Graphical Markov models: overview

We describe how graphical Markov models started to emerge in the last 40 years, based on three essential concepts that had been developed independently more than a century ago. Sequences of joint or single regressions and their regression graphs are singled out as being best suited for analyzing longitudinal data and for tracing developmental pathways. Interpretations are illustrated using two sets of data and some of the more recent, important results for sequences of regressions are summarized.Comment: 22 pages, 9 figure

Sequences of regressions and their independences

Ordered sequences of univariate or multivariate regressions provide statistical models for analysing data from randomized, possibly sequential interventions, from cohort or multi-wave panel studies, but also from cross-sectional or retrospective studies. Conditional independences are captured by what we name regression graphs, provided the generated distribution shares some properties with a joint Gaussian distribution. Regression graphs extend purely directed, acyclic graphs by two types of undirected graph, one type for components of joint responses and the other for components of the context vector variable. We review the special features and the history of regression graphs, derive criteria to read all implied independences of a regression graph and prove criteria for Markov equivalence that is to judge whether two different graphs imply the same set of independence statements. Knowledge of Markov equivalence provides alternative interpretations of a given sequence of regressions, is essential for machine learning strategies and permits to use the simple graphical criteria of regression graphs on graphs for which the corresponding criteria are in general more complex. Under the known conditions that a Markov equivalent directed acyclic graph exists for any given regression graph, we give a polynomial time algorithm to find one such graph.Comment: 43 pages with 17 figures The manuscript is to appear as an invited discussion paper in the journal TES

Graphical Markov models, unifying results and their interpretation

Graphical Markov models combine conditional independence constraints with graphical representations of stepwise data generating processes.The models started to be formulated about 40 years ago and vigorous development is ongoing. Longitudinal observational studies as well as intervention studies are best modeled via a subclass called regression graph models and, especially traceable regressions. Regression graphs include two types of undirected graph and directed acyclic graphs in ordered sequences of joint responses. Response components may correspond to discrete or continuous random variables and may depend exclusively on variables which have been generated earlier. These aspects are essential when causal hypothesis are the motivation for the planning of empirical studies. To turn the graphs into useful tools for tracing developmental pathways and for predicting structure in alternative models, the generated distributions have to mimic some properties of joint Gaussian distributions. Here, relevant results concerning these aspects are spelled out and illustrated by examples. With regression graph models, it becomes feasible, for the first time, to derive structural effects of (1) ignoring some of the variables, of (2) selecting subpopulations via fixed levels of some other variables or of (3) changing the order in which the variables might get generated. Thus, the most important future applications of these models will aim at the best possible integration of knowledge from related studies.Comment: 34 Pages, 11 figures, 1 tabl

Concepts and a case study for a flexible class of graphical Markov models

With graphical Markov models, one can investigate complex dependences, summarize some results of statistical analyses with graphs and use these graphs to understand implications of well-fitting models. The models have a rich history and form an area that has been intensively studied and developed in recent years. We give a brief review of the main concepts and describe in more detail a flexible subclass of models, called traceable regressions. These are sequences of joint response regressions for which regression graphs permit one to trace and thereby understand pathways of dependence. We use these methods to reanalyze and interpret data from a prospective study of child development, now known as the Mannheim Study of Children at Risk. The two related primary features concern cognitive and motor development, at the age of 4.5 and 8 years of a child. Deficits in these features form a sequence of joint responses. Several possible risks are assessed at birth of the child and when the child reached age 3 months and 2 years.Comment: 21 pages, 7 figures, 7 tables; invited, refereed chapter in a boo

Star graphs induce tetrad correlations: for Gaussian as well as for binary variables

Tetrad correlations were obtained historically for Gaussian distributions when tasks are designed to measure an ability or attitude so that a single unobserved variable may generate the observed, linearly increasing dependences among the tasks. We connect such generating processes to a particular type of directed graph, the star graph, and to the notion of traceable regressions. Tetrad correlation conditions for the existence of a single latent variable are derived. These are needed for positive dependences not only in joint Gaussian but also in joint binary distributions. Three applications with binary items are given.Comment: 21 pages, 2 figures, 5 table

Matrix representations and independencies in directed acyclic graphs

For a directed acyclic graph, there are two known criteria to decide whether any specific conditional independence statement is implied for all distributions factorized according to the given graph. Both criteria are based on special types of path in graphs. They are called separation criteria because independence holds whenever the conditioning set is a separating set in a graph theoretical sense. We introduce and discuss an alternative approach using binary matrix representations of graphs in which zeros indicate independence statements. A matrix condition is shown to give a new path criterion for separation and to be equivalent to each of the previous two path criteria.Comment: Published in at http://dx.doi.org/10.1214/08-AOS594 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org