, H. & Joo, H. (2012). The time has come: Bayesian

methods for data analysis in the organizational sciences. Organizational

Research Methods, 15, 722-752.

Lau, R. S. & Cheung, G. W. (2012). Estimating and comparing specific

mediation effects in complex latent variable models, Organizational


Ledermann, T. & Macho, S. (2009). Mediation in dyadic data at the level of

the dyads: A Structural Equation Modeling approach. Journal of Family

Psychology, 23, 661-670.

Ledermann, T., Macho, S. & Kenny, D. A. (2011). Assessing mediation in

dyadic data using the actor-partner interdependence model. Structural

Equation Modeling, 18, 595-612.

Lee, D. & Hershberger, S. (1990). A simple rule for generating equivalent

models in covariance structure modeling. Multivariate Behavioral

Research, 25, 313-334.

Maassen, G. D. & Bakker, A. B. (2001). Suppressor variables in path models:

Definitions and interpretations. Sociological Methods and Research, 30,

241-270.

MacCallum, R. C., Wegener, D. T., Uchiono, B. N. & Fabrigar, L. R. (1993).

The problem of equivalent models in applications of covariance structure

analysis. Psychological Bulletin, 114, 185-199.

Macho, S. & Ledermann, T. (2011). Estimating, testing, and comparing

specific effects in structural equation models: The phantom model

approach. Psychological Methods, 16, 34-43.

MacKinnon, D. P. (2000). Contrasts in multiple mediator models. In J. Rose,

L. Chassin, C. C. Presson & S. J. Sherman (Eds.), Multivariate

applications in substance use research: New methods for new questions

(141-160). Mahwah, NJ: Lawrence Erlbaum.

MacKinnon, D. P. (2008). Introduction to statistical mediation analysis. New

York: Erlbaum.

MacKinnon, D. P., Krull, J. L. & Lockwood, C. M. (2000). Equivalence of the

mediation, confounding and suppression effect. Prevention Science, 1,

173-181.

MacKinnon, D. P., Lockwood, C. M. & Williams, J. (2004). Confidence limits

for the indirect effect: Distribution of the product and resampling

methods. Multivariate Behavioral Research, 39, 99-128.

MacKinnon, D. P. & Pirlott, A. G. (2015). Statistical approaches for enhancing

causal interpretation of the M to Y relation in mediation analysis.

Personality and Social Psychology Review, 19, 30-43.


MacKinnon, D. P., Warsi, G. & Dwyer, J. H. (1995). A simulation study of

mediated effect measures. Multivariate Behavioral Research, 30, 41-62.

Muthén, L. K. & Muthén, B. O. (1998-2012). Mplus user's guide. Seventh

Edition. Los Angeles, CA: Muthén & Muthén.

Pearl, J. (2014). Interpretation and identification of causal mediation.

Psychological Methods, 19, 459-481.

Perera, H. N. (2013). A novel approach to estimating and testing specific

mediation effects in educational research: Explication and application of

Macho and Ledermann‟s (2011) Phantom Model approach. International

Journal of Quantitative Research in Education, 1, 39-60.

Perkinson-Gloor, N., Lemola, S. & Grob, A. (2013). Sleep duration, positive

attitude toward life, and academic achievement: The role of daytime

tiredness, behavioral persistence, and school start times. Journal of

Adolescence, 36, 311-318.

Preacher, K. J. & Hayes, A. F. (2008). Asymptotic and resampling strategies

for assessing and comparing indirect effects in multiple mediator models.

Behavior Research Methods, 40, 879-891.

Preacher, K. J. & Kelley, K. (2011). Effect size measures for Mediation

models: Quantitative strategies for communicating indirect effects.

Psychological Methods, 16, 93-115

Preacher, K. J., Zyphur, M. J. & Zhang, Z. (2010). A general multilevel SEM

framework for assessing multilevel mediation. Psychological Methods, 15,

209-233.

Quinn, R. P. & Staines, G. L. (1977). Quality of Employment Survey, 1977:

Cross-section [Computer file]. Conducted by University of Michigan,

Survey Research Center: ICPSRO7689-v1. Ann Arbor, MI: Inter-

university Consortium for Political and Social Research [producer and

distributor], 2000.

Rao, C. P. (1973). Linear statistical inference and its applications (2nd ed.).

New York: Wiley.

Raykov, T., Brennan, M., Reinhardt, J. P. & Horowitz, A. (2008). Comparison

of mediated effects: A correlation structure modeling approach. Structural


Raykov, T. & Marcoulides, G. A. (2004). Using the Delta Method for

Approximate Interval Estimation of Parameter Functions in SEM.

Structural Equation Modeling, 11, 621-637.

Reardon, S. F., Unlu, F., Zhu, P. & Bloom, H. S. (2014). Bias and bias

correction in multisite instrumental variables analysis of heterogeneous


mediator effects. Journal of Educational and Behavioral Statistics, 39, 53-

86.

Rosseel, Y. (2012). lavaan: An R package for structural equation modeling.

Journal of Statistical Software, 48, 1-36.

Rucker, D. D., Preacher, K. J., Tormala, Z. L. & Petty, R. E. (2011).

Mediation analysis in social psychology: Current practices and new

recommendations. Social and Personality Psychology Compass, 5, 359-

371.

Shrout, P. E. & Bolger, N. (2002). Mediation in experimental and

nonexperimental studies: New procedures and recommendations.


Sobel, M. E. (1982). Asymptotic confidence intervals for indirect effects in

structural equation models. In S. Leinhart (Ed.), Sociological methodology

(290-312). San Francisco: Jossey-Bass.

Sobel, M. E. (1986). Some new results on indirect effects and their standard

errors in covariance structure models. In N. Tuma (Ed.), Sociological

methodology (159-186). Washington, DC: American Sociological

Association.

Spencer, S. J., Zanna, M. P. & Fong, G. T. (2005). Establishing a causal chain:

Why experiments are often more effective in examining psychological

process than mediational analyses. Journal of Personality and Social

Psychology, 89, 845-851.

Taylor, A. B., MacKinnon, D. P. & Tein, J.-Y. (2008). Tests of the three-path

mediated effect. Organizational Research Methods, 11, 241-269.

Williams, J. & MacKinnon, D. P. (2008). Resampling and distribution of the

product methods for testing indirect effects in complex models. Structural


Wood, R. E., Goodman, J. S., Beckmann, N. & Cook, A. (2008). Mediation

testing in management research: A review and proposals. Organizational


Yuan, Y. & MacKinnon, D. P. (2009). Bayesian mediation analysis.


Yuan, Y. & MacKinnon, D. P. (2014). Robust mediation analysis based on

median regression. Psychological Methods, 19, 1-20.

INDEX

A

academic performance, 36, 65

access, 5, 7, 9, 21, 28, 30, 61, 78

accountability, vii, 35, 36, 38, 40, 41, 47,

51, 57, 60, 61, 67

accountability ratings, 40

accounting, 14, 52

achievement test, 40

adjustment, 2, 3, 65, 97

administrators, vii, 35, 37, 38, 40, 59, 60, 61

adolescents, 68

African-American, 63

age, 37, 80

agencies, 5, 6

algorithm, 42

American Educational Research

Association, 62, 64, 66

American Psychological Association, 98

anxiety, 62

APC, 9

appraisals, 71

aptitude, 36

arithmetic, 15

Asia, 3, 31

assessment, 25, 60, 63, 70, 73, 79, 87

assets, 6, 7, 8, 9, 12, 20, 21, 26, 30

asymmetric information, 2, 24, 28, 30

asymmetry, 2, 8, 9, 21

atmosphere, 37

attitudes, 59, 67

B

banking, 10, 29

bankruptcy, 2, 5, 6, 8, 11, 13, 21, 24, 28

banks, 4, 5, 9, 10, 28

barriers, 40, 60

Bayesian estimation, 98

Bayesian methods, 99

behavioral sciences, 63, 70, 98

behaviors, 13

benefits, 8, 21, 38, 40

benign, 21

bias, 79, 82, 83, 86, 100

blame, 71

bond market, 5

bonds, 5

bullying, 38

burnout, 64

C

calorie, 74

capital gains, 19

case study, 63

cash, 9, 10, 11

cash flow, 9, 10, 11

causal inference, 87

causal interpretation, 99

Index 104

CEE, 3

Chicago, 39, 62, 64, 68, 96, 98

child development, 64

child poverty, 40, 60

children, 36, 71, 97

China, vii, 1, 4, 5, 6, 7, 10, 14, 19, 21, 24,

25, 26, 29, 32, 34

Chinese firms, 4, 5, 19, 24, 29

Chinese government, 5, 24

City, 63

clarity, 68

classes, vii, 35, 42, 43, 44, 45, 46, 49, 50,

51, 52, 53, 54, 58, 66

classification, 42, 43, 45, 46, 50, 52, 53, 54,

56, 58, 59

classroom, 38, 47, 49, 65

classroom management, 47

climate, vii, 35, 36, 37, 38, 39, 40, 41, 42,

47, 48, 50, 51, 52, 53, 56, 58, 59, 60, 61,

62, 63, 64, 65, 66, 67, 68

cluster analysis, 41, 42, 43, 54, 62, 64, 66,

68

cluster model, 46

clustering, 42, 43, 44, 50, 62

collaboration, 36, 41

collateral, 7, 8, 10, 26

commercial, 10

commercial bank, 10

communication, 62

community(s), 37, 39, 60, 61

complexity, 9

complications, 4

computation, 50

computer, 43

computing, 16, 77

conflict, 97

construction, 3

consumption, 11, 74

controversial, 7, 10, 36

cooperation, 38

corporate finance, 2

correlation, 100

cost, 8, 9, 10, 11

creditors, 21

criminal acts, 36

criticism, 42

crowds, 4

CT, 64, 67

culture, 37, 65, 68

curriculum, 61

customers, 11

D

data analysis, 99

data set, 3, 4, 14, 25, 42, 47, 48, 70, 96

database, 41, 47, 60

debts, 30

delinquency, 64

Delta, 88, 100

Department of Education, 41, 50, 51, 63, 67

depreciation, 8

depression, 38

derivatives, 88

developed countries, 4, 19, 20, 24, 29

developing countries, 3, 19

deviation, 11, 75

disclosure, 6

disorder, 63, 64

disposition, 39

dissatisfaction, 54

distress, 11, 24

distribution, 14, 15, 25, 43, 44, 46, 50, 58,

78, 101

distribution function, 78

E

earnings, 9, 11, 21, 24

Eastern Europe, 3

ecology, 37

education, 68

educational research, 100

educational system, 61

educators, 38, 62

elementary school, vii, 35, 39, 42, 47, 51,

52, 53, 54, 58, 59

emerging markets, 5

empirical studies, 8, 14

Index 105

energy, 74

enforcement, 40, 49

entropy, 46, 52, 53, 54, 62

environment(s), 3, 4, 37, 38, 47, 49, 54, 59,

61, 64

EPR, 8

EPS, 9, 12, 17, 18, 22, 26

equity, 5, 7, 9, 10, 11, 13, 19, 20, 21, 29

equity market, 5

estimation problems, 78

ethnicity, 37

Europe, 3

European Union (EU), 3, 32, 33

evidence, 3, 8, 20, 41, 45, 84, 87

examinations, viii, 36

experimental design, 87

external financing, 11

F

factor analysis, 4, 40, 46, 48

families, 62

family life, 83, 84, 85

family system, 97

fear, 10

financial, 3, 5, 7, 9, 10, 11, 14, 16, 20, 24,

28

financial condition, 28

financial crisis, 24

financial distress, 11, 24

financial institutions, 5

financial sector, 5, 10

Finland, 68

firm size, 25, 28, 29, 30

flexibility, 51

food, 74

force, 84

formula, 82, 83, 86, 89

France, 2

free trade, 6

freedom, 38

funding, 60

funds, 11

G

Germany, 2

governments, 6

GPA, 39

grades, 39, 40, 47, 67

group membership, 41, 45

grouping, 4, 42

growth, 5, 6, 8, 21, 26, 29, 57, 66

H

happiness, 85, 86, 87

health, 63, 67, 68, 72, 98

health care, 98

health education, 68

high school, 49, 63, 65, 98

history, 41

homogeneity, 44

Hong Kong, 1, 30

House, 65

human, 37

hypothesis, 2, 8, 9, 21, 53, 59

hypothesis test, 53

I

identification, 16, 100

improvements, 38, 39, 40

income, 6, 9, 13, 59, 85, 86, 87

income tax, 6, 13

independence, 44

independent variable, 71, 98

indirect effect, viii, 69, 70, 71, 73, 74, 75,

76, 77, 78, 79, 80, 84, 85, 87, 89, 90, 91,

92, 93, 94, 95, 96, 97, 99, 100, 101

individuals, 43, 45

industrial sectors, 14

industry(s), 3, 5, 14, 20, 25, 26, 28, 29, 30

inefficiency, 4

insider trading, 6

institutions, 5, 10

interdependence, 99

interest rates, 10

Index 106

internal financing, 9, 21

internalizing, 71

interparental conflict, 97

interpretability, 44, 46, 54

intervention, 41, 61

investment(s), 5, 8, 11, 21, 29

investment capital, 29

investors, 4, 8, 28

Iowa, 39

IPO, 19

issues, 65, 98

J

Japan, 2, 19

job satisfaction, 38, 61, 65, 80, 84, 85

K

Korea, 3, 32

L

labor force, 84

laws, 29

lead, 14, 78

leadership, 38, 39, 49, 66

learners, 59, 60

learning, 36, 37, 38, 47, 49, 59, 60, 61, 64,

67

learning environment, 37, 47, 49, 59, 61, 64

learning outcomes, 38

legislation, 41

lending, 10

life satisfaction, 80, 84

light, 76, 88

Likert scale, 48

linear model, 3

liquid assets, 9, 21

liquidity, 6, 10, 21, 26, 29

liquidity ratio, 10

loans, 4, 5, 9, 21, 28, 29, 30

local government, 6

locus, 60

long-term debt, 5, 7, 9, 19, 20, 21, 24, 26,

28, 29, 30

LTA, 9, 12, 17, 18, 22, 25, 26

M

magnitude, 74

majority, 5, 26

management, 5, 9, 10, 19, 20, 47, 101

manufacturing, 14, 26

marriage, 80, 81, 84

materials, 49

mathematics, 39, 40, 41

matrix, 15, 44, 78, 88

measurement, 14, 15, 16, 20, 46, 48, 52, 74,

85

median, 79, 101

mediation, vii, viii, 69, 70, 71, 72, 73, 74,

75, 76, 77, 78, 79, 80, 81, 83, 86, 87, 88,

89, 90, 91, 95, 97, 98, 99, 100, 101

mediational analyses, 101

Medicaid, 51

medicine, 33

membership, 41, 44, 45, 46, 58, 59

messages, 66

meta-analysis, 60

methodology, 4, 42, 101

misunderstanding, 75

mixing, 45

model specification, 44, 45

models, vii, viii, 2, 43, 44, 45, 46, 51, 52,

53, 59, 63, 64, 65, 66, 67, 69, 70, 71, 72,

73, 74, 75, 77, 80, 84, 85, 87, 88, 89, 90,

91, 92, 93, 94, 95, 99, 100, 101

moderators, 98

monopoly, 5

morale, 40

motivation, 36, 38

multidimensional, vii, 35, 37

multiple fit, 52

multivariate analysis, 96

Index 107

N

NAEP, 40

National Center for Education Statistics, 63

negative affectivity, 98

negative effects, 21, 28

negative mood, 74

negative relation, 11, 39

net profit margin, 9

New Zealand, 31

No Child Left Behind, 36

normal distribution, 15

normative behavior, 37

O

obstacles, 60

operations, 28

opportunities, 8, 29

ownership, 5, 6, 10, 24, 25, 26, 29

ownership structure, 5, 6, 10, 25, 26, 29

P

parallel, viii, 69, 70

parameter estimates, 20, 25, 46, 75, 88, 89

parental involvement, 40, 67

parents, vii, 35, 36, 37, 38, 40, 41, 47, 48,

49, 59, 61

participants, 39

path model, 99

pathways, 73, 88

performance indicator, 58

personality, 37

Philadelphia, 63, 66

physical environment, 49

physical health, 72

pluralism, 63

policy, vii, 6, 11, 24, 35, 36, 61, 62

policy makers, vii, 35, 36, 61

population, 43, 78

positive relationship, 8, 28

poverty, vii, 35, 39, 40, 41, 42, 50, 52, 58,

59, 60, 61, 65, 66

prevention, 41, 64

prior knowledge, 48

private firms, 20, 25, 28, 30

private sector, 4, 5

probability, 11, 43, 44, 45, 46, 54, 56, 59

probability distribution, 43

problem behavior, 39

profit, 9

profit margin, 9

profitability, 6, 9, 21, 26, 29

programming, 63

project, 64

property rights, 29

psychological stress, 72

psychology, 62, 98, 101

psychosocial climate, 68

public schools, 36

Q

qualifications, 36

quality of life, 80

quick ratio, 10

R

reading, 39, 40

reality, 30

recommendations, 101

reform, 5, 10, 40, 61, 62, 98

regression, 4, 14, 15, 20, 25, 40, 50, 79, 101

regression analysis, 4, 14, 25

regression equation, 14

regression model, 25

regulations, 5

reputation, 21

requirements, viii, 10, 41, 51, 69, 87

researchers, 5, 14, 37, 39, 41, 42, 43, 44, 46,

48, 50, 61, 70, 75, 97

residuals, 71

resources, 4

response, 16, 41

restrictions, 44

retained earnings, 9

Index 108

revaluation, 8

revenue, 9, 11

rights, 6, 10, 29

risk(s), 5, 6, 11, 16, 19, 20, 24, 28, 29, 30

root(s), 77, 88

rules, 37, 40

S

safety, 37, 40, 47, 49, 63, 64, 67

sanctions, 19

SAS, 25, 48

school, vii, 35, 36, 37, 38, 39, 40, 41, 42,

47, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59,

60, 61, 62, 63, 64, 65, 66, 67, 68, 74, 98,

100

school achievement, 39, 59, 64, 74

school activities, 49

school climate, vii, 35, 36, 37, 38, 39, 40,

41, 42, 50, 53, 56, 58, 59, 60, 61, 62, 63,

64, 65, 66, 67, 68

school culture, 37, 65

school improvement, 39, 60, 61, 62, 64, 65,

67

school learning, 67

school performance, vii, 35, 41, 59, 60, 61

school success, 36, 38

school work, 65

schooling, 59

science, 41, 65

secondary schools, 36

security(s), 21, 50

self esteem, 86

self-esteem, 38, 85, 86, 87

sensitivity, 88

shareholders, 6, 9, 10

showing, 41, 53, 54, 56, 58

signals, 9, 28

signs, 73

simulation, 66, 100

social costs, 28

social psychology, 98, 101

social sciences, 42

society, 38

socioeconomic status, 37, 65

software, 43, 48, 51, 75, 77, 79, 97

solution, 42, 44, 45, 46, 49, 52, 53, 54, 75

South Korea, 3, 32

Soviet Union, 3

specifications, 45

spending, 10

stakeholders, 24, 61

standard deviation, 11, 50, 75

standard error, viii, 46, 69, 70, 77, 80, 82,

83, 84, 85, 86, 88, 94, 101

standardization, 75, 78

state(s), vii, 2, 5, 10, 19, 20, 24, 25, 26, 28,

29, 30, 35, 36, 40, 41, 47, 51, 54, 57, 58,

59, 60

state control, 10

state-owned banks, 5, 28

statistical inference, 100

statistics, viii, 18, 22, 23, 26, 27, 51, 56, 62,

69, 79

stimulus, 24

stock, 5, 14

stock exchange, 14

stress, 38, 61, 72

structural characteristics, 67

structural equation modeling, vii, 1, 4, 15,

96, 101

structure, vii, 1, 2, 3, 4, 5, 6, 7, 8, 10, 14, 16,

21, 24, 25, 26, 29, 30, 35, 37, 48, 49, 50,

64, 65, 67, 78, 99, 100, 101

structuring, 61

student achievement, 36, 39, 40, 41, 58, 62,

63, 65, 67

substance use, 38, 99

substitutes, 8

substitution(s), 10, 11

suppliers, 11

suppression, 73, 99

surplus, 28

suspensions, 59

Switzerland, 69

T

target, 2, 3, 18

tax policy, 6

Index 109

tax system, 24

teachers, vii, 35, 36, 37, 38, 39, 40, 41, 42,

47, 48, 49, 50, 56, 58, 59, 60, 61, 65

techniques, 16, 40, 43, 79

technology, 6

test anxiety, 62

test scores, 38, 39, 41, 47, 51, 57

testing, vii, viii, 69, 70, 73, 75, 77, 78, 88,

98, 99, 100, 101

textbooks, 49

third dimension, 49

thoughts, 47

trade, 2, 3, 6, 8, 9, 11, 13, 21, 24

trade-off, 2, 3, 8, 9, 11, 13, 21, 24

training, 60

trajectory, 66

transaction costs, 9

transformation, 5

turnover, 41, 59, 61

U

uniform, 7

unique features, 5

United, 2, 3, 31, 32, 36, 63

United Kingdom (UK), 2, 31, 33, 68

United States, 2, 3, 32, 36, 63

urban, 40, 65, 67

urbanicity, 40

USA, 63

V

validation, 62, 63

variables, vii, viii, 13, 14, 15, 16, 25, 35, 36,

37, 38, 41, 42, 43, 44, 46, 50, 53, 54, 56,

57, 59, 60, 62, 69, 70, 71, 72, 74, 75, 76,

77, 78, 79, 80, 84, 85, 87, 88, 90, 91, 93,

96, 97, 98, 99, 100

variance-covariance matrix, 44, 88

vector, 15, 44, 88

Vietnam, 3, 33

violence, 41, 64

volatility, 11, 24

voting, 10

W

waiver, 51, 52, 58

Washington, 63, 98, 101

web, 36, 66

weight loss, 74

workers, 11, 84, 98

working conditions, 84

workplace, 65

Y

yield, 7

structural equation modeling (sem): concepts, applications and...

Documents