Collapsibility for Directed Acyclic Graphs

Abstract.  Collapsibility means that the same statistical result of interest can be obtained before and after marginalization over some variables. In this paper, we discuss three kinds of collapsibility for directed acyclic graphs (DAGs): estimate collapsibility, conditional independence collapsibility and model collapsibility. Related to collapsibility, we discuss removability of variables from a DAG. We present conditions for these three different kinds of collapsibility and relationships among them. We give algorithms to find a minimum variable set containing a variable subset of interest onto which a statistical result is collapsible.     

Collapsibility of Graphical CG-Regression Models

Abstract.  CG-regressions are multivariate regression models for mixed continuous and discrete responses that result from conditioning in the class of conditional Gaussian (CG) models. Their conditional independence structure can be read off a marked graph. The property of collapsibility, in this context, means that the multivariate CG-regression can be decomposed into lower dimensional regressions that are still CG and are consistent with the corresponding subgraphs. We derive conditions for this property that can easily be checked on the graph, and indicate computational advantages of this kind of collapsibility. Further, a simple graphical condition is given for checking whether a decomposition into univariate regressions is possible.     

A Note on Collapsibility in DAG Models of Contingency Tables

Abstract.  Necessary and sufficient conditions for collapsibility of a directed acyclic graph (DAG) model for a contingency table are derived. By applying the conditions, we can easily check collapsibility over any variable in a given model either by using the joint probability distribution or by using the graph of the model structure. It is shown that collapsibility over a set of variables can be checked in a sequential manner. Furthermore, a DAG is compared with its moral graph in the context of collapsibility.     

Average Collapsibility of Distribution Dependence and Quantile Regression Coefficients

Abstract. The Yule–Simpson paradox notes that an association between random variables can be reversed when averaged over a background variable. Cox and Wermuth introduced the concept of distribution dependence between two random variables X and Y, and gave two dependence conditions, each of which guarantees that reversal of qualitatively similar conditional dependences cannot occur after marginalizing over the background variable. Ma, Xie and Geng studied the uniform collapsibility of distribution dependence over a background variable W, under stronger homogeneity condition. Collapsibility ensures that associations are the same for conditional and marginal models. In this article, we use the notion of average collapsibility, which requires only the conditional effects average over the background variable to the corresponding marginal effect and investigate its conditions for distribution dependence and for quantile regression coefficients.     

Log‐mean Linear Parameterization for Discrete Graphical Models of Marginal Independence and the Analysis of Dichotomizations

We extend the log‐mean linear parameterization for binary data to discrete variables with arbitrary number of levels and show that also in this case it can be used to parameterize bi‐directed graph models. Furthermore, we show that the log‐mean linear parameterization allows one to simultaneously represent marginal independencies among variables and marginal independencies that only appear when certain levels are collapsed into a single one. We illustrate the application of this property by means of an example based on genetic association studies involving single‐nucleotide polymorphisms. More generally, this feature provides a natural way to reduce the parameter count, while preserving the independence structure, by means of substantive constraints that give additional insight into the association structure of the variables. © 2014 Board of the Foundation of the Scandinavian Journal of Statistics     

Conditions for Non-confounding and Collapsibility without Knowledge of Completely Constructed Causal Diagrams

In this paper, we discuss several concepts in causal inference in terms of causal diagrams proposed by Pearl (1993 , 1995a , b ), and we give conditions for non-confounding, homogeneity and collapsibility for causal effects without knowledge of a completely constructed causal diagram. We first introduce the concepts of non-confounding, conditional non-confounding, uniform non-confounding, homogeneity, collapsibility and strong collapsibility for causal effects, then we present necessary and sufficient conditions for uniform non-confounding, homegeneity and collapsibilities, and finally we show sufficient conditions for non-confounding, conditional non-confounding and uniform non-confounding.     

Statistical Inference in Context Specific Interaction Models for Contingency Tables

Abstract.  Context specific interaction models is a class of interaction models for contingency tables in which interaction terms are allowed to vanish in specific contexts given by the levels of sets of variables. Such restrictions can entail conditional independencies which only hold for some values of the conditioning variables and allows also for irrelevance of some variables in specific contexts. A Markov property is established and so is an iterative proportional scaling algorithm for maximum likelihood estimation. Decomposition of the estimation problem is treated and model selection is discussed.     

Labelled Graphical Models

A class of log‐linear models, referred to as labelled graphical models (LGMs), is introduced for multinomial distributions. These models generalize graphical models (GMs) by employing partial conditional independence restrictions which are valid only in subsets of an outcome space. Theoretical results concerning model identifiability, decomposability and estimation are derived. A decision theoretical framework and a search algorithm for the identification of plausible models are described. Real data sets are used to illustrate that LGMs may provide a simpler interpretation of a dependence structure than GMs.     

Quasi‐Symmetric Graphical Log‐Linear Models

Abstract. We propose an extension of graphical log‐linear models to allow for symmetry constraints on some interaction parameters that represent homologous factors. The conditional independence structure of such quasi‐symmetric (QS) graphical models is described by an undirected graph with coloured edges, in which a particular colour corresponds to a set of equality constraints on a set of parameters. Unlike standard QS models, the proposed models apply with contingency tables for which only some variables or sets of the variables have the same categories. We study the graphical properties of such models, including conditions for decomposition of model parameters and of maximum likelihood estimates.     

基于图方法的MA和ARMA模型系数检验

应用图模型方法来讨论传统的MA和ARMA模型,证明了MA和ARMA模型的系数为去掉其他时间序列分量线性效应的条件下的偏相关系数,且利用图模型推断算法提出了一种新的参数估计和检验方法。     

Collapsibility of contingency tables based on conditional models

Strict collapsibility and model collapsibility are two important concepts associated with the dimension reduction of a multidimensional contingency table, without losing the relevant information. In this paper, we obtain some necessary and sufficient conditions for the strict collapsibility of the full model, with respect to an interaction factor or a set of interaction factors, based on the interaction parameters of the conditional/layer log-linear models. For hierarchical log-linear models, we present also necessary and sufficient conditions for the full model to be model collapsible, based on the conditional interaction parameters. We discuss both the cases where one variable or a set of variables is conditioned. The connections between the strict collapsibility and the model collapsibility are also pointed out. Our results are illustrated through suitable examples, including a real life application.     

Fitting DNA sequences through log-linear modelling with linear constraints

For some discrete state series, such as DNA sequences, it can often be postulated that its probabilistic behaviour is given by a Markov chain. For making the decision on whether or not an uncharacterized piece of DNA is part of the coding region of a gene, under the Markovian assumption, there are two statistical tools that are essential to be considered: the hypothesis testing of the order in a Markov chain and the estimators of transition probabilities. In order to improve the traditional statistical procedures for both of them when stationarity assumption can be considered, a new version for understanding the homogeneity hypothesis is proposed so that log-linear modelling is applied for conditional independence jointly with homogeneity restrictions on the expected means of transition counts in the sequence. In addition we can consider a variety of test-statistics and estimators by using φ-divergence measures. As special case of them the well-known likelihood ratio test-statistics and maximum-likelihood estimators are obtained.     

Conditional Prior Proposals in Dynamic Models

ABSTRACT. Dynamic models extend state space models to non-normal observations. This paper suggests a specific hybrid Metropolis–Hastings algorithm as a simple device for Bayesian inference via Markov chain Monte Carlo in dynamic models. Hastings proposals from the (conditional) prior distribution of the unknown, time-varying parameters are used to update the corresponding full conditional distributions. It is shown through simulated examples that the methodology has optimal performance in situations where the prior is relatively strong compared to the likelihood. Typical examples include smoothing priors for categorical data. A specific blocking strategy is proposed to ensure good mixing and convergence properties of the simulated Markov chain. It is also shown that the methodology is easily extended to robust transition models using mixtures of normals. The applicability is illustrated with an analysis of a binomial and a binary time series, known in the literature.     

A Graphical Diagnostic for Identifying Influential Model Choices in Bayesian Hierarchical Models

Abstract. Real‐world phenomena are frequently modelled by Bayesian hierarchical models. The building‐blocks in such models are the distribution of each variable conditional on parent and/or neighbour variables in the graph. The specifications of centre and spread of these conditional distributions may be well motivated, whereas the tail specifications are often left to convenience. However, the posterior distribution of a parameter may depend strongly on such arbitrary tail specifications. This is not easily detected in complex models. In this article, we propose a graphical diagnostic, the Local critique plot, which detects such influential statistical modelling choices at the node level. It identifies the properties of the information coming from the parents and neighbours (the local prior) and from the children and co‐parents (the lifted likelihood) that are influential on the posterior distribution, and examines local conflict between these distinct information sources. The Local critique plot can be derived for all parameters in a chain graph model.     

Conditional Difference Asymmetry Model for Square Contingency Tables with Nominal Categories

This paper proposes a model, which is an extension-of-symmetry model, for square contingency tables with the same nominal row and column classifications. The model states that the absolute values of difference between the conditional probability that an observation will fall in cell (i, j) on condition that it falls in cell (i, j) or (j, i) and the conditional probability that it falls in cell (j, i) on the same condition, are constant for every i≠j. The model describes a structure of asymmetry (not symmetry), and it is applied to the data on a nominal scale. An example is given.     

Graphical models for skew-normal variates

This paper explores the usefulness of the multivariate skew-normal distribution in the context of graphical models. A slight extension of the family recently discussed by Azzalini & Dalla Valle (1996 ) and Azzalini & Capitanio (1999 ) is described, the main motivation being the additional property of closure under conditioning. After considerations of the main probabilistic features, the focus of the paper is on the construction of conditional independence graphs for skew-normal variables. Necessary and sufficient conditions for conditional independence are stated, and the admissible structures of a graph under restriction on univariate marginal distribution are studied. Finally, parameter estimation is considered. It is shown how the factorization of the likelihood function according to a graph can be rearranged in order to obtain a parameter based factorization.     

Objective Bayes Factors for Gaussian Directed Acyclic Graphical Models

Abstract. We propose an objective Bayesian method for the comparison of all Gaussian directed acyclic graphical models defined on a given set of variables. The method, which is based on the notion of fractional Bayes factor (BF), requires a single default (typically improper) prior on the space of unconstrained covariance matrices, together with a prior sample size hyper‐parameter, which can be set to its minimal value. We show that our approach produces genuine BFs. The implied prior on the concentration matrix of any complete graph is a data‐dependent Wishart distribution, and this in turn guarantees that Markov equivalent graphs are scored with the same marginal likelihood. We specialize our results to the smaller class of Gaussian decomposable undirected graphical models and show that in this case they coincide with those recently obtained using limiting versions of hyper‐inverse Wishart distributions as priors on the graph‐constrained covariance matrices.     

Functionally Compatible Local Characteristics for the Local Specification of Priors in Graphical Models

Abstract.  The local specification of priors in non-decomposable graphical models does not necessarily yield a proper joint prior for all the parameters of the model. Using results concerning general exponential families with cuts, we derive specific results for the multivariate Gamma distribution (conjugate prior for Poisson counts) and the Wishart distribution (conjugate prior for Gaussian models). These results link the existence of a locally specified joint prior to the solvability of a related marginal problem over the cliques of the graph.     

Graphical Models for Composable Finite Markov Processes

Abstract.  Composable Markov processes were introduced by Schweder (1970) in order to capture the idea that a process can be composed of different components where some of these only depend on a subset of the other components. Here we propose a graphical representation of this kind of dependence which has been called 'local dependence'. It is shown that the graph allows to read off further independencies characterizing the underlying Markov process. Also, some standard methods for inference are adapted to exploit the graphical representation, e.g. for testing local independence.     

A Goodness-of-Fit Test for a Class of Autoregressive Conditional Duration Models

This article develops a method for testing the goodness-of-fit of a given parametric autoregressive conditional duration model against unspecified nonparametric alternatives. The test statistics are functions of the residuals corresponding to the quasi maximum likelihood estimate of the given parametric model, and are easy to compute. The limiting distributions of the test statistics are not free from nuisance parameters. Hence, critical values cannot be tabulated for general use. A bootstrap procedure is proposed to implement the tests, and its asymptotic validity is established. The finite sample performances of the proposed tests and several other competing ones in the literature, were compared using a simulation study. The tests proposed in this article performed well consistently throughout, and they were either the best or close to the best. None of the tests performed uniformly the best. The tests are illustrated using an empirical example.