Skip to main content Accessibility help
×
Hostname: page-component-7c8c6479df-27gpq Total loading time: 0 Render date: 2024-03-28T17:57:50.319Z Has data issue: false hasContentIssue false

2 - Maximum likelihood estimation in latent class models for contingency table data

from Part I - Contingency tables

Published online by Cambridge University Press:  27 May 2010

Paolo Gibilisco
Affiliation:
Università degli Studi di Roma 'Tor Vergata'
Eva Riccomagno
Affiliation:
Università degli Studi di Genova
Maria Piera Rogantin
Affiliation:
Università degli Studi di Genova
Henry P. Wynn
Affiliation:
London School of Economics and Political Science
Get access

Summary

Abstract

Statistical models with latent structure have a history going back to the 1950s and have seen widespread use in the social sciences and, more recently, in computational biology and in machine learning. Here we study the basic latent class model proposed originally by the sociologist Paul F. Lazarfeld for categorical variables, and we explain its geometric structure. We draw parallels between the statistical and geometric properties of latent class models and we illustrate geometrically the causes of many problems associated with maximum likelihood estimation and related statistical inference. In particular, we focus on issues of non-identifiability and determination of the model dimension, of maximisation of the likelihood function and on the effect of symmetric data. We illustrate these phenomena with a variety of synthetic and real-life tables, of different dimension and complexity. Much of the motivation for this work stems from the ‘100 Swiss Francs’ problem, which we introduce and describe in detail.

Introduction

Latent class (LC) or latent structure analysis models were introduced in the 1950s in the social science literature to model the distribution of dichotomous attributes based on a survey sample from a populations of individuals organised into distinct homogeneous classes on the basis of an unobservable attitudinal feature. See (Anderson 1954, Gibson 1955, Madansky 1960) and, in particular, (Henry and Lazarfeld 1968). These models were later generalised in (Goodman 1974, Haberman 1974, Clogg and Goodman 1984) as models for the joint marginal distribution of a set of manifest categorical variables, assumed to be conditionally independent given an unobservable or latent categorical variable, building upon the then recently developed literature on log-linear models for contingency tables.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×