How does AMOS handle binary observed variables, whether exogenous or endogenous? For exogenous binary variables, I suppose that a multiple group model could be built, with the binary variable (or combinations of such variables) acting as the group variable(s), but I would like to include the binary variables directly in a single-group model.
Resolving The Problem
The treatment of binary variables in AMOS depends on whether they are fixed or random.
Fixed Exogenous Variables
The distributional assumptions for AMOS models are discussed on pages 39-40 of the AMOS 7 .0 User's Guide. The section notes that multivariate normality is a standard distributional assumption of many structural equation applications but also describes a more general situation where maximum likelihood estimation could be carried out. If some exogenous variables are fixed (known beforehand or measured without error), their distributions may have any shape, provided that:
1. For any value of the fixed variables, the random variables in the model have a conditional normal distribution (i.e. conditional on treatment=1, the nonfixed variables have an MVN distribution).
2. The conditional variance-covariance matrix of the random variables is the same for every pattern of the fixed variable(s) (i.e. for every combination of fixed variable values).
3. The conditional expected values of the random variables depend linearly on the values of the fixed variable(s).
Actually testing the above conditions would require a multi-group analysis with the binary variable as the group variable, but the main analysis could then be a single group analysis with the fixed binary variables entered as observed exogenous variables in the model, provided that the researcher was satisfied that the above assumptions had been met.
A typical example of such a variable is a treatment variable where cases are classified as treatment (1) or control (0). Gender would also qualify as such a fixed variable. See Example 9 and Example 16 in the AMOS User's Guide. Example 9 presents Treatment as a binary predictor in a single-group analysis while Example 16 presents a multiple-group analysis of the same data. The assumptions employed in Example 9 are tested directly in Example 16. There is also an example of the inclusion of such binary treatment variables in the SPSS training course guide, "Structural Equation Modeling with AMOS". Exercise 3 for Chapter 5 ("The General Model" ) involves a model where the latent variable SES (Social Economic Status) and the binary predictor Headstrt (participation in the Head Start preschool program) both predict the latent variable Cognitive Ability.
Random Observed Variables
Binary variables that are endogenous (dependent) variables or indicators of latent variables are treated as random variables and presumed to have an underlying numeric scale with a normal distribution. Observed binary exogenous variables could also be treated in this way. For example, voting for a particular piece of legislation could be treated as a measure of a particular attitude, where the attitude was believed to have an underlying continuous scale. Binary variables that fit these descriptions, as opposed to the fixed exogenous variables described above, should be declared as ordered-categorical, as in Example 33 in the AMOS User's Guide. The categorical observed variables are rescaled in a manner akin to the transformations in the Categories module of SPSS. This treatment of ordered categorical variables is only available in AMOS with Bayesian Estimation. There is a statement in that chapter that indicates that additional parameter constraints are required if the variable is dichotomous and the user is referred to the Help topic "Parameter identification with dichotomous variables". From
Help->Contents in AMOS, choose the Index tab, type "Parameter identification with dichotomous variables" and click Display.
The ordered categorical rescaling of variables is available in AMOS only with Bayesian Estimation. . Bootstrapping is an additional approach to modeling with data that do not fit the distributional assumptions of AMOS models for maximum likelihood estimation. See videos that illustrate the inclusion of ordered categorical variables at
Random Unobserved Variables
In AMOS 7, both independent and dependent observed variables could be ordered categorical, including binary, but the latent variables were assumed to be continuous. A new feature in AMOS 16 is the ability to perform Latent Mixture Modeling (LMM). . Latent Mixture Modeling is only available with Bayesian Estimation. Rather than drawing a circle in the diagram that represents a latent class, multiple group analysis is used for LMM, where the group can be predicted (by checking the 'Assign cases to groups' checkbox in AMOS' Data Files dialog) and where the group membership variable may be unobserved for some or all cases. The user must specify the number of groups, so if you proposed a binary latent class variable, you would indicate that there were 2 groups in the Manage Groups dialog. LMM is covered in Examples 34 to 36 in the AMOS 16.0 User's Guide. There is also a set of videos that demonstrate the analysis at
16 April 2020