J. Extension to continuous variables and other scale types

Course Text: Pages 22-24

The field of finite mixture (FM) modeling developed as mixtures from K latent populations of normally distributed variables. Hence, the extension from traditional LC to LC models with continuous observed variables (utilizing the normal distribution) formalizes the close connection between LC modeling and finite mixture (FM) modeling. In traditional LC modeling with nominal indicators, the multinomial distribution is used. For continuous variables, the joint distribution used is the multivariate normal. For count variables, a different distribution is used (Poisson, binomial count) to form the likelihood function.

To make clear that the T indicators / response/ dependent variables may be quantitative, some of the reading materials change the notation for the indicators to Y1, Y2, …, YT.

An important difference from traditional LC modeling with categorical indicators, is that with some other scale types, fewer indicators are required to achieve identification. In traditional LC models, 3 dichotomous indicators are required for a 2-class model to be identified. In contrast, a LC model with K classes is identifiable with only a single indicator for any value of K when that indicator is continuous or a Poisson count variable. As an example of a LC analysis of a single continuous dependent variable defined as a mixture of two normally distributed variables, see Updated Sage Article, section 4.1 (pages 25-27, Figure 3, Table 9).

Relationship to K-Means

There is a close connection between the iterative maximum likelihood (ML) algorithm used in estimating the LC Cluster model and the iterative K-means algorithm used in cluster analysis, the latter being the most widely used technique for performing cluster analysis currently. While the K-means algorithm uses Euclidean distance to group cases that are close to each other based on their values on continuous (or at least, quantitative) variables, the LC approach utilizes probabilities to measure distance, and thus is not limited to quantitative variables. The LC approach may be viewed as a way of formalizing the K-means approach in terms of a statistical model, and extending it in many directions.

Cluster Analysis – 2 Approaches PowerPoint presentation. This is a non-technical presentation:

“Session 1 Cluster Analysis.ppt”

Assigned Reading:

LatentGOLD Technical Guide, Section 3.5

Latent Class Models Article:

1. Latent class models for clustering (pages 2-9)

Reference:

Magidson and Vermunt “Latent class models for clustering: A comparison with K-

means”, Canadian Journal of Marketing Research , Vol. 20.1, 2002.

This article presents a more technical comparison.

 LatentGOLD Technical Guide

J1:  Finite Mixture models for Continuous Response Variables, Section 3.5 (pages 105-106)

J2:  LC Cluster models for mixed mode data, Section 3.6 (page 107)