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An introduction to latent variable models. (English) Zbl 0583.62049
Monographs on Statistics and Applied Probability. London - New York: Chapman and Hall. VIII, 107 p. (1984).
Latent variables, such as ”public opinion” or ”social class,” are hypothetical constructs that are not directly observable. However, certain of their effects on measurable (manifest) variables, such as test scores or responses to certain questions, are observable, and hence subject to study. Latent variable models explore the structure of covariance and correlation matrices in terms of unobservable constructs and assess whether observed relationships between a set of manifest variables can be accounted for by a small number of latent variables.
This monograph gives an excellent introduction to three types of latent variable models: the factor analysis model, the LISREL model for linear structural relationships, and the latent class model in which the manifest variables are dichotomies.
Chapter 1 gives a general introduction to latent variables and latent variables models. Chapter 2 focuses on the factor analysis model in which the correlations between a set of observable variables are explained in terms of a small number of latent variables (i.e., the factors). Issues of identification of the factor structure, estimation of parameters in the factor analysis model, goodness-of-fit tests and factor rotation are discussed and illustrated with several examples from the behavioral and social sciences.
Chapter 3 discusses LISREL models, which are models for linear structural relationships that are studied by Jöreskog and others. A LISREL model consists of two parts: a structural equation model which specifies the relationship between dependent and independent latent variables, and measurement equations which specify how the two sets of latent variables are related to the measured variables. This model includes the factor analysis model as a special case.
The parameters in a LISREL model are estimated by fitting the covariance matrices that are implied by this model to the corresponding observed covariance matrices. Assuming that the observations have a multivariate normal distribution one can obtain maximum likelihood estimates by minimizing a certain nonlinear discrepancy (fitting) function. Convenient LISREL computer programs, such as the one developed by Jöreskog and Sörbom at the University of Uppsala, can be used to obtain the estimates.
The goodness-of-fit of an estimated model is assessed by a chi-square statistic which is related to the minimum value of the fitting function. If the model is correct and the sample size sufficiently large, the chi- square is the likelihood ratio test statistic for testing the model against the alternative that the covariance matrix is unconstrained. Several very interesting examples illustrate these techniques on real data.
Chapter 4 discusses latent variable models for categorical data. A latent class model with a single latent variable having two or more classes is postulated for data in which the manifest variables are dichotomies. Concluding remarks on goodness-of-fit measures and a brief discussion of the limitations of latent variable models are given in chapter 5.
This book provides an excellent introduction to latent variable models. Applied statisticians and researchers interested in the practical application of latent variable procedures should find this monograph very useful.
Reviewer: J.Ledolter

62H25 Factor analysis and principal components; correspondence analysis
62-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics
62H05 Characterization and structure theory for multivariate probability distributions; copulas
62P15 Applications of statistics to psychology
62H15 Hypothesis testing in multivariate analysis
62H12 Estimation in multivariate analysis
62P25 Applications of statistics to social sciences