Model Specification

For $K$ latent classes and $J$ binary indicators, the marginal probability of response pattern $\mathbf{y} = (y_1, \ldots, y_J)$ is:

\[P(\mathbf{y}) = \sum_{k=1}^{K} \pi_k \prod_{j=1}^{J} \rho_{jk}^{y_j}(1 - \rho_{jk})^{1 - y_j}\]

where $\pi_k$ is the class prevalence ($\sum_k \pi_k = 1$) and $\rho_{jk}$ is the conditional probability of a positive response on item $j$ given class $k$.

EM Algorithm

Parameters are estimated by Expectation-Maximization:

E-step — compute posterior class probabilities for each person $n$:

\[\hat{P}(k \mid \mathbf{y}_n) = \frac{\pi_k \prod_j \rho_{jk}^{y_{nj}}(1-\rho_{jk})^{1-y_{nj}}}{\sum_{k'} \pi_{k'} \prod_j \rho_{jk'}^{y_{nj}}(1-\rho_{jk'})^{1-y_{nj}}}\]

M-step — update $\pi_k = N^{-1}\sum_n \hat{P}(k \mid \mathbf{y}n)$ and $\rho{jk} = \sum_n \hat{P}(k \mid \mathbf{y}n)y{nj} / \sum_n \hat{P}(k \mid \mathbf{y}_n)$.

Model Selection

The number of classes $K$ is selected using information criteria:

Criterion	Formula
BIC	$-2\ell + q\ln N$
AIC	$-2\ell + 2q$
Entropy $R^2$	$1 - H(\hat{P})/\ln N$

BIC is preferred for class enumeration. Entropy $R^2$ near 1 indicates clean separation among posterior class assignments. Multiple random starts are required to avoid local maxima in the log-likelihood.