Latent Variable Models and Factor Analysis A Unified Approach

by Bartholomew, David J.; Knott, Martin; Moustaki, Irini

Edition: 3rd

ISBN13: 9780470971925

ISBN10: 0470971924

Format: Hardcover

Pub. Date: 2011-08-08

Publisher(s): Wiley

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Summary

This book provides a comprehensive and unified approach to factor analysis and latent variable modeling and theory, providing a unified and coherent treatment from a statistical perspective. A general framework is presented to enable the derivation of the commonly used models. Updated numerical examples are provided as well as the software to carry them out. Written by leading experts in the field, Latent Variable Models and Factor Analysis: Includes new topics such as, covariate effects and non-linear terms, multiple population analysis and univariate and bivariate margins. Provides a new section on structural equation models (SEM) and Markov Chain Monte Carlo methods, along with illustrative examples. Looks at estimation methods, goodness-of-fit, non-linear models, covariates, longitudinal data and multilevel modeling along with updated examples throughout. Unifies many different streams of latent variable modeling and probability modeling. An introductory section is provided, which looks at the nature and interpretation of a latent variable, motivating discussions of closely related methods which make little or no explicit use of latent variables. Principal components are discussed in more depth, exploring its relationship to factor analysis in both historical and contemporary and theoretically and empirically. Furthermore, the book explores The Bonds' Model for abilities, a model which has a correlation structure which is identical to that of the factor model and hence cannot be distinguished from it and does not involve latent variables. No prior acquaintance with latent variable modeling is needed although a broad understanding of statistical theory is necessary.

Author Biography

David Bartholomew, Martin Knott and Irini Moustaki, Department of Statistics, The London School of Economics and Political Science, London, UK

Preface	p. xi
Acknowledgements	p. xv
Basic ideas and examples	p. 1
The statistical problem	p. 1
The basic idea	p. 3
Two examples	p. 4
Binary manifest variables and a single binary latent variable	p. 4
A model based on normal distributions	p. 6
A broader theoretical view	p. 6
Illustration of an alternative approach	p. 8
An overview of special cases	p. 10
Principal components	p. 11
The historical context	p. 12
Closely related fields in statistics	p. 17
The general linear latent variable model	p. 19
Introduction	p. 19
The model	p. 19
Some properties of the model	p. 20
A special case	p. 21
The sufficiency principle	p. 22
Principal special cases	p. 24
Latent variable models with non-linear terms	p. 25
Fitting the models	p. 27
Fitting by maximum likelihood	p. 29
Fitting by Bayesian methods	p. 30
Rotation	p. 33
Interpretation	p. 35
Sampling error of parameter estimates	p. 38
The prior distribution	p. 39
Posterior analysis	p. 41
A further note on the prior	p. 43
Psychometric inference	p. 44
The normal linear factor model	p. 47
The model	p. 47
Some distributional properties	p. 48
Constraints on the model	p. 50
Maximum likelihood estimation	p. 50
Maximum likelihood estimation by the E-M algorithm	p. 53
Sampling variation of estimators	p. 55
Goodness of fit and choice of q	p. 58
Model selection criteria	p. 58
Fitting without normality assumptions: least squares methods	p. 59
Other methods of fitting	p. 61
Approximate methods for estimating ¿	p. 62
Goodness of fit and choice of q for least squares methods	p. 63
Further estimation issues	p. 64
Consistency	p. 64
Scale-invariant estimation	p. 65
Heywood cases	p. 67
Rotation and related matters	p. 69
Orthogonal rotation	p. 69
Oblique rotation	p. 70
Related matters	p. 70
Posterior analysis: the normal case	p. 71
Posterior analysis: least squares	p. 72
Posterior analysis: a reliability approach	p. 74
Examples	p. 74
Binary data: latent trait models	p. 83
Preliminaries	p. 83
The logit/normal model	p. 84
The probit/normal model	p. 86
The equivalence of the response function and underlying variable approaches	p. 88
Fitting the logit/normal model: the E-M algorithm	p. 90
Fitting the probit/normal model	p. 93
Other methods for approximating the integral	p. 93
Sampling properties of the maximum likelihood estimators	p. 94
Approximate maximum likelihood estimators	p. 95
Generalised least squares methods	p. 96
Goodness of fit	p. 97
Posterior analysis	p. 100
Fitting the logit/normal and probit/normal models: Markov chain Monte Carlo	p. 102
Gibbs sampling	p. 102
Metropolis-Hastings	p. 105
Choosing prior distributions	p. 108
Convergence diagnostics in MCMC	p. 108
Divergence of the estimation algorithm	p. 109
Examples	p. 109
Polytomous data: latent trait models	p. 119
Introduction	p. 119
A response function model based on the sufficiency principle	p. 120
Parameter interpretation	p. 124
Rotation	p. 124
Maximum likelihood estimation of the polytomous logit model	p. 125
An approximation to the likelihood	p. 126
One factor	p. 127
More than one factor	p. 130
Binary data as a special case	p. 134
Ordering of categories	p. 136
A response function model for ordinal variables	p. 136
Maximum likelihood estimation of the model with ordinal variables	p. 138
The partial credit model	p. 140
An underlying variable model	p. 140
An alternative underlying variable model	p. 144
Posterior analysis	p. 147
Further observations	p. 148
Examples of the analysis of polytomous data using the logit model	p. 149
Latent class models	p. 157
Introduction	p. 157
The latent class model with binary manifest variables	p. 158
The latent class model for binary data as a latent trait model	p. 159
K latent classes within the GLLVM	p. 161
Maximum likelihood estimation	p. 162
Standard errors	p. 164
Posterior analysis of the latent class model with binary manifest variables	p. 166
Goodness of fit	p. 167
Examples for binary data	p. 167
Latent class models with unordered polytomous manifest variables	p. 170
Latent class models with ordered polytomous manifest variables	p. 171
Maximum likelihood estimation	p. 172
Allocation of individuals to latent classes	p. 174
Examples for unordered polytomous data	p. 174
Identifiability	p. 178
Starting values	p. 180
Latent class models with metrical manifest variables	p. 180
Maximum likelihood estimation	p. 181
Other methods	p. 182
Allocation to categories	p. 185
Models with ordered latent classes	p. 185
Hybrid models	p. 186
Hybrid model with binary manifest variables	p. 186
Maximum likelihood estimation	p. 187
Models and methods for manifest variables of mixed type	p. 191
Introduction	p. 191
Principal results	p. 192
Other members of the exponential family	p. 193
The binomial distribution	p. 193
The Poisson distribution	p. 194
The gamma distribution	p. 194
Maximum likelihood estimation	p. 195
Bernoulli manifest variables	p. 196
Normal manifest variables	p. 197
A general E-M approach to solving the likelihood equations	p. 199
Interpretation of latent variables	p. 200
Sampling properties and goodness of fit	p. 201
Mixed latent class models	p. 202
Posterior analysis	p. 203
Examples	p. 204
Ordered categorical variables and other generalisations	p. 208
Relationships between latent variables	p. 213
Scope	p. 213
Correlated latent variables	p. 213
Procrustes methods	p. 215
Sources of prior knowledge	p. 215
Linear structural relations models	p. 216
The LISREL model	p. 218
The structural model	p. 218
The measurement model	p. 219
The model as a whole	p. 219
Adequacy of a structural equation model	p. 221
Structural relationships in a general setting	p. 222
Generalisations of the LISREL model	p. 223
Examples of models which are indistinguishable	p. 224
Implications for analysis	p. 227
Related techniques for investigating dependency	p. 229
Introduction	p. 229
Principal components analysis	p. 229
A distributional treatment	p. 229
A sample-based treatment	p. 233
Unordered categorical data	p. 235
Ordered categorical data	p. 236
An alternative to the normal factor model	p. 236
Replacing latent variables by linear functions of the manifest variables	p. 238
Estimation of correlations and regressions between latent variables	p. 240
Q-Methodology	p. 242
Concluding reflections of the role of latent variables in statistical modelling	p. 244
Software appendix	p. 247
References	p. 249
Author index	p. 265
Subject index	p. 271
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