Michael D. Perlman

Alternative Markov properties for chain graphs

Graphical Markov models use graphs to represent possible dependences among statistical variables. Lauritzen, Wermuth, and Frydenberg (LWF) introduced a Markov property for chain graphs (CG): graphs that can be used to represent both structural and associative dependences simultaneously and that include both undirected graphs (UG) and acyclic directed graphs (ADG) as special cases. Here an alternative Markov property (AMP) for CGs is introduced and shown to be the Markov property satisfied by a block-recursive linear system with multivariate normal errors. This model can be decomposed into a collection of conditional normal models, each of which combines the features of multivariate linear regression models and covariance selection models, facilitating the estimation of its parameters. In the general case, necessary and sufficient conditions are given for the equivalence of the LWF and AMP Markov properties of a CG, for the AMP Markov equivalence of two CGs, for the AMP Markov equivalence of a CG to some ADG or decomposable UG, and for other equivalences. For CGs, in some ways the AMP property is a more direct extension of the ADG Markov property than is the LWF property.

Bayesian model averaging and model selection for markov equivalence classes of acyclic digraphs

Acyclic digraphs (ADGs) are widely used to describe dependences among variables in multivariate distributions. In particular, the likelihood functions of ADG models admit convenient recursive factorizations that often allow explicit maximum likelihood estimates and that are well suited to building Bayesian networks for expert systems. There may, however, be many ADGs that determine the same dependence (= Markov) model. Thus, the family of all ADGs with a given set of vertices is naturally partitioned into Markov-equivalence classes, each class being associated with a unique statistical model. Statistical procedures, such as model selection or model averaging, that fail to take into account these equivalence classes, may incur substantial computational or other inefficiencies. Recent results have shown that each Markov-equivalence class is uniquely determined by a single chain graph, the essential graph, that is itself Markov-equivalent simultaneously to all ADGs in the equivalence class. Here we propose t...

Power of the Noncentral F-Test: Effect of Additional Variates on Hotelling's T2-Test

Abstract We show that the power of the noncentral F-test increases with the number of denominator d.f. and decreases with the numerator d.f. Hence, increasing the dimension of the observed vectors may decrease the power of Hotellings T 2-test for means, unless the additional non-centrality provided by the extra variates is sufficiently large. A function determining the exact increase in noncentrality required for increased power is studied and tabulated We present procedures designed to test, on the basis of a preliminary sample, whether the increase in noncentrality is sufficiently large to justify inclusion of the extra variates.

The emergence of lineage-specific chromosomal topologies from coordinate gene regulation

Although the importance of chromosome organization during mitosis is clear, it remains to be determined whether the nucleus assumes other functionally relevant chromosomal topologies. We have previously shown that homologous chromosomes have a tendency to associate during hematopoiesis according to their distribution of coregulated genes, suggesting cell-specific nuclear organization. Here, using the mathematical approaches of distance matrices and coupled oscillators, we model the dynamic relationship between gene expression and chromosomal associations during the differentiation of a multipotential hematopoietic progenitor. Our analysis reveals dramatic changes in total genomic order: Commitment of the progenitor results in an initial increase in entropy at both the level of gene coregulation and chromosomal organization, which we suggest represents a phase transition, followed by a progressive decline in entropy during differentiation. The stabilization of a highly ordered state in the differentiated cell types results in lineage-specific chromosomal topologies and is related to the emergence of coherence—or self-organization—between chromosomal associations and coordinate gene regulation. We discuss how these observations may be generally relevant to cell fate decisions encountered by progenitor/stem cells.

Combining Independent Chi-Squared Tests

Abstract Classes of Bayes tests for combining n independent noncentral chi-squared statistics Ti ∼ x2 ki(θi) are derived, including the simple sum test based on Σ Ti , and are compared in power to the common “omnibus” procedures such as Fishers based on II Pi , the product of the attained significance levels. Linear Bayes statistics Σ biTi with appropriate weights bi are found to yield more powerful tests against prespecified alternatives (θ1, …, θ n ) than weighted Fisher procedures advocated by others, provided each ki , > 2. Over the range of alternatives considered, the test based on II Pi minimizes the maximum shortcoming in power relative to the other tests studied when each ki ≥ 2, while the sum test has this property when each ki = 1.

On the Markov Equivalence of Chain Graphs, Undirected Graphs, and Acyclic Digraphs

Graphical Markov models use undirected graphs (UDGs), acyclic directed graphs (ADGs), or (mixed) chain graphs to represent possible dependencies among random variables in a multivariate distribution. Whereas a UDG is uniquely determined by its associated Markov model, this is not true for ADGs or for general chain graphs (which include both UDGs and ADGs as special cases). This paper addresses three questions regarding the equivalence of graphical Markov models: when is a given chain graph Markov equivalent (1) to some UDG? (2) to some (at least one) ADG? (3) to some decomposable UDG? The answers are obtained by means of an extension of Frydenberg’s (1990) elegant graph-theoretic characterization of the Markov equivalence of chain graphs.

How Likely Is Simpson's Paradox?

What proportion of all 2×2×2 contingency tables exhibit Simpson’s Paradox? An exact answer is obtained for large sample sizes and extended to 2×2×ℓ tables by Monte Carlo approximation. Conditional probabilities of the occurrence of Simpson’s Paradox are also derived. If the observed cell proportions satisfy a Simpson reversal, the posterior probability that the population parameters satisfy the same reversal is obtained. This Bayesian analysis is applied to the well-known Simpson reversal of the 1995–1997 batting averages of Derek Jeter and David Justice.

The size distribution for Markov equivalence classes of acyclic digraph models

Bayesian networks, equivalently graphical Markov models determined by acyclic digraphs or ADGs (also called directed acyclic graphs or dags), have proved to be both effective and efficient for representing complex multivariate dependence structures in terms of local relations. However, model search and selection is potentially complicated by the many-to-one correspondence between ADGs and the statistical models that they represent. If the ADGs/models ratio is large, search procedures based on unique graphical representations of equivalence classes of ADGs could provide substantial computational efficiency. Hitherto, the value of the ADGs/models ratio has been calculated only for graphs with n = 5 or fewer vertices. In the present study, a computer program was written to enumerate the equivalence classes of ADG models and study the distributions of class sizes and number of edges for graphs up to n = 10 vertices. The ratio of ADGs to numbers of classes appears to approach an asymptote of about 3.7. Distributions of the classes according to number of edges and class size were produced which also appear to be approaching asymptotic limits. Imposing a bound on the maximum number of parents to any vertex causes little change if the bound is sufficiently large, with four being a possible minimum. The program also includes a new variation of orderly algorithm for generating undirected graphs.

Jensen's inequality for a convex vector-valued function on an infinite-dimensional space

Jensens inequality f(EX) ≤ Ef(X) for the expectation of a convex function of a random variable is extended to a generalized class of convex functions f whose domain and range are subsets of (possibly) infinite-dimensional linear topological spaces. Convexity of f is defined with respect to closed cone partial orderings, or more general binary relations, on the range of f. Two different methods of proof are given, one based on geometric properties of convex sets and the other based on the Strong Law of Large Numbers. Various conditions under which Jensens inequality becomes strict are studied. The relation between Jensens inequality and Fatous Lemma is examined.

On the relation between conditional independence models determined by finite distributive lattices and by directed acyclic graphs

The relations among the classes of multivariate conditional independence models determined by directed acyclic graphs (DAG), undirected graphs (UDG), decomposable graphs (DEC), and finite distributive lattices (LCI) are investigated. First, LCI models that admit positive joint densities are characterized in terms of an appropriate factorization of the density. This factorization is then recognized as a particular form of the recursive factorization that characterizes DAG models, thereby establishing that the LCI models comprise a subclass of the class of DAG models. Precisely, the class of LCI models coincides with the subclass of transitive DAG models. Furthermore, the class of LCI models has nontrivial intersection with the class of DEC models. A series of examples illustrating these relations are presented.