Mixed or Mixture Models and Saltus models

Rasch models are grounded in the concept of the unidimensional latent variable, i.e., the items defining the latent variable operate in the same way for all members of the target population. Of course, this is a fiction. But reality can often be made to cooperate.

 

But there are occasions when a population is comprised of different classes of persons with the items comprising a different latent variable for each class. The classes are called "Latent Classes".

 

Standard Rasch "latent trait" models can be extended to allow for latent classes. These are called "Mixture Models" (Rost, 1990). The Saltus model (Mark Wilson, 1989) is a mixed model in which segments of items are modeled to shift their difficulties together, and by the same amount, for different latent classes. In these models, the different latent variables are defined by item difficulties, but individual respondents are not assigned to a particular class, but rather the probability that each respondent belongs to each class is reported.

 

In a finite mixture model, the number of classes is decided in advance. For an infinite model, the number of classes is infinite. The actual number of active classes depends on the data. Observed classes have their probability distributions inferred from the data. Unobserved classes keep their Bayesian distributions.

 

Winsteps does not do a mixture or Saltus analysis directly, but it can provide much of the same information, and also can indicate whether a more rigorous latent class analysis is likely to be productive.

 

Here is an approach:

 

Step 1. Identify meaningful potential respondent classes, e.g., male/female, high/low performers. The Winsteps PCA analysis (e.g., Table 24) may help identify potential person classes. and Table 23 may help to identify item classes.

 

Step 2. Mark in the person label the class codes. The Microsoft Word "rectangle copy" function may be useful. High and low performers do not need to be flagged, instead the MA2 function can be used.

 

Step 3. Perform DIF analysis based on the class codes. Items displaying strong DIF may be exhibiting class-related behavior.

 

Step 4. Flag the items by class in the item identification.

 

Step 5. Look for item-classification by person-classification interactions (differential classification-grouped functioning, DGF, Table 33). These would approximate the Saltus findings.

 

Rost, Jürgen. (1990) Rasch Models in Latent Classes: An Integration of Two Approaches to Item Analysis, Applied Psychological Measurement, 14, 271-282

 

Wilson, M. (1989). Saltus: A psychometric model for discontinuity in cognitive development. Psychological Bulletin, 105, 276-289.