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European Association of Methodology

Department of methodology and evaluation research

Jena University


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Contributions: Abstract

The observed association structure from graphical log-linear models with a binary latent variable

M. Fatima Salgueiro
ISCTE Business School, Lisbon
Portugal
Peter W.F. Smith John W. McDonald
University of Southampton
United Kingdom

We investigate the observed association structure between two and three binary manifest variables arising from a latent class model with a single binary latent variable. We represent and parameterise the latent class model as a graphical log-linear model. These models are a family of parametric statistical models used for modelling discrete data cross-classified in contingency tables (see, for example, Whittaker, 1990).

We prove that marginalizing over the latent variable will give the saturated model for the manifest variables (see Salgueiro, 2003). However, when performing data analysis, type II errors (i.e. false acceptances of the null hypotheses) can occur, and therefore the observed association structure will often not correspond to this saturated model. A simulation study is used to estimate the statistical power of the first step of a backward elimination model selection procedure and, hence, the probability of a latent class model being chosen. In this context the overall power is the probability of selecting the true saturated model given the specified true model parameters.

Results show that, for certain combinations of observed odds ratios and marginal cell probabilities, an association structure that is not necessarily the one corresponding to the saturated model of the manifest variables may be observed, even when the true underlying model is a latent class model. The importance of the total sample size is also discussed.

References

Salgueiro, M.F. (2003) Distributions of test statistics for edge exclusion for graphical models. Ph.D. Thesis, University of Southampton.

Whittaker, J. (1990) Graphical models in applied multivariate statistics. Chichester: John Wiley & Sons.