workshop on meta-analytic structural equation …meta-analytic structural equation modeling (masem)...

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Workshop onMeta-analytic Structural Equation Modeling

Mike W.-L. Cheung, PhDDepartment of Psychology

National University of Singaporehttp://courses.nus.edu.sg/course/psycwlm/internet/masem.zip

28 August 200928 August 2009

Acknowledgment: I thank the Centre for Instructional Technology (CIT) at NUS for providing the technical support.

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Copyright (c) Mike W.-L. Cheung 2009 2

Outline of the workshop• What is SEM?

– A general technique to model multivariate data• What is MASEM?

– A technique combining meta-analysis and SEM to synthesize research findings

• What approaches are available for MASEM?– Univariate (r and z) vs. multivariate (GLS and

TSSEM) approaches• What is TSSEM?

– A “pure” SEM approach to conduct MASEM• What is SEM-based meta-analysis?

– A SEM approach for general meta-analysis

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Structural Equation Modeling (SEM)

• Some of its special cases: Regression analysis, path analysis, factor analysis and latent growth models, etc.

• Most models can be represented by path diagrams.

Multiple regression Path analysis Confirmatory factor analysis Structural equation model

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Why SEM?• It can be used to test whether or not a proposed model

(theory) is consistent with the data.• Many advanced techniques have been integrated into

SEM, e.g.,– analysis of categorical data;– mixture modeling;– multilevel modeling;– analysis of missing data with maximum likelihood;– meta-analysis, etc.

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Steps in conducting SEM• Inputs: Raw data, covariance or correlation

matrices• Procedures: Model fitting and model modification• Outputs:

– Overall goodness-of-fit indices: 2, GFI, CFI, NNFI, RMSEA, etc.

– Parameter estimates and their standard errors (SEs)

5.33.3 6 . 43.0 2 . 8 4 . 54 .1 3 . 2 2. 6 5. 9 D1

.75**

.59**

-.31**-.24**

-.15**

Results: 2(21)=32.3, p=.05, CFI=.98, RMSEA=.034

-.24**

P1

D2

P2

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Problems in individual studies• Low statistical power of most empirical studies in

psychology;• Confirmation bias: reluctant to consider alternative

models in SEM; and• Different researchers may propose different models

supported by their own data

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Need to synthesize research findings• A major problem in scientific research: how to

compare and combine research findings?• Conducting more empirical studies does not

necessarily decrease the uncertainty of a particular topic if these findings are inconsistent;

• There is a need to integrate the research findings for the advance of theory

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Three objectives in meta-analysis

• To test if the effect sizes are consistent across studies;

• To estimate a pooled effect size;• To identify potential moderators.

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Meta-analytic structural equation modeling (MASEM)

• MASEM combines techniques of meta-analysis and SEM (e.g., Hunter & Schmidt, 2004; Viswesvaran & Ones, 1995).

• It can be used to synthesize studies in SEM.

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Advantages of MASEM

• To address the generalizability of the findings across settings;

• To identify potential moderators that influence the structure of the model;

• To obtain more precise estimates by increasing the sample size.

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An example using MASEM

• Brown and Stayman (1992): Antecedents and consequences of attitudes toward the advertisement (Ad)– No. of variables: 5– No. of studies: 47– Pooled sample size across studies: +4,600

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Ad cognitions Ad attitude

Brand cognitions Brand attitude Purchase intention

Dual mediation hypothesis



Affect transfer hypothesis



Reciprocal mediation hypothesis



Independent influences hypothesis

.52

.57.34

.20 .73

Four alternative models of Ad attitude

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Another example in cross-cultural research

• Cheung, Leung, and Au (2006) are interested in the whether the factor structure of social axioms is applicable across different cultural groups.

• 7,590 university students from 40 cultural groups were collected (Leung & Bond, 2004).

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• Items:– Social cynicism (CYN): 11 items – Social complexity (COM): 6 items– Reward for application (REW): 9 items – Religiosity (REL): 7 items– Fate control (FAT): 6 items

cyn1

cyn11

CYN

com1

com6

COM

rew1

rew9

REW

rel1

rel7

REL

fat1

fat6

FAT

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Results for MASEM

– First stage: Homogeneity of correlation matrices• 2(28,899, N=7,590) = 41,132, p < .0001• RMSEA = 0.047

– Second stage: Fitting a five-factor model• 2(692, N=7,590) = 6,653, p < .0001• RMSEA = 0.034, SRMR = 0.046

– The results support that the five-factor model fits well across 40 cultural groups.

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Steps of conducting MASEM

• Two-stage approach:– First stage: Similar to conventional meta-analysis

• To test the homogeneity of correlation matrices;• To estimate the pooled correlation matrix

– Second stage: Similar to conventional SEM• To fit and compare models based on the pooled

correlation matrix

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An artificial example

1 . 0. 60 1. 0. 43 .44 1 . 0. 36 .22 . 19 1. 0

1 . 0-- --. 32 -- 1.0.31 -- .14 1 .0 1 . 0

. 49 1 .0

. 35 . 37 1. 0

. 40 . 27 .15 1 . 0Stage 1: Are they homogeneous?

1 . 0. 58 1 .0. 37 . 40 1. 0. 38 . 24 .18 1 . 0

Stage 2: Which model fits the data best?

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Approaches for MASEM

• Stage 1: Univariate approaches– Based on individual correlation coefficients;– Repeat it for all correlation coefficients;– Hunter and Schmidt (2004): Averaging

correlation coefficients (univariate r)– Hedges and Olkin (1985): Averaging Fisher’s z

scores (univariate z)

1 . 0. 60 1. 0. 43 . 44 1 . 0. 36 . 22 . 19 1. 0

1 . 0-- --. 32 -- 1. 0.31 -- .14 1 . 01 . 0

. 49 1 .0

. 35 . 37 1 . 0

. 40 . 27 . 15 1 . 0 1 . 0

. 58 1 . 0

. 37 . 40 1 . 0

. 38 . 24 . 18 1 . 0

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Stage 1: Univariate r approach

• Estimate a pooled correlation with weighted correlations:

rij=∑k=1

g

nk rijk

∑k=1

g

nk

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Stage 1: Univariate z approach

• Estimate a pooled correlation with weighted Fisher-z scores:

zijk =0.5 log

1rijk

1−rijk

zij=∑k=1

g

nk −3 z ijk

∑k=1

g

nk −3

rij=exp 2∗zij −1exp 2∗zij 1

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Advantages and limitations of the univariate approaches

• Advantage:– Simple to learn.

• Limitations:– Do not acknowledge the dependence of the

correlation coefficients;– Empirical performance in simulation studies is

not very good (see Cheung & Chan, 2005a)

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• Correlation matrix is used in fitting SEM (Cudeck, 1989)– Chi-square statistics and/or standard errors may be incorrect for

some models.• The pooled correlation matrix may be non-positive

definite.• Different results for different sample sizes (N):

– Arithmetic mean; – Median;– Harmonic mean; – Total.

Advantages and limitations of the univariate approaches

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• Stage 1: Multivariate approaches – Generalized least squares (GLS; e.g., Becker,

1992; Cheung, S.-F., 2000; Furlow & Beretvas, 2005; Hafdahl, 2001)

– Two-stage structural equation modeling (TSSEM; Cheung, 2002; Cheung & Chan, 2005a; 2009)

– Based on all correlation coefficients and their dependence

1 . 0. 60 1. 0. 43 . 44 1 . 0. 36 . 22 . 19 1. 0

1 . 0-- --. 32 -- 1. 0.31 -- .14 1 . 01 . 0

. 49 1 .0

. 35 . 37 1 . 0

. 40 . 27 . 15 1 . 0 1 . 0

. 58 1 . 0

. 37 . 40 1 . 0

. 38 . 24 . 18 1 . 0

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Stage 1: GLS approach• Model:

•• where K is the design matrix and var(e) is the

sampling covariance matrix of r (Olkin & Siotani, 1976)

• The studies are “stacked” together.• Parameter estimates:

r=K e

R=1. 0. 46 1. 0.31 .55 1.0 r=.46

.31

.55 =var e=0.00610.0008 0.00800.0020 0.0037 0.0091

=K '−1K −1K '−1 r

K=1. 0 0 00 1 .0 00 0 1. 0

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Advantages and limitations of the multivariate approaches

• Advantage:– Theoretically appealing as they consider the

dependence among the correlation coefficients.• Limitations:

– GLS: empirical performance is not very good in simulation studies (see Cheung & Chan, 2005a)

– TSSEM: More difficult to learn as a full SEM approach is used.

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What is TSSEM?

• Two-stage structural equation modeling (TSSEM) uses a “pure” SEM approach to conduct MASEM.

• It has good statistical properties when compared to other conventional approaches.

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Analysis of correlation matrix in SEM• Correlation coefficients can be analyzed as a

confirmatory factor analytic model.• Between- and within- comparisons on correlation

coefficients can be made (Cheung & Chan, 2004).

1.00

F1

x 1

SD1

0.00

F2

x 2

1.00

SD2

0.00

F3

x 3

1.00

SD3

0.00

f12 f23

f13

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Stage 1 of TSSEM

• Multiple-group SEM;• Advantage: Missing data are handled effectively

with maximum likelihood method (Muthén, Kaplan, & Hollis, 1987).

1.0.19.22.361.0.44.43

1.0.601.0

1.0.14.311.0.32

1.0

1.0.15.27.401.0.37.35

1.0.491.0

1.0.18.24.381.0.40.37

1.0.581.0

x1

1.00

f1

0.00

x2

f2

0.00

1.00

x3

f3

0.00

1.00

x4

f4

0.00

1.00

x1

1.00

f1

0.00

x2

f2

0.00

1.00

x3

f3

0.00

1.00

x4

f4

0.00

1.00

x1

f1

x3

f3

x4

f4

1.00

0.00 0.00

1.00

0.00

1.00

x1

1.00

f1

0.00

x2

f2

0.00

1.00

x3

f3

0.00

1.00

x4

f4

0.00

1.00

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• After the stage 1 analysis, we have a pooled correlation matrix and its asymptotic covariance matrix. For example,– A 4x4 pooled correlation matrix– A 6x6 asymptotic covariance matrix

• The asymptotic covariance matrix indicates the precision of the pooled correlation matrix.

P=1. 0. 58 1 . 0. 37 . 40 1 . 0. 38 . 24 .18 1 .0

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Stage 2 of TSSEM

• SEM with weighted least squares (WLS) method, also known as asymptotically distribution free (ADF) method, as the estimation method;

• The asymptotic covariance matrix is used as the inverse of the weight matrix;

• The total sample sizes is used as the sample size in fitting SEM.

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Data and programs for TSSEM• Data requirement for MASEM:

– Raw data;– Correlation matrices (Cheung & Chan, 2005a);– Covariance matrices (Cheung & Chan, 2009)

• LISREL student version: http://www.ssicentral.com/lisrel/student.html

• A TSSEM program:– A program to generate LISREL code and to do the

necessary data manipulations– http://courses.nus.edu.sg/course/psycwlm/internet/tssem.zip

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A hand-on exercise• A cross-cultural data set on work-related attitudes

(Inter-University Consortium for Political and Social Research, 1989).

• Individuals aged 18 years and older were sampled from 11 countries.

• The minimum and maximum sample sizes per country were 319 and 1,047, respectively.

• For demonstration, only 4 cultural groups were selected.

• Missing data were also introduced.

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A proposed model for testing

X1: Job security

X3: Advancement

X8: Flexible working hours

X2: Income

X7: Useful to the society

X4: Interesting job

X9: Lots of leisure time

X5: Independent work

X6: Help other people

Job Prospect

Job Nature

Time Demand

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Steps in TSSEM• Prepare a data file for correlation/covariance matrices,

– For example, cor1.dat• Edit the sample cor.cfg file

– Note. If you use “abc.cfg” as the filename, the outputs will be “abc1.ls8” and “abc2.ls8”

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• Data file: cor.dat• Missing variances: 1• Missing correlation: 0

Study Missing variables

Sample size

1 (Assumed no missing data)

591

2 X1, X2 656

3 X3, X4, X5 832

4 X8, X9 823

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Steps in TSSEM• Stage 1 analysis:

– Run: tssem.exe -1 cor.cfg• Output: cor1.ls8

– Run: lisrel88s.exe cor1.ls8 cor1.out• Output: cor1.out, cor1.cor and cor1.ack

• Stage 2 analysis:– Run: tssem.exe -2 cor.cfg

• Output: cor2.ls8, cor2.cor and cor2.ack– Run: lisrel88s.exe cor2.ls8 cor2.out

• Output: cor2.out

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• Let's try our first MASEM now ...

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• Stage 1 results: cor1.out •• The chi-square statistic and goodness-of-fit indices

indicate whether the correlation matrices are homogenous.

• The correlation matrices are reasonably homogeneous.

2 57=172, p.001, RMSEA=0.054,CFI=0.97, NNFI=0.92

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• Stage 2 results: cor2.out•• The proposed model fits the pooled data quite well.

2 24=379, p.001, RMSEA=0.078, CFI=1.00, NNFI=1.01

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X1: Job security

X3: Advancement

X8 Flexible working hours

X2: Income

X7: Useful to the society

X4: Interesting job

X9: Lots of leisure time

X5: Independent work

X6: Help other people

Job Prospect

Job Nature

Time Demand

0.52

0.57

0.59

0.71

0.58

0.75

0.70

0.61

0.33

.54

.39

.48

The fitted model:

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Common issues (FAQs) on MASEM• Should I use correlation or covariance matrices in

MASEM?– Cudeck (1989) criticized the use of correlation matrices

in SEM.– TSSEM approach analyzes correlation matrices

correctly.– Correlation matrices are usually preferred in MASEM.– Covariance matrices may also be used if the measures

are the same across studies (see Cheung & Chan, 2009; Cheung, Leung, & Au, 2006).

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Common issues (FAQs) on MASEM• Should I continue to pool the correlation matrices

if the goodness-of-fit indices at Stage 1 are very poor?– It may not be a good idea to pool the studies together if

they do not share a common population correlation matrix.

– One option is to group the studies according to some study characteristics (moderators).

– Another option is to group the studies empirically with cluster analytic techniques (Cheung & Chan, 2005b).

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Common issues (FAQs) on MASEM• Can I use a random-effects model for MASEM?

– Yes (see Becker, 1992).– However, the empirical performance has yet to be

evaluated.

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SEM-based meta-analysis• TSSEM integrates meta-analysis into SEM

because correlation/covariance matrices may also be analyzed in SEM.

• SEM-based meta-analysis can be used to conduct “conventional” meta-analysis (Cheung, 2008, 2009, in press)– Any type of effect size, e.g., standardized mean

difference, odds ratio, risk ratio, etc.– Univariate and multivariate meta-analysis;– Fixed-, random-, and mixed-effects meta-

analysis.– It can be done in Mplus.

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Model for univariate meta-analysis• Model: • Fixed- and random-effects estimates:

yi=iei var ei=i2

Fix=∑i=1

k

wi yi

∑i=1

k

wi

Ran=∑i=1

k

wi yi

∑i=1

k

wi

wi=1

2 i2wi=

1 i

2

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Formulating a meta-analysis as a SEM• Meta-analytic data cannot be directly analyzed in

SEM because the effect sizes are distributed with a known variance.

• We can transform the effect sizes such that their variances are the same with a value of 1.0

wi0.5 yi=i∗wi

0.5wi0.5ei wi

0.5= 1i

var wi0.5ei=1.0

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An illustration

• A summary of 20 simulated studies (Hox, 2002) is listed in the next table.

• Effect size: Hedges’s d• Moderator: duration of the experimental

intervention in terms of weeks

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Study d var(d) weeks W^0.5 d* Intercept* weeks*

1 -0.264 0.086 3 3.41 -0.9002 3.41 10.2299

2 -0.23 0.106 1 3.0715 -0.7064 3.0715 3.0715

3 0.166 0.055 2 4.264 0.7078 4.264 8.528

4 0.173 0.084 4 3.4503 0.5969 3.4503 13.8013

5 0.225 0.071 3 3.7529 0.8444 3.7529 11.2588

6 0.291 0.078 6 3.5806 1.0419 3.5806 21.4834

7 0.309 0.051 7 4.4281 1.3683 4.4281 30.9965

8 0.435 0.093 9 3.2791 1.4264 3.2791 29.5122

9 0.476 0.149 3 2.5906 1.2331 2.5906 7.7719

10 0.617 0.095 6 3.2444 2.0018 3.2444 19.4666

11 0.651 0.11 6 3.0151 1.9628 3.0151 18.0907

12 0.718 0.054 7 4.3033 3.0898 4.3033 30.1232

13 0.74 0.081 9 3.5136 2.6001 3.5136 31.6228

14 0.745 0.084 5 3.4503 2.5705 3.4503 17.2516

15 0.758 0.087 6 3.3903 2.5699 3.3903 20.3419

16 0.922 0.103 5 3.1159 2.8728 3.1159 15.5794

17 0.938 0.113 5 2.9748 2.7904 2.9748 14.8741

18 0.962 0.083 7 3.4711 3.3392 3.4711 24.2974

19 1.522 0.1 9 3.1623 4.813 3.1623 28.4605

20 1.844 0.141 9 2.6631 4.9108 2.6631 23.9681

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-1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5Standardized mean difference

12

7

8

2

20

1

13

19

6

14

9

18

3

5

16

4

10

15

11

17

Fixed effect model

Random effects model

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50

A fixed-effects meta-analysis

Key points:1. Error variance on y* is fixed

at 1;1. The intercept is represented

by X0*

Results:1. The weighted effect size b0 is

0.550 (0.065).

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51

A fixed-effects meta-analysis with a covariate

Key points:1. X0*: intercept2. X1*: moderator (weeks)

Results:1. The intercept and the slope (for weeks) are -0.204 (0.170) and 0.135 (0.028).2. When “weeks” increases oneunit, the effect size increases by0.135 unit.

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52

A random-effects meta-analysis

Key points:1. u is the random effect2. Random slope analysis is

required (Mehta & Neale, 2005 )3. b0 is the weighted effect size4. m is the variance component (theamount of heterogeneity).

Results:1. The weighted effect size b0 is

0.579 (0.107) 2. The variance component m is 0.132

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53

A mixed-effects meta-analysis with a covariate

Results:1. The intercept and the slope are -0.214 (0.171) and 0.139 (0.036).2. The residual variance component is 0.023.

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54

Advantages and extensions of the SEM-based meta-analysis

• Handing missing covariates with maximum likelihood estimation method;

• Constructing confidence intervals on parameter estimates and heterogeneity indices;

• Addressing heterogeneity with mixture models;• Testing mediation and moderation models for

multivariate effect sizes.

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Conclusion

• TSSEM and SEM-based meta-analysis are two new approaches for research findings;

• They integrate meta-analysis into the general SEM framework.

• New opportunities for methodological development and substantive applications...

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References• Becker, B. J. (1992). Using results from replicated studies to estimate linear models. Journal of Educational Statistics, 17, 341-

362.

• Brown, S. P., & Stayman, D. M. (1992). Antecedents and consequences of attitude toward the ad: A meta-analysis. Journal of Consumer Research, 19, 34-51.

• Cheung, M.W.L. (2002). Meta-analysis for structural equation modeling: A two-stage approach. Unpublished doctoral dissertation, Chinese University of Hong Kong, Hong Kong.

• Cheung, M.W.L. (2008). A model for integrating fixed-, random-, and mixed-effects meta-analyses into structural equation modeling. Psychological Methods, 13, 182-202.

• Cheung, M.W.L. (May, 2009). Modeling multivariate effect sizes with structural equation models. Paper presented at the Association for Psychological Science 21st Annual Convention, San Francisco, CA, USA.

• Cheung, M.W.L. (in press). Fixed-effects meta-analyses as multiple-group structural equation models. Structural Equation Modeling.

• Cheung, M.W.L., & Chan, W. (2004). Testing dependent correlation coefficients via structural equation modeling. Organizational Research Methods, 7, 206-223.

• Cheung, M.W. L., & Chan, W. (2005a). Meta-analytic structural equation modeling: A two-stage approach. Psychological Methods, 10, 40-64.

• Cheung, M.W.L., & Chan, W. (2005b). Classifying correlation matrices into relatively homogeneous subgroups: A cluster analytic approach. Educational and Psychological Measurement, 65, 954-979.

• Cheung, M.W.L., Leung, K., & Au, K. (2006). Evaluating multilevel models in cross-cultural research: An Illustration with Social Axioms. Journal of Cross-Cultural Psychology, 37, 522-541.

• Cheung, S.F. (2000). Examining solutions to two practical issues in meta-analysis: Dependent correlations and missing data in correlation matrices. Unpublished doctoral dissertation, Chinese University of Hong Kong, Hong Kong.

• Cudeck, R. (1989). Analysis of correlation matrices using covariance structure models. Psychological Bulletin, 105, 317-327.

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References• Furlow, C.F. &, Beretvas, S.N. (2005). Meta-analytic methods of pooling correlation matrices for structural equation modeling

under different patterns of missing data. Psychological Methods, 10, 227-254.

• Hafdahl, A.R. (2001). Multivariate meta-analysis for exploratory factor analytic research. Unpublished doctoral dissertation, University of North Carolina at Chapel Hill.

• Hedges, L.V., & Olkin, I. (1985). Statistical methods for meta-analysis. Orlando, FL: Academic Press.

• Hox, J.J. (2002). Multilevel analysis: Techniques and applications. Mahwah, N.J.: Lawrence Erlbaum Associates.

• Hunter, J.E., & Schmidt, F.L. (2004). Methods of meta-analysis: Correcting error and bias in research findings (2nd ed.). Thousand Oaks, CA: Sage.

• Inter-University Consortium for Political and Social Research. (1989). International Social Survey Program: Work orientation. Ann Arbor, MI: Author.

• Leung, K., & Bond, M.H. (2004). Social axioms: A model for social beliefs in multicultural perspective. Advances in Experimental Social Psychology, 36, 119-197.

• Mehta, P.D., & Neale, M.C. (2005). People are variables too: Multilevel structural equation modeling. Psychological Methods, 10, 259-284.

• Muthén, B., Kaplan, D., & Hollis, M. (1987). On structural equation modeling with data that are not missing completely at random. Psychometrika, 51, 431-462.

• Olkin, I., & Siotani, M. (1976). Asymptotic distribution of functions of a correlation matrix. In S. Ideka (Ed.), Essays in probability and statistics (pp. 235-251). Tokyo: Shinko Tsusho.

• Premack, S.L., & Hunter, J.E. (1988). Individual unionization decisions. Psychological Bulletin, 103, 223-234.

• Viswesvaran, C., & Ones, D.S. (1995). Theory testing: Combining psychometric meta-analysis and structural equations modeling. Personnel Psychology, 48, 865-885.

workshop on meta-analytic structural equation …meta-analytic structural equation modeling (masem)...

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