introduction to multi-level modeling

Introduction to Multi-Level Modeling

Juliet Aiken

Design and Statistical Analysis Laboratory

November 13, 2009

What to Expect Slides will be posted on DaSAL website

(http://blog.umd.edu/statconsulting/) after the presentation (but not immediately)

Not all questions will be answered right now Ask anyway! We’ll write them down and follow up with them before

posting the slides online Terminology:

Random Coefficient Modeling: A statistical technique Hierarchical Linear Modeling: A piece of software which can

be used to do random coefficient modeling It’s okay if some of this is information overload! I do not care if you get up to get more food in the

middle of this

Understand basics of random coefficient multi-level modeling (RCM) When to use and when not to use RCM What is a random coefficient? Slopes vs. intercepts as outcomes

Provide a platform for further reading and investigation into proper use of RCM Materials and references for later use

Overview

What is multi-level modeling? Multi-level modeling examples What is multi-level modeling?, revisited Why do we care? Multi-level statistical models

Justification for use Two types of models

Overview MLM: Random Coefficient Modeling

Distinguishing Random Coefficients from Random Effects

Kinds of RC Modeling that can be conducted HLM Show and Tell with HLM

Overview of Example 20 teams, 100 people (5 per team), 4

observations of performance per person Self efficacy is measured at each time point

per person Measures of team empowerment from each

individual with respect to their team Does self-efficacy predict performance?

Within individuals? Across individuals?

Does team empowerment predict performance?

What Is Multi-Level Modeling? Family of statistical models for data analysis Theory, design, and measurement

Multi-Level Theoretical Models:Modeling with Correlated Errors (V-C)

OrganizationalSupport

Work-FamilyConflict

OrganizationalSupport

Work-FamilyConflict

Partner 1

Partner 2

Multi-Level Theoretical Models:N-Level Modeling (V-C)

Self-EfficacyIndividual

Performance

Team Empowerment

GroupPerformance

Level 2: Group-Level

Level 1: Individual-Level

Multi-Level Theoretical Models:Multi-Level Modeling (V-C)

Performance

TeamEmpowerment

GroupPerformance

Multi-Level Theoretical Models:Cross-Level Modeling (RCM)

Performance

TeamEmpowerment

Multi-Level Theoretical Models:Frog-in-Pond Modeling (RCM)

Self-EfficacyRelative to

Group Mean

Individual Performance

TeamEmpowerment

Multi-Level Modeling: Levels

Organization (School)—Group (Class) Group (Class)—Individual Organization (School)—Individual Individual—Time

Two levels, three levels, four levels, more!

What do they have in common? Hierarchical Non-randomly Nested

Implies sampling method that begins at the highest level

Random nesting (e.g. experimental conditions) does not violate independence of observations assumptions

Why do we care?

Nesting violates independence of observations Results in heteroscedasticity Violates assumptions of “regular” regression Results in incorrectly estimated standard errors, and

consequently, “wrong” results Thus multi-level modeling more powerful and more

honest

Multi-Level Statistical Models Must justify use of multi-level modeling

techniques Group-level variables (e.g. same supervisor,

same organization) Justification for aggregation

Theoretical Design and Measurement (referent) Statistical (ICC1, ICC2, rwg, AD, others)

Multi-Level Statistical Models Regression-Based

Random Coefficient Modeling Variance-Covariance-Based

Latent Growth Analysis (in SEM)

RCM vs. VC

RCM Path analysis harder Measured variables Easy to include

interactions Hard to include

correlated errors Copes with missing data Easy to add levels

VC Factor analysis Path analysis easy Latent variables Interactions harder Different errors easy to

add Harder to cope with

missing data Harder to add levels Goodness of fit

information Model suggestions to

improve fit

Random Coefficient Modeling Random Effects

Experimental conditions (e.g. for medicine) Fixed: Can infer about treatments used in the experiment Random: For purposes of generalization

Random Variables Fixed: Variable with values that are known (e.g. gender) Random: Variable with values selected from a probability

distribution and are measured with error (e.g. IQ)

Random Coefficient Modeling Random Coefficients

Fixed: Coefficients (e.g. slopes or intercepts) do not vary across people/teams/etc.

Random: Coefficients in which values estimated are assumed to be distributed as a probability function

Random coefficients do NOT correspond to random effects or variables Equations versus design/experimental

manipulations

Random Coefficient Modeling Extension of the general/generalized linear

model Outcome of interest measured at the lowest

level “Intercepts as outcomes” “Slopes as outcomes”

Random Coefficient Modeling: Intercepts as OutcomesH1: Group team empowerment impacts average individual performance

Performance

TeamEmpowerment

Individual Self-Efficacy

Team 1 has high team empowerment, and thus, higher average individual performance than Team 5

However, Team 1 and Team 5 exhibit the SAME relationship between individual self-efficacy and performance

Random Coefficient Modeling: Slopes as OutcomesH1: Group team empowerment moderates the relationship between self-

efficacy and individual performance

Performance

TeamEmpowerment

Random Coefficient Modeling: Intercepts as OutcomesH1: Group team empowerment moderates the relationship between self-efficacy

and individual performance

Team 1 has high team empowerment, BUT still the same average individual performance as Team 5

However, for the high team empowerment team, Team 1, self-efficacy is positively related to performance, whereas for the low- team empowerment, Team 5, self-efficacy is negatively related to performance

Random Coefficient Modeling: Slopes and Intercepts as OutcomesH1: Group team empowerment moderates the relationship between self-

efficacy and individual performance AND impacts average individual performance directly

Performance

TeamEmpowerment

Random Coefficient Modeling: Slopes and Intercepts as OutcomesH1: Group team empowerment moderates the relationship between self-efficacy

and individual performance AND impacts average individual performance directly

Team 1 has high team empowerment, and, consequently, a higher average individual performance than Team 5

Additionally, for the high team empowerment team, Team 1, self-efficacy is positively related to performance, whereas for the low- team empowerment team, Team 5, self-efficacy is negatively related to performance

Random Coefficient Modeling: Word of Warning Large models can be unstable

Small changes in the model may result in large changes in the result of the analysis

Might be due to multicollinearity in cross-level interactions/high correlations in parameter estimates

Mainly a problem when few observations on the highest level

Unbalanced samples may have too-small estimated standard errors Makes hypothesis tests too liberal

Except for fixed coefficients, your df is tied to the number of observations at the highest level of predictor variables

Random Coefficient Modeling: Software Available

Intercept? Equations Estimation Missing Data

Cross-Classification

HLM Yes One for each level

RML is default (ML)

Level 1 only (Pair-wise)

MLwiN X1 = 1 One for each level

MCMC is default (ML/QML)

Imputes with each iteration

Yes—easier (no level spec. requirement)

SAS Proc

Optional, default is yes

One RML Pair-wise Yes

SPSS Mixed

Optional, default is no

Point-and-click or One

RML (ML) Pair-wise Yes

Random Coefficient Modeling: Software Available

# Datafiles # Levels Random Effects?

Centering GUI?

HLM Multiple Up to 3 Level 2 and above assumed fixed

Group and Grand

MLwiN 1 Unlimited (Default = 5)

Yes, can specify

Done Manually

SAS 1 Unlimited Yes, can specify

Done Manually

SPSS Mixed

1 Up to 3 Yes, can specify

Done Manually

Random Coefficient Modeling: HLM Assumptions

Observations at highest level are independent Linear models Level 1—normal random errors Level 2—multivariate normal random errors Level 1(2) predictors are independent of Level 1(2)

residuals Variance of residual errors is the same at all levels Variances of residual errors is the same across units at

Level 1 Independent errors across and within levels

Random Coefficient Modeling: HLM Options

RCM Multivariate RCM 2- or 3- levels Cross-classified models (2 levels only)

HLM: Options

Random Coefficient Modeling: HLM Data preparation

“Down” format Sort in ascending order

Names truncate to 8 letters ID variables on all levels No missing data on higher levels

Random Coefficient Modeling: HLM Preparation

Random Coefficient Modeling: Preparation

Random Coefficient Modeling: HLM Preparation

Random Coefficient Modeling: HLM Everyone with HLM on their machine, please

open it now.

Random Coefficient Modeling: HLM

Random Coefficient Modeling: HLM Centering

None Grand-mean:

Just re-centering Makes intercept more meaningful Helps reduce multicollinearity

Group-mean: For frog-in-pond effect studies Completely different model

Random Coefficient Modeling: HLM Model Specification

Different equations for each level and coefficient Level 1: Yij = β0j + βjXij + rij

Level 2: β0j = γ00 + u0j

β1j = γ10 + uij

Other programs would write the same model in one statement: Yij = γ00 + γ10Xij + γ11Xij + u0j + u1jXij + rij

Random Coefficient Modeling: HLM Robust Standard Errors

If response variable does not have normal distribution

AND N >= 100

Random Coefficient Modeling: Analysis

Random Coefficient Modeling: Major Points Multi-level modeling always comes back to theory and

measurement For some variables, need to justify aggregation Hierarchical designs should employ hierarchical

statistics for maximum power and more accurate analyses

Random coefficients are NOT random effects Your df is only as high as the number of units at the

highest level of your predictors HLM software is not the only (or even the best!)

available

ReferencesBooks

Bryk, A. S., & Raudenbush, S. W. (1992). Hierarchical linear models: Applications and data analysis methods. Newbury Park, CA: Sage.

Hox, J. (2002). Multilevel analyses: Techniques and applications. Mahwah, NJ: Erlbaum.

Kreft, I. & de Leeuw, J. (1998). Introducing Multilevel Modeling. London: Sage.

ReferencesHow-To and General Reference Articles

Bliese, P.D. & Ployhart, R. E. (2002). Growth modeling using random coefficient models: Model building, testing, and illustrations. Organizational Research Methods, 5, 362-387.

Bliese, P. D., & Hanges, P. J. (2004). Too liberal and too conservative: The perils of treating grouped data as though they were independent. Organizational Research Methods, 7, 400-417.

Hofmann, D. A. (1997). An overview of the logic and rationale of hierarchical linear models. Journal of Management, 23, 723-744.

Kreft, I. G. G., de Leeuw, J., & Aiken, L. S. (1995). The effect of different forms of centering in hierarchical linear models. Multivariate Behavioral Research, 30, 1-21.

ReferencesHow-To and General Reference Articles

Kreft, I. G. G., de Leeuw, J., & van der Leeden, R. (1994). Review of five multilevel analysis programs: BMDP-5V, GENMOD, HLM, ML3, and VARCL. The American Statistician, 48, 324-335.

Singer, J. D. (1998). Using SAS PROC MIXED to fit multi-level models, hierarchical models, and individual growth models. Journal of Educational and Behavioral Statistics, 23, 323-355.

Zhou, X., Perkins, A. J., & Hui, S. L. (1999). Comparisons of software packages for generalized linear multilevel models. The American Statistician, 53, 282-290.

ReferencesTheoretical Issues

Chan, D. (1998). Functional relations among constructs in the same content domain at different levels of analysis: A typology of composition models. Journal of Applied Psychology, 83, 234-246.

Klein, K. J. & Kozlowski, S. W. (2000). From micro to meso: Critical steps in conceptualizing and conducting multilevel research. Organizational Research Methods, 3, 211-236.

Morgeson, F. P. & Hofmann, D. A. (1999). The structure and function of collective constructs: Implications for multilevel research and theory development. Academy of Management Review, 24, 249-265.

Ostroff, C. (1993). Comparing correlations based on individual-level and aggregated data. Journal of Applied Psychology, 78, 569-582.

ReferencesEmpirical Examples

Atwater, L., Wang, M., Smither, J. W., & Fleenor, J. W. (2009). Are cultural characteristics associated with the relationship between self and others’ ratings of leadership? Journal of Applied Psychology, 94, 876-886. (Uses MPLUS)

Chen, G., Kirkman, B. L., Kanfer, R., Allen, D., & Rosen, B. (2007). A multilevel study of leadership, empowerment, and performance in teams. Journal of Applied Psychology, 92, 331-346. (Uses S-PLUS/R)

Klein, K. J., Lim, B., Saltz, J. L., & Mayer, D. M. (2004). How do they get there? An examination of the antecedents of centrality in team networks. Academy of Management Journal, 47, 952-963. (Uses SAS)

Liao, H., & Chuang, A. (2004). A multilevel investigation of factors influencing employee service performance and customer outcomes. Academy of Management Journal, 47, 41-58. (Uses HLM)

Thank You!

Paul Hanges, Mo Wang, Kevin O’Grady, and DaSAL consultants (Marsha Sargeant, Laura Sherman, Brandi Stupica, and Tracy Tomlinson) for comments and suggestions

Songqi Liu for theoretical & empirical references

Everyone for coming out today!

introduction to multi-level modeling

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