A Hypothesis Test for Equality of Bayesian Network Models

Graphical models play a central role in modelling genomic data, largely because the pathway structure governing the interactions of cellular components induces statistical dependence naturally described by directed or undirected graphs [1–3]. These models vary in their formal structure. While a Boolean network can be interpreted as a set of state transition rules, Bayesian or Markov networks reduce to static multivariate densities on random vectors extracted from genomic data. Such densities are designed to model coexpression patterns resulting from functional cooperation. Our concern will be with this type of multivariate model. Although the ideas presented here extend naturally to various forms of genomic data, to fix ideas we will refer specifically to multivariate samples of microarray gene expression data.In this paper, we consider the problem of comparing network models for a common set of genes under varying phenotypes. In principle, separately fit models can be directly compared. This approach is discussed in [3] and is based on distances definable on a space of graphs. Significance levels are estimated using replications of random graphs similar in structure to the estimated models.The algorithm proposed below differs significantly from the direct graph approach. We will formulate the problem as a two-sample test in which significance levels are estimated by randomly permuting phenotypes. This requires only the minimal assumption of independence with respect to subjects.Our strategy will be to confine attention to Bayesian network models (Section 2). Fitting Bayesian networks is computationally difficult, so a simplified model is developed for which a polynomial-time algorithm exists for maximum likelihood calculations. A two-sample hypotheses test based on the general likelihood ratio test statistic is introduced in Section 3. In Section 4, we discuss the application of sequential testing principles to permutation replications. This may be done in a way wh