Local Dependence Latent Structure Models

Abstract

A local independence latent structure model, which assumes m latent classes, requires a minimum of 2 m -1 items for the solution of the 2 m 2 latent parameters. If one adds 3 items to the test and if one assumes local “dependence” between pairs of items, thereby adding \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left( {\begin{array}{*{20}c} {2m + 2} \\ 2 \\ \end{array} } \right)$$ \end{document} additional latent parameters, ξ ij , representing the association between items i and j , then it is possible to obtain estimates for all of the latent parameters: latent class frequencies latent probabilities, and measures of association between pairs of items. The solution consists of (1) forming ( m + 1) × ( m + 1) matrices of manifest data, which are singular, (2) solving for the ξ ij in equations that result from the singularity of the data matrices, (3) “correcting” the manifest data by removing the “contamination” due to local dependence, and (4) estimating the remaining latent parameters from the corrected data, using methods outlined in earlier literature.