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

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

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

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

Effects of an unobserved confounder on a system with an intermediate outcome

In der Theorie graphischer Markov Modelle, in denen Beziehungen zwischen vielen Variablen über konditionale Interdependenzen vereinfacht werden, spielen azyklische Graphen eine spezielle Rolle. Sie können dazu benutzt werden, um statistische Modelle zu representieren, in denen Daten schrittweise generiert werden. Responses und intermediäre Variablen können event histories sein . Wir diskutieren ein derartiges System mit sequentieller Behandlung und einem Confounder, das ist eine Variable mit Auswirkungen auf den endgültigen Output und eine der erklärenden Vaiablen. Es werden Verfahren aufgezeigt, wie mit diesem Problem umgegangen werden kann. (Lo)'In the theory of graphical Markov models in which relations between many variables are simplified via conditional independencies a special rote is played by directed acyclic graphs. They can be used to represent statistical models in which data are generated in a stepwise fashion. Responses and intermediate variables may be event histories. We discuss such a system with sequentially administered treatments and a confounder, that is a variable which affects both the final outcome and one of its explanatory variables. The effect of not observing the confounder is to obtain the final and an intermediate outcome as Joint responses and leads to the important observation by Robins and Wasserman (1997) that any univariate conditional distribution for the final outcome will be inappropriate for analysis no matter whether the intermediate outcome is conditioned on or not. It means in particular that the independence structure of the observed variables can no longer be fully described by a directed acyclic graph, that criteria for reading independencies off graphs have to be modified and that joint instead of univariate regression models are needed. These modifications resolve directly the puzzling situation which has been discussed by the above authors for randomized clinical trials as a case in which a true hypothesis of no treatment effect is always falsely rejected. Joint response models provide an alternative route for avoiding this unpleasant situation.' (author's abstract)

A sweep algorithm for triangular matrices and its statistical applications

Ein Sweepoperator ist definiert für schrittweise sequentielle Invertierung von triangulären Matrizen und ihre Werte werden verglichen mit denen des Sweepoperators zur Invertierung symmetrischer Matrizen. Der Algorithus wird angewendet, um Folgeverteilungen zu untersuchen, die über einen azyklischen Graphen gebildet wurden. Drei Hauptanwendungen werden abgeleitet. Die erste dient der Prüfung einer einfachen Form für die Folgeverteilung, die nach Marginalisierung über und Konditionalisierung von beliebigen Variablensubsets in solch einem linearen System resultiert. Die zweite Anwendung betrifft die Ausweitung der Ergebnisse für lineare Systeme auf generelle Verteilungen durch die Interpretation von strukturellen Nullen in den Matrizen durch Ausdrücke für fehlende Ecken in assoziierten Graphen und symbolische Matrixtransformationen als Grafenmodifikation. Die dritte dient dem Nachweis der Equivalenz von mehreren Kriterien für das Herauslesen von Unabhängigkeitsfeststellungen aus gerichteten azyklischen Graphen. (OH)A sweep operator is defined for stepwise sequential inversion of triangular matrices and its properties are compared to those of the sweep operator for inverting symmetric matrices. The algorithm is used to study joint distributions generated over a directed acyclic graph. Three main applications are derived. The first is to prove a simple form for the joint distribution resulting after marginalising over and conditioning on arbitrary subsets of variables in such a linear system. The second is to extend the results for linear systems to general distributions by interpreting structural zeros in matrices in terms of missing edges in associated graphs and symbolic matrix transformations as modifications of graphs. The third is to show the equivalence of several criteria for reading off independence statements from directed acyclic graphs.' (author's abstract)