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Structural equation modeling

Rex B Kline Concordia University

Montréal

ISTQL Set A Concepts, models, tools

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Resources

o Kline, R, B. (2012). Assumptions of structural equation

modeling. In R. Hoyle (Ed.), Handbook of structural

equation modeling (pp. 111–125). New York: Guilford. o Kline, R. B. (2013). Exploratory and confirmatory factor

analysis. In Y. Petscher & C. Schatschneider (Eds.), Applied quantitative analysis in the social sciences (pp. 171–207). New York: Routledge.

o Kline, R. B. (2013). Reverse arrow dynamics: Feedback loops and formative measurement. In G. R. Hancock and R. O. Mueller, (Eds.), Structural equation

modeling: A second course (2nd ed.) (pp. 39–76). Greenwich, CT: IAP.

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Topics

o Mon: Concepts, models, tools

o Tues: Data, path models

o Weds: Estimation

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Topics

o Thurs: Model testing

o Fri: CFA models

o Sat: SR models

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Practice

o Weds: LISREL SIMPLIS syntax

o Thurs: LISREL Path Diagram

o Fri: Ωnyx demo

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four-variable example

observed variables

neg_str cur_prob prob_sol depress

covariance matrix

76.913

55.668 249.324

33.757 115.685 478.297

21.775 53.614 60.695 37.700

sample size is 205

relationships

cur_prob = neg_str

prob_sol = neg_str cur_prob

depress = neg_str cur_prob prob_sol

LISREL output: ND = 3 SC RS MI

path diagram

end of problem

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exercise, hardy, fitness, stress, illness

4422.250

-75.810 1444.000

477.204 48.944 338.560

-111.388 -292.790 -80.132 1122.250

-332.394 -379.878 -333.393 711.647 3903.750

Exercise

Hardiness

Illness

DIl

1

Stress

1 DSt

Fitness

1 DFi

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Worst practices

o MacCallum, R. C., & Austin, J. T. (2000).

Applications of structural equation modeling in psychological research. Annual Review of

Psychology, 51, 201–226.

o Shah, R., & Goldstein, S. M. (2006). Use of structural equation modeling in operations management research: Looking back and forward. Journal of Operations Management, 24, 148–169.

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Best practices

1. Justify specifications

2. Report psychometrics

3. Verify assumptions

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Best practices

4. Report summary statistics

5. Describe method

6. Model-table-text agree?

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Best practices

7. Report unstandardized

8. Describe residuals

9. Equivalent models

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Best practices

10. Rethink :

Limited role

Not a criterion

Path to folly

*

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Keep your eyes on the prize

1

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Any monkey can find a

model that fits the data

2

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Closer to fit is not closer to

truth

3

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Directionality is assumed, not

tested

4

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Never forget equivalent

models

5

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Smart modeling

o Know your area

o Simple is good

o Build, not trim

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Smart modeling

o Advantages of building:

1. Avoid identification issues

2. Prioritize hypotheses

3. Back-up list

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Smart modeling

o Add from list

o Done:

1. List exhausted

2. No citations for rest

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2. Identification

3. Data collection

4. Analysis

5. Respecification

6. Reporting

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no

yes

4a. Model fit adequate?

6. Report results

4c. Consider equivalent or near-equivalent models

4b. Interpret estimates

yes

Justifiable respecification?

no

5. Respecify model

no yes 2. Model identified?

3. Select measures, collect data

1. Specify model

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Smart modeling

o Goals:

Parsimony

Respect theory

Avoid HARKing

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Hello, SEM

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SEM

o Family

o Integration of MR, FA

o Flexible, extensible

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SEM o Inherits from MR ( ):

Multiple predictors

B, R2 effect sizes

Means, too

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SEM

o Inherits from FA ( ):

Observed vs. latent

Latents as predictors

Measurement error

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SEM

o Inherits from both ( ):

Capitalizes on chance

Specification error

Misuse through

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SEM

o Synergistic ( ):

Latents as outcomes

Means of latents

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SEM

o Synergistic ( ):

Indirect effects (mediation)

Error covariance structure

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Models

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Core models

o PA

o CFA

o SR

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Path models

o Structural model

o Observed variables only

o Single indicators only

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Path models

o Structural model:

Causal effects

Noncausal associations

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Path models

o Causal effects:

Direct

Indirect

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Path models

o Indirect effects:

Part of mediation

Indirect ⇒/ mediation

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Path models

o Noncausal effects:

Common cause (spurious)

Correlated with cause

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CFA models

o Multiple indicators only

o L → M only

o Factors covary only

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CFA models

o Classical measurement theory

o Convergent validity

o Discriminant validity

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SR models o Single, multiple indicators

o L → M or M → L

o L as predictors, outcomes

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SR models o Highest level model

o Advanced = SR variation

o Know and love

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Examples

o Roth, D. L., Wiebe, D. J., Fillingim, R. B., & Shay, K. A. (1989).

Life events, fitness, hardiness, and health: A simultaneous analysis of proposed stress-resistance effects. Journal of Personality and Social Psychology,

57, 136–142. o Kaufman, A. S., & Kaufman, N. L. (1983). K-ABC

administration and scoring manual. Circle Pines, MN: American Guidance Service.

o Shen, B.-J., & Takeuchi, D. T. (2001). A structural model of acculturation and mental health status among Chinese Americans. American Journal of Community

Psychology, 29, 387–418.

CFA

SR

PA

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Exercise

Hardiness

Fitness

1

DFi

Illness

1

DIl

Stress

1 DSt

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1

1 1 1 1 1 1 1 1

1

Sequential

EHM

Hand Movements

Number Recall

ENR

Word Order

EWO

Simultaneous

ETr

Triangles Spatial

Memory

ESM

Matrix Analogies

EMA

Gestalt Closure

EGC

Photo Series

EPS

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Acculturation

EGS

1

General Status

1

Acculturation

Scale

EAS

1

Percent Life U.S.

EPL

1

1

Job

EJo

1

Interpersonal

EInt

1

Stress

DSt

1

Depression Scale

DDS

1

SES

1

Education

EEd

1

Income

EInc

1

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Tools

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Tools

o Commercial vs. free

o Stand-alone vs. environment

o User interface

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Interaction modes

Computer tool Free Environment

needed Batch

(syntax) Wizard

(template) Drawing

editor

Stand-alone programs

Amos

EQS

LISREL

Mplus

Ωnyx

Packages, procedures, or commands in larger environments

sem, lavaan, lava,

systemfit R

OpenMx R

CALIS SAS/STAT

Builder, sem, gsem Stata

SEPATH STATISTICA

RAMONA SYSTAT

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QQQ

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GUI liabilities

o Complex models

o Multiple-samples analysis

o Multi-level analyses

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GUI liabilities

o Syntax may be easier, faster

o Diagram as archive

o Publication quality graphic

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LISREL

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Amos

CurrentProblems

NegativeLife Stress

ProblemSolving

Depression

D_PSD_De

1

1

D_CP

1

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EQS

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Mplus

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Stata

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Exercise

Hardiness

Illness

DIl

1

Stress

1 DSt

Fitness

1 DFi

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Tool support

o Amos:

Blunch, N. (2013). Introduction to structural equation

modeling using IBM SPSS Statistics and Amos (2nd ed.). Thousand Oaks, CA: Sage.

Byrne, B. M. (2010). Structural equation modeling

with Amos: Basic concepts, applications, and

programming (2nd ed.). New York: Routledge.

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Tool support

o EQS:

Byrne, B. M. (2006). Structural equation

modeling with EQS: Basic concepts,

applications, and programming (2nd ed.). New York: Routledge.

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Tool support

o lavaan:

Beaujean, A. A. (2014). Latent variable

modeling using R: A step-by-step guide. New York: Routledge.

http://lavaan.ugent.be/

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Tool support

o LISREL:

Vieira, A. L. (2011). Interactive LISREL in

practice: Getting started with a SIMPLIS

approach. New York: Springer.

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Tool support

o Mplus:

Geiser, C. (2013). Data analysis with Mplus. New York: Guilford.

Wang, J., & Wang, X. (2012). Structural equation

modeling: Applications using Mplus. Chichester, UK: Wiley.

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Tool support

o Stata:

Acock, A. C. (2013). Discovering structural

equation modeling using Stata 13. College Station, TX: Stata Press.

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Statistics

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Continuous

o Theoretically infinite scores

o In practice, range > 15

o Symmetrical distribution

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Likert scale

o Items

o E.g., 1 = disagree, 2 = not sure 3 = agree

o Ordinal, ordered-categorical

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Scales vs. items

o Scale: Σ score, continuous

o Item: Noncontinuous

o Proper method

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Other outcomes

o Dichotomous (binary), k = 2

o Nominal, k > 2

o Logit or probit link function

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Other outcomes

o Agresti, A. (2007). An

introduction to categorical

data analysis. Hoboken, NJ: Wiley.

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Other outcomes

o Count variables

o Poisson distribution

o Mean ≈ variance

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Covariance

o Continuous only

o Linear only

o covXY = rXY SDX SDY

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rXY = .55, SDX = 3.5, SDY = 2.0

covXY = .55 (3.5) (2.0) = 8.1

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Raw data

Case X W Y

A 3 65 24

B 8 50 20

C 10 40 22

D 15 70 32

E 19 75 27

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Covariance matrix

38.500

42.500 212.500

17.500 51.250 22.000

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Correlation matrix + SDs

1.000

.470 1.000

.601 .750 1 .000

6.205 14.577 4.690

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Raw data not needed

comment spss, y on x, w, matrix input.

matrix data variables=x w y/contents=mean sd n corr

/format=lower nodiagonal.

begin data

11.000 60.000 25.000

6.205 14.577 4.690

5 5 5

.470

.601 .750

end data.

regression matrix=in(*)/variables=x w y/dependent=y

/enter.

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MR

o Y, X1, X2

o 1 1 2 2Y B X B X A= + +

o ˆYYR r=

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Coefficients

B1 = 2.30, B2 = 6.35, A = 10.50

b1 = .65, b2 = .30

Capitalization on chance

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Comparisons

Predictors Samples B b

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MR assumptions

o Y is continuous

o Linear only

o No interactions

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MR assumptions

o r11 = r22 =1.00

o No indirect effects

o No specification error

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No specification error

No irrelevant predictors

No omitted predictors that covary with measured predictors

Correct functional form

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Heartbreak of L.O.V.E

Y = suicide attempts

X1 Therapy

X2 Depression

rY1 = .19 rY2 = .49 r12 = .70

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Heartbreak of L.O.V.E

Y = suicide attempts

X1 Therapy

X2 Depression

rY1 = .19 rY2 = .49 r12 = .70

b1 = −.30 b2 = .70 R = .54

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Results depend on

What is measured (data)

What is not (omitted variables)

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Predictor entry

Rational (e.g., HMR)

Statistical (e.g., stepwise)

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Stepwise flaws

Results are wrong

Will not replicate

Banned

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SEM version

MIs only

Same problems

Think for yourself