Measurement Models: Identification and Estimation

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Measurement ModelsIdentification and Estimation

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

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Identification and Estimation

• Identification is concerned with whether the
  parameters of the model are uniquely determined.
• Estimation involves using sample data to make
  estimates of population parameters.

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A Confirmatory Factor Analysis Model of
Psychological Disorders
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Mathematical Specification
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Mathematical Specification
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Assumptions

• Each observed variable (X) loads on only one
  latent variable
• Each observed variable (X) is also affected by a
  single residual or unique factor
• Curved arrows correspond to correlations among
  latent variables
• Variances and covariances of the residual factors
  are contained in the Theta matrix

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Conditions for Identification

• Necessary
• Sufficient
• Necessary and Sufficient

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Degrees of Freedom
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Degrees of Freedom
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Models
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Model A
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Model B
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Model C
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Model D
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Model E
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Model F
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Model G
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Model H
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Model Identification
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Estimation

• Unweighted Least Squares (ULS) Minimizes the
  trace or the sum of the diagonal elements of
  tr(S-Sigma)2 ULS makes no distributional
  assumption so there are no tests of significance.
  Also ULS is scale dependent.

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Estimation

• Generalized Least Squares (GLS) In the fitting
  function, differences between S and Sigma are
  weighted by elements of S-1 The fitting function
  is tr(S-Sigma)S-12
• Maximum Likelihood (ML) minimizes the fitting
  function tr(SSigma-1) log lSigmal
  
• - log lSl q
• If X has a multivariate normal distribution, both
  GLS and ML have desirable asymptotic properties.

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Estimates with Different Constraints
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Estimation of Model Md with ULS, GLS and ML
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Model Building

• Models are nested when one model can be obtained
  from the other by imposing one or more free
  parameters. Therefore, model Mg is nested in Md
  and Mg is nested in Mh.
• When models are nested, the difference in Chi
  Square value is also distributed as Chi Square so
  the models can be compared statistically.

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Comparing the Fit of Nested Models
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Standardization

• The observed variables (X) can be standardized so
  S is a correlation matrix.
• The latent variables can be standardized by
  constraining the diagonal elements of the phi
  matrix to be 1.0.

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Estimates of Md and Mg with Standardized Observed
Variables and/or Standardized Latent Variables
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Effects of Standardization

• For Md the decision to analyze the covariance or
  correlation matrix or to set the metric by fixing
  loadings or variances makes no difference when
  scale free estimators such as GLS or ML are used.
• For Mg which involves equality constraints,
  analyzing the correlation matrix versus the
  covariance matrix can have substantive effects on
  the results obtained.

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Improper Solutions

• Nonpositive definite matrices
• Nonconvergence
• Heywood cases
• Improper sign in nonrecursive models
• Binary variables

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Information to Report on CFA Models

• Model specification
• List the indicators for each factor
• Indicate how the metric of each factor was
  defined
• Describe all fixed and constrained parameters
• Demonstrate that the model is identified
• Input data
• Description of sample characteristics and size
• Description of the type of data (e.g., nominal,
  interval, and scale range of indicators)
• Tests of assumptions
• Extent and method of missing data management
• Provide correlations, means, and SDs

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Information to Report on CFA Models

• Model estimation
• Indicate software and version
• Indicate type of data matrix analyzed
• Indicate estimation method used
• Model evaluation
• Report chi square with df and p value
• Report multiple fit indices (e.g., RMSEA, CFI,
  and confidence intervals if applicable)
• Report strategies used to assess strains in the
  solution (e.g., MIs, standardized residuals)
• If model is specified, provide a substantive
  rational for added or removed parameters

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Information to Report on CFA Models

• Parameter estimates
• Provide all parameter estimates (e.g., factor
  loadings, error variances, factor variances)
• Include the standard errors of the parameter
  estimates
• Consider the clinical as well as the statistical
  significance of the parameter estimates
• Substantive conclusions
• Discuss the CFA results in regard to their
  substantive implications
• Interpret the findings in the context of the
  study limitations

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Reference

• J.S. Long, Confirmatory Factor Analysis, Series
  Quantitative Applications in the Social Sciences,
  No. 33, Sage publications, 1983.