Multiple Sample Models

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Multiple Sample Models

• James G. Anderson, Ph.D.
• Purdue University

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Rationale of Multiple Sample SEMs

• Do estimates of model parameters vary across
  groups?
• Another way of asking this question is Does
  group membership moderate the relationships
  specified in the model?

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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
• Estimate separately the parameters for multiple
  samples
• Test whether specified parameters are equivalent
  across these groups.
• Test whether there are group mean differences for
  the indicator variables and/or for the structural
  equations

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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 after
  imposing cross-group equality constraints on
  parameters

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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
  unconstrained model, then individual path
  coefficients should be compared across samples.

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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. Journal of Abnormal Psychology,
  102, 187-196.

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Structural Model ExampleData

• Covariance matrices for White (n214) and African
  American (n181) male adolescents
• Total observations n395
• Degrees of Freedom 2 5(6) 30
• 2
• 7 parameters constrained to be equal
• Variances and covariances allowed to vary between
  groups.

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Fit Statistics for the Multiple Sample Model

• ?2 11.68
• df 7
• NS
• ?2/df 1.67
• NFI 0.96
• NNFI 0.95

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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
  difference on the Achievement to Delinquency path
  and the Social Class to Achievement path.

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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 comparisons
  between or across groups.

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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 has a
  larger role in preventing the development of
  delinquency for African-Americans that for Whites

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Use of Multiple Sample CFAs

• Test for measurement invariance, whether a set of
  indicators assesses the same latent variables in
  different groups.
• Examine a test for construct bias, whether a test
  measures something different in one group than in
  another.

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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 after
  imposing cross-group equality constraints
  parameters

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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 across samples to
  determine the extent of partial measurement
  invariance..

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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 Measurement Errors. Psychology
  Bulletin, 83, 52-56.

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Confirmatory Factor AnalysisData

• Covariance matrices are for two samples (n1865
  and n2900) of candidates who took the SAT in
  January 1971.
• Total observations n1765
• Degrees of Freedom 2 4(5) 20
• 2

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Confirmatory Factor AnalysisResults

• The factor loadings are the same for the two
  groups.
• The error variances differ between the two
  groups.

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Model A

• Parameters for the two groups
• Factor Loadings Equal
• Factor Correlations Equal
• Error Variances Equal
• Model Fit
• Chi Square 34.89
• df 11
• p lt 0.00011

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Model B

• Parameters for the two groups
• Factor Loadings Unequal
• Factor Correlations Equal
• Error Variances Equal
• Model Fit
• Chi Square 29.67
• df 7
• p lt 0.00011

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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 lt 0.26
• Chi Square difference 29-67-4.03 25.03
• df difference 7-34

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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 lt 0.14
• Chi Square difference 34.89-10.87 24.01
• df difference 11-74

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