Collapsibility of Conditional Graphical Models

Abstract. In this paper, we consider two kinds of collapsibility, that is, the model‐collapsibility and the estimate‐collapsibility, of conditional graphical models for multidimensional contingency tables. We show that these two definitions are equivalent, and propose a sufficient and necessary condition for them in terms of the interaction graph, which allows the collapsibility to be characterized and judged intuitively and conveniently.     

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.     

Relationships Between Full and Layer Models with Applications to Level Merging

Analysis of a large dimensional contingency table is quite involved. Models corresponding to layers of a contingency table are easier to analyze than the full model. Relationships between the interaction parameters of the full log-linear model and that of its corresponding layer models are obtained. These relationships are not only useful to reduce the analysis but also useful to interpret various hierarchical models. We obtain these relationships for layers of one variable, and extend the results for the case when layers of more than one variable are considered. We also establish, under conditional independence, relationships between the interaction parameters of the full model and that of the corresponding marginal models. We discuss the concept of merging of factor levels based on these interaction parameters. Finally, we use the relationships between layer models and full model to obtain conditions for level merging based on layer interaction parameters. Several examples are discussed to illustrate the results.     

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.     

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.     

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.     

Whittemore's notion of collapsibility in multidimensional contingency tables

We develop simple necessary and sufficient conditions for a hierarchical log linear model to be strictly collapsible in the sense defined by Whittemore (1978). We then show that collapsibility as defined by Asmussen & Edwards (1983) can be viewed as equivalent to collapsibility as defined by Whittemore (1978) and illustrate why Bishop, Fienberg, & Holland's (1975, p.47) conditions for collapsibility are sufficient but not necessary. Finally, we discuss how collapsibility facilitates interpretation of certain hierarchical log linear models and formulation of hypotheses concerning marginal distributions associated with multidimensional contingency tables.     

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.     

A class of log-linear models with constrained marginal distributions

Summary In the log-linear model for bivariate probability functions the conditional and joint probabilities have a simple form. This property make the log-linear parametrization useful when modeling these probabilities is the focus of the investigation. On the contrary, in the log-linear representation of bivariate probability functions, the marginal probabilities have a complex form. So the log-linear models are not useful when the marginal probabilities are of particular interest. In this paper the previous statements are discussed and a model obtained from the log-linear one by imposing suitable constraints on the marginal probabilities is introduced. This work was supported by a M.U.R.S.T. grant.     

Disaggregated spatial modelling for areal unit categorical data

Summary.  We consider joint spatial modelling of areal multivariate categorical data assuming a multiway contingency table for the variables, modelled by using a log-linear model, and connected across units by using spatial random effects. With no distinction regarding whether variables are response or explanatory, we do not limit inference to conditional probabilities, as in customary spatial logistic regression. With joint probabilities we can calculate arbitrary marginal and conditional probabilities without having to refit models to investigate different hypotheses. Flexible aggregation allows us to investigate subgroups of interest; flexible conditioning enables not only the study of outcomes given risk factors but also retrospective study of risk factors given outcomes. A benefit of joint spatial modelling is the opportunity to reveal disparities in health in a richer fashion, e.g. across space for any particular group of cells, across groups of cells at a particular location, and, hence, potential space–group interaction. We illustrate with an analysis of birth records for the state of North Carolina and compare with spatial logistic regression.     

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.     

Algebraic exact inference for rater agreement models

In recent years, a method for sampling from conditional distributions for categorical data has been presented by Diaconis and Sturmfels. Their algorithm is based on the algebraic theory of toric ideals which are used to create so called “Markov Bases”. The Diaconis-Sturmfels algorithm leads to a non-asymptotic Monte Carlo Markov Chain algorithm for exact inference on some classes of models, such as log-linear models. In this paper we apply the Diaconis-Sturmfels algorithm to a set of models arising from the rater agreement problem with special attention to the multi-rater case. The relevant Markov bases are explicitly computed and some results for simplify the computation are presented. An extended example on a real data set shows the wide applicability of this methodology. Partially supported by MIUR Cofin03 (G. Consonni) and by INdAM projectAlgebraic Statistics.     

Classical multilevel and Bayesian approaches to population size estimation using multiple lists

One of the major objections to the standard multiple-recapture approach to population estimation is the assumption of homogeneity of individual 'capture' probabilities. Modelling individual capture heterogeneity is complicated by the fact that it shows up as a restricted form of interaction among lists in the contingency table cross-classifying list memberships for all individuals. Traditional log-linear modelling approaches to capture–recapture problems are well suited to modelling interactions among lists but ignore the special dependence structure that individual heterogeneity induces. A random-effects approach, based on the Rasch model from educational testing and introduced in this context by Darroch and co-workers and Agresti, provides one way to introduce the dependence resulting from heterogeneity into the log-linear model; however, previous efforts to combine the Rasch-like heterogeneity terms additively with the usual log-linear interaction terms suggest that a more flexible approach is required. In this paper we consider both classical multilevel approaches and fully Bayesian hierarchical approaches to modelling individual heterogeneity and list interactions. Our framework encompasses both the traditional log-linear approach and various elements from the full Rasch model. We compare these approaches on two examples, the first arising from an epidemiological study of a population of diabetics in Italy, and the second a study intended to assess the 'size' of the World Wide Web. We also explore extensions allowing for interactions between the Rasch and log-linear portions of the models in both the classical and the Bayesian contexts.     

Bayesian Model Choice in Exponential Survival Models

ABSTRACT

Log-linear models for the distribution on a contingency table are represented as the intersection of only two kinds of log-linear models. One assuming that a certain group of the variables, if conditioned on all other variables, has a jointly independent distribution and another one assuming that a certain group of the variables, if conditioned on all other variables, has no highest order interaction. The subsets entering into these models are uniquely determined by the original log-linear model. This canonical representation suggests considering joint conditional independence and conditional no highest order association as the elementary building blocks of log-linear models.     

The Identifiability of Dependent Competing Risks Models Induced by Bivariate Frailty Models

In this paper, we propose to use a special class of bivariate frailty models to study dependent censored data. The proposed models are closely linked to Archimedean copula models. We give sufficient conditions for the identifiability of this type of competing risks models. The proposed conditions are derived based on a property shared by Archimedean copula models and satisfied by several well‐known bivariate frailty models. Compared with the models studied by Heckman and Honoré and Abbring and van den Berg, our models are more restrictive but can be identified with a discrete (even finite) covariate. Under our identifiability conditions, expectation–maximization (EM) algorithm provides us with consistent estimates of the unknown parameters. Simulation studies have shown that our estimation procedure works quite well. We fit a dependent censored leukaemia data set using the Clayton copula model and end our paper with some discussions. © 2014 Board of the Foundation of the Scandinavian Journal of Statistics     

Estimation and Asymptotic Properties in Periodic GARCH(1, 1) Models

This article studies the probabilistic structure and asymptotic inference of the first-order periodic generalized autoregressive conditional heteroscedasticity (PGARCH(1, 1)) models in which the parameters in volatility process are allowed to switch between different regimes. First, we establish necessary and sufficient conditions for a PGARCH(1, 1) process to have a unique stationary solution (in periodic sense) and for the existence of moments of any order. Second, using the representation of squared PGARCH(1, 1) model as a PARMA(1, 1) model, we then consider Yule-Walker type estimators for the parameters in PGARCH(1, 1) model and derives their consistency and asymptotic normality. The estimator can be surprisingly efficient for quite small numbers of autocorrelations and, in some cases can be more efficient than the least squares estimate (LSE). We use a residual bootstrap to define bootstrap estimators for the Yule-Walker estimates and prove the consistency of this bootstrap method. A set of numerical experiments illustrates the practical relevance of our theoretical results.     

Bayesian inference for Poisson and multinomial log-linear models

Categorical data frequently arise in applications in the Social Sciences. In such applications, the class of log-linear models, based on either a Poisson or (product) multinomial response distribution, is a flexible model class for inference and prediction. In this paper we consider the Bayesian analysis of both Poisson and multinomial log-linear models. It is often convenient to model multinomial or product multinomial data as observations of independent Poisson variables. For multinomial data, Lindley (1964) [20] showed that this approach leads to valid Bayesian posterior inferences when the prior density for the Poisson cell means factorises in a particular way. We develop this result to provide a general framework for the analysis of multinomial or product multinomial data using a Poisson log-linear model. Valid finite population inferences are also available, which can be particularly important in modelling social data. We then focus particular attention on multivariate normal prior distributions for the log-linear model parameters. Here, an improper prior distribution for certain Poisson model parameters is required for valid multinomial analysis, and we derive conditions under which the resulting posterior distribution is proper. We also consider the construction of prior distributions across models, and for model parameters, when uncertainty exists about the appropriate form of the model. We present classes of Poisson and multinomial models, invariant under certain natural groups of permutations of the cells. We demonstrate that, if prior belief concerning the model parameters is also invariant, as is the case in a ‘reference’ analysis, then the choice of prior distribution is considerably restricted. The analysis of multivariate categorical data in the form of a contingency table is considered in detail. We illustrate the methods with two examples.     

Sufficiency of variables in discrete discriminant analysis

In discrete discriminant analysis the high dimensional estimation problem makes it necessary to restrict oneself to the most effective variables. Conditions are derived to determine whether subsets of variables yield the same optimal allocation rule as the original set. In the discrete case the conditions turn out to be sufficient but not necessary. Tests are derived in the framework of log-linear models. Based on the concept of adequate (α) discriminant sets two variable selection procedures are considered.     

Algebraic Markov Bases and MCMC for Two-Way Contingency Tables

ABSTRACT.  The Diaconis–Sturmfels algorithm is a method for sampling from conditional distributions, based on the algebraic theory of toric ideals. This algorithm is applied to categorical data analysis through the notion of Markov basis. An application of this algorithm is a non-parametric Monte Carlo approach to the goodness of fit tests for contingency tables. In this paper, we characterize or compute the Markov bases for some log-linear models for two-way contingency tables using techniques from Computational Commutative Algebra, namely Gröbner bases. This applies to a large set of cases including independence, quasi-independence, symmetry, quasi-symmetry. Three examples of quasi-symmetry and quasi-independence from Fingleton ( Models of category counts , Cambridge University Press, Cambridge, 1984) and Agresti ( An Introduction to categorical data analysis , Wiley, New York, 1996) illustrate the practical applicability and the relevance of this algebraic methodology.