Diagnostic Classification Models

DCMs as Confirmatory Latent Class Models

The attributes are operationalized, or thoroughly defined, as part of designing a diagnostic assessment; thus, the latent classes into which examinees will be classified are defined prior to analyses of response data collected from the diagnostic assessment. This feature of DCMs make them a special case of a larger family of models known as latent class models: DCMs are confirmatory latent class models because the number and the nature of the latent classes are hypothesized and specified prior to analyses.

The general latent class model defines the probability of a scored item response vector (denoted xe) for a given examinee e as a function of the attribute profile c of the examinee (αe = αc) as:

$$P\left(X_e=x_e\right)={\displaystyle \sum }_{c=1}^{2^A}{\upsilon }_c{\displaystyle \prod }_{i=1}^I{\pi }_{i|{\alpha }_e}^{x_{ei}}{\left(1-{\pi }_{i|{\alpha }_e}\right)}^{1-x_{ei}}.$$

This equation has two main components: the structural component, which describes the relationships and distributions of the attributes, and the measurement component, which specifies the relationships between the attributes and items. The structural parameter υc represents the [Page 508]proportion of examinees who are members of latent class c. Because the classes, defined by attribute patterns, are exhaustive and mutually exclusive, these proportions sum to 1 $\left({\displaystyle \sum }_{c=1}^{2^A}{\vartheta }_c=1\right)$. The structural model is commonly parameterized by using a log-linear model where attributes are predictors of the class proportions or by specifying a higher order structure where attributes are predictors of one or more higher order continuous factors. Using either method, marginal proportions, or base rates, of mastery for individual attributes, as well as correlations of attribute pairs, can be derived.

...