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Special Modeling Issues

515

CHAPTER 14

SPECIAL MODELING ISSUES

In this chapter, the following special modeling issues are discussed:

 Model estimation

 Multiple group analysis

 Missing data

 Categorical mediating variables

 Calculating probabilities from probit regression coefficients

 Calculating probabilities from logistic regression coefficients

 Parameterization of models with more than one categorical latent

variable

In the model estimation section, technical details of parameter

specification and model estimation are discussed. In the multiple group

analysis section, differences in model specification, differences in data

between single-group analysis and multiple group analysis, and testing

for measurement invariance are described. In the missing data section,

estimation of models when there is missing data and special features for

data missing by design are described. There is a section that describes

how categorical mediating variables are treated in model estimation.

There is a section on calculating probabilities for probit regression

coefficients. In the section on calculating probabilities for logistic

regression coefficients, a brief background with examples of converting

logistic regression coefficients to probabilities and odds is given. In the

section on parameterization with multiple categorical latent variables,

conventions related to logistic and loglinear parameterizations of these

models are described.

MODEL ESTIMATION

There are several important issues involved in model estimation beyond

specifying the model. The following general analysis considerations are

discussed below:

 Parameter default settings

 Parameter default starting values

CHAPTER 14

516

 User-specified starting values for mixture models

 Multiple solutions for mixture models

 Convergence problems

 Model identification

 Numerical integration

PARAMETER DEFAULT SETTINGS

Default settings are used to simplify the model specification. In order to

minimize the information provided by the user, certain parameters are

free, constrained to be equal, or fixed at zero as the default. These

defaults are chosen to reflect common practice and to avoid

computational problems. These defaults can be overridden. Because of

the extensive default settings, it is important to examine the analysis

results to verify that the model that is estimated is the intended model.

The output contains parameter estimates for all free parameters in the

model, including those that are free by default and those that are free

because of the model specification. Parameters that are fixed in the

input file are also listed with these results. Parameters fixed by default

are not included. In addition, the TECH1 option of the OUTPUT

command shows which parameters in the model are free to be estimated

and which are fixed.

Following are the default settings for means/intercepts/thresholds in the

model when they are included:

 Means of observed independent variables are not part of the model.

The model is estimated conditioned on the observed independent

variables.

 In single group analysis, intercepts and thresholds of observed

dependent variables are free.

 In multiple group analysis and multiple class analysis, intercepts and

thresholds of observed dependent variables that are used as factor

indicators for continuous latent variables are free and equal across

groups or classes. Otherwise, they are free and unequal in the other

groups or classes except for the inflation part of censored and count

variables in which case they are free and equal.

 In single group analysis, means and intercepts of continuous latent

variables are fixed at zero.

Special Modeling Issues

517

 In multiple group analysis and multiple class analysis, means and

intercepts of continuous latent variables are fixed at zero in the first

group and last class and are free and unequal in the other groups or

classes except when a categorical latent variable is regressed on a

continuous latent variable. In this case, the means and intercepts of

continuous latent variables are fixed at zero in all classes.

 Logit means and intercepts of categorical latent variables are fixed at

zero in the last class and free and unequal in the other classes.

Following are the default settings for variances/residual variances/scale

factors:

 Variances of observed independent variables are not part of the

model. The model is estimated conditioned on the observed

independent variables.

 In single group analysis and multiple group analysis, variances and

residual variances of continuous and censored observed dependent

variables and continuous latent variables are free. In multiple class

analysis, variances/residual variances of continuous and censored

observed dependent variables and continuous latent variables are

free and equal across classes.

 In single group analysis using the Delta parameterization, scale

factors of latent response variables for categorical observed

dependent variables are fixed at one. In multiple group analysis

using the Delta parameterization, scale factors of latent response

variables for categorical observed dependent variables are fixed at

one in the first group and are free and unequal in the other groups.

 In single group analysis using the Theta parameterization, variances

and residual variances of latent response variables for categorical

observed dependent variables are fixed at one. In multiple group

analysis using the Theta parameterization, variances and residual

variances of latent response variables for categorical observed

dependent variables are fixed at one in the first group and are free

and unequal in the other groups.

Following are the default settings for covariances/residual covariances:

 Covariances among observed independent variables are not part of

the model. The model is estimated conditioned on the observed

independent variables.

CHAPTER 14

518

 In single group analysis and multiple group analysis, covariances

among continuous latent independent variables are free except when

they are random effect variables defined by using ON or XWITH in

conjunction with the | symbol. In these cases, the covariances

among continuous latent independent variables are fixed at zero. In

multiple class analysis, free covariances among continuous latent

independent variables are equal across classes.

 In single group analysis and multiple group analysis, covariances

among continuous latent independent variables and observed

independent variables are fixed at zero.

 Covariances among observed variables not explicitly dependent or

independent are fixed at zero.

 Residual covariances among observed dependent variables and

among continuous latent dependent variables are fixed at zero with

the following exceptions:

 In single group analysis and multiple group analysis,

residual covariances among observed dependent variables

are free when neither variable influences any other variable,

when the variables are not factor indicators, and when the

variables are either continuous, censored (using weighted

least squares), or categorical (using weighted least squares).

In multiple class analysis, free residual covariances among

observed dependent variables are equal across classes.

 In single group analysis and multiple group analysis,

residual covariances among continuous latent dependent

variables that are not indicators of a second-order factor are

free when neither variable influences any other variable

except its own indicators, except when they are random

effect variables defined by using ON or XWITH in

conjunction with the | symbol. In these cases, the

covariances among continuous latent independent variables

are fixed at zero. In multiple class analysis, free residual

covariances among continuous latent dependent variables

are equal across classes.

Following are the default settings for regression coefficients:

 Regression coefficients are fixed at zero unless they are explicitly

mentioned in the MODEL command. In multiple group analysis,

free regression coefficients are unequal in all groups unless they

involve the regression of an observed dependent variable that is used

Special Modeling Issues

519

as a factor indicator on a continuous latent variable. In this case,

they are free and equal across groups. In multiple class analysis,

free regression coefficients are equal across classes.

PARAMETER DEFAULT STARTING VALUES

If a parameter is not free by default, when the parameter is mentioned in

the MODEL command, it is free at the default starting value unless

another starting value is specified using the asterisk (*) followed by a

number or the parameter is fixed using the @ symbol followed by a

number. The exception to this is that variances and residual variances

for latent response variables corresponding to categorical observed

dependent variables cannot be free in the Delta parameterization. They

can be free in the Theta parameterization. In the Theta parameterization,

scale factors for latent response variables corresponding to categorical

observed dependent variables cannot be free. They can be free in the

Delta parameterization.

GENERAL DEFAULTS

Following are the default starting values:

Means/intercepts of continuous and 0 or sample mean

censored observed variables depending on the

analysis

Means/intercepts of count observed variables 0

Thresholds of categorical observed variables 0 or determined by the

sample proportions

depending on the

analysis

Variances/residual variances of .05 or 1 depending on

continuous latent variables the analysis

Variances/residual variances of .5 of the sample

continuous and censored observed variables variance

Variances/residual variances of 1

latent response variables for categorical

observed variables

Scale factors 1

Loadings for indicators of continuous 1

latent variables

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