Multiple Sample Models
Rationale of Multiple
Sample SEMs
• Do estimates of model parameters vary across groups?
Uses of Multiple Sample SEMs
• Use for analysis of cross-sectional, longitudinal, experimental and
quasi-experimental data and to test for measurement variance.
• This procedure allows the investigator to:
1) Estimate separately the parameters for multiple samples
2) Test whether specified parameters are equivalent across these groups.
3) Test whether there are group mean differences for the indicator variables and/or for the structural
Analytical Procedure
• Estimate the parameters of the model with
no constraints (i.e., allow the parameters to differ among groups)
• Compute chi-square as a measure of fit. • Re-estimate the parameters of the model
Analytical Procedure (2)
• Determine the chi square difference is significant
• If the relative fit of the constrained model is significantly worse than that of the
Structural Model Example
Lyman, DR., Moff, HT,
Stouthamer-Loeber,M. (1993). “Explaining the Relation Between IQ and Delinquency: Class,
Race, Test Motivation or Self-Control.”
Structural Model Example:
Data
• Covariance matrices for White (n=214) and African American (n=181) male adolescents • Total observations: n=395
• Degrees of Freedom: 2 * 5(6) = 30 2
• 7 parameters constrained to be equal
Fit Statistics for the Multiple
Sample Model
• Χ2 = 11.68
df = 7 NS
• Χ2/df =1.67
Modification Indices for
Equality-Constrained Parameters
• MI values estimate the amount by which the overall chi square value would decrease if the associated parameters were estimated
separately in each group.
• Statistical significance of a modification index indicates a group difference on that parameter • For example, there is a statistically significant
Additional Analysis
• Path coefficients were estimated separately for each sample
• Standardized values can only be used for comparisons within a group.
• Unstandardized values are used for
Results
• In both samples, Verbal Ability has a significant effect on Achievement.
• Verbal Ability is the only significant predictor of Delinquency in the White sample.
• Achievement is the only significant predictor of Delinquency in the African-American sample. • Conclusion: Among male adolescents, school
Use of Multiple Sample CFAs
• Test for measurement invariance, whether a set of indicators assesses the same
latent variables in different groups.
Analytical Procedure
• Estimate the parameters of the model with
no constraints (i.e., allow the factor loadings and error variances to differ among groups).
• Compute chi square as a measure of fit. • Re-estimate the parameters of the model
Analytical Procedure (2)
• Determine if the chi square difference is significant
• If the relative fit of the constrained model is significantly worse than that of the
unconstrained model, then individual factor loadings should be compared
Confirmatory Factor Analysis:
Example
Werts, CE, Rock, DA, Linn, RL and
Joreskog, KG. (1976). “A Comparison of Correlations, Variances, Covariances and Regression Weights With or Without
Confirmatory Factor Analysis:
Data
• Covariance matrices are for two samples (n1=865 and n2=900) of candidates who
took the SAT in January 1971. • Total observations: n=1765
Confirmatory Factor Analysis:
Results
• The factor loadings are the same for the two groups.
Model A
• Parameters for the two groups
– Factor Loadings Equal – Factor Correlations Equal – Error Variances Equal
Model Fit
Chi Square = 34.89 df= 11
Model B
• Parameters for the two groups
– Factor Loadings Unequal – Factor Correlations Equal – Error Variances Equal
Model Fit
Model C
• Parameters for the two groups
– Factor Loadings Unequal – Factor Correlations Equal – Error Variances Unequal Model Fit
Chi Square = 4.03 df= 11
p < 0.26
Model D
• Parameters for the two groups
– Factor Loadings Equal – Factor Correlations Equal – Error Variances Unequal Model Fit
Chi Square = 10.87 df= 7
p < 0.14
