03b Measurement Models Id and Est

Measurement Models:
Identification and Estimation
James G. Anderson, Ph.D.
Purdue University

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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Example 8
Factor analysis: Girls' sample
Holzinger and Swineford (1939)
No. of independent pieces of information =3
No. of paramters estimated =4
DF =-1

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Example 8
Factor analysis: Girls' sample
Holzinger and Swineford (1939)
No. of independent pieces of information =6

No. of paramters estimated =6
DF = 0

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Example 8
Factor analysis: Girls' sample
Holzinger and Swineford (1939)
No. of independent pieces of information =6
No. of paramters estimated =7
DF =-1

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Example 8
Factor analysis: Girls' sample
Holzinger and Swineford (1939)
No. of independent pieces of information = 21
No. of paramters estimated = 13

DF = 8

A Confirmatory Factor Analysis
Model of Psychological Disorders

Mathematical Specification

Mathematical Specification

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

Conditions for Identification
• Necessary
• Sufficient
• Necessary and Sufficient

Degrees of Freedom

Degrees of Freedom

Models

Model A
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Example 8
Factor analysis: Girls' sample
Holzinger and Swineford (1939)
No. of independent pieces of information = 10
No. of parameters to be estimated =11
DF=-1

Model B
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A

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Example 8
Factor analysis: Girls' sample
Holzinger and Swineford (1939)
No. of independent pieces of information = 10
No. of parameters to be estimated =9
DF=1

Model C
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Example 8
Factor analysis: Girls' sample
Holzinger and Swineford (1939)
No. of independent pieces of information = 10
No. of parameters to be estimated = 9
DF=1

Model D
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Example 8
Factor analysis: Girls' sample
Holzinger and Swineford (1939)
No. of independent pieces of information = 10
No. of parameters to be estimated =9
DF=1

Model E
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Example 8
Factor analysis: Girls' sample
Holzinger and Swineford (1939)
No. of independent pieces of information = 10
No. of parameters to be estimated =8
DF=2

Model F
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Example 8
Factor analysis: Girls' sample
Holzinger and Swineford (1939)
No. of independent pieces of information = 10
No. of parameters to be estimated =9
DF=1

Model G
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Example 8
Factor analysis: Girls' sample
Holzinger and Swineford (1939)
No. of independent pieces of information = 10
No. of parameters to be estimated =7
DF=3

Model H
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Example 8
Factor analysis: Girls' sample
Holzinger and Swineford (1939)
No. of independent pieces of information = 10
No. of parameters to be estimated =8
DF=2

Model Identification
Model
Ma
Mb

Identification
No
No (scale
indeterminacy)

Mc
Md
Me
Mf
Mg
Mh

Yes
Yes
Yes
No
Yes
Yes

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.

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-1]2
• 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.

Estimates with Different Constraints

Estimation of Model Md with ULS,
GLS and ML

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.

Comparing the Fit of Nested Models

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.

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

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.

Improper Solutions
•
•
•
•
•

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

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

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

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

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