Distributions of test statistics for edge exclusion for graphical models

Abstract

Three test statistics for single edge exclusion from the saturated model are considered: the likelihood ratio test, the Wald test and the efficient score test. Non-signed and signed square-root versions are used. Their distributions are investigated, in particular under the alternative hypothesis that the saturated model holds. The delta-method is used to derive approximating asymptotic normal distributions. A non-central chi-square approximation is also proposed.

The power of the three test statistics for single edge exclusion is studied in detail, both for graphical Gaussian models with p variables and for graphical log-linear models with two and three binary variables. Theoretical asymptotic power functions are derived for the non-signed and the signed square-root versions of each test statistic. The normal and the non-central chi-square approximations, previously derived, are used. The quality of the approximation is assessed by simulation.

The single-factor model and the latent class model (with all variables binary) are parameterised, within the framework of graphical models, as a graphical Gaussian model and as a graphical log-linear model, respectively. The implications of such parameterisations are discussed, in particular concerning the parameter space and its admissible regions. It is proved that marginalising over the latent variable, both in a single-factor graphical Gaussian model and in a single-factor graphical log-linear model, induces no conditional independencies between manifest variables and, therefore, an independence graph that is complete. Consequently, starting with the saturated model and performing backwards elimination model selection is suggested as the most appropriate way for the data analyst to detect the presence of a latent variable. However, since model selection is subject to type II error, it is recommended that the power of the test statistics for single edge exclusion from the saturated model is taken into account.

A parallel is made between results obtained for graphical Gaussian models and for graphical log-linear models and some guidelines/recommendations are given to the data analyst.

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