Estimating multiple classification latent class models
by E. Maris (1999)
This paper presents a new class of models for persons-by-items data. The essential new feature of this class is the representation of the persons: every person is represented by its membership to multiple latent classes, so-called multiple classification latent class models (MCLCMs). Two types of MCLCMs were introduced: the conjunctive and the disjunctive. Three kinds of extensions were included in the conjunctive MCLCMs: (a) formulate different probability density function for the latent responses conditional on Z which indicates a person’s membership to some class of latent classifications; (b) assume that the latent responses to be polytomous, or even continuous, instead of dichotomous; and (c) formulate other condensation rule besides the conjunctive one. Demonstrating the steps for using two algorithms: EM-algorithm and hybrid algorithm (EM- and Newton-type) to get the maximum likelihood estimation (MLE) and the maximum a posterior (MAP). The simulation study was conducted to examine the properties of the MAP estimators, such as the uniqueness and the goodness-of-recovery. The models also applied to a real data from Tatsuoka (1984).
Comments
The MCLCMs were conducted for sequential administration testing, for example, every examinee was classified into one of a set of latent classes after administered one item; thus, called multiple classifications. In the contrast, the traditional latent class analysis (Goodman, 1974; Lazarsfeld & Henry, 1968) every person is represented by its membership to one of a set of latent classes; this situation likes the paper-and-pencil testing, every examinee was classified after finished a test. In this paper, the conjunctive MCLCMs like the prototype of the cognitive diagnostic models, such as the deterministic-input, noisy-and-gate (DINA) model and the noisy-input, deterministic-and-gate (NIDA) model. And the disjunctive MCLCMs are the prototype of the deterministic input, noisy-or-gate (DINO) model and the noisy input, deterministic-or-gate (NIDO) models.