On rank correlation measures for non-continuous random variables

For continuous random variables, many dependence concepts and measures of association can be expressed in terms of the corresponding copula only and are thus independent of the marginal distributions. This interrelationship generally fails as soon as there are discontinuities in the marginal distribution functions. In this paper, we consider an alternative transformation of an arbitrary random variable to a uniformly distributed one. Using this technique, the class of all possible copulas in the general case is investigated. In particular, we show that one of its members—the standard extension copula introduced by Schweizer and Sklar—captures the dependence structures in an analogous way the unique copula does in the continuous case. Furthermore, we consider measures of concordance between arbitrary random variables and obtain generalizations of Kendall's tau and Spearman's rho that correspond to the sample version of these quantities for empirical distributions.     

Hierarchical subspace models for contingency tables

For the statistical analysis of multiway contingency tables, we propose modeling interaction terms in each maximal compact component of a hierarchical model. By this approach we can search for parsimonious models with smaller degrees of freedom than the usual hierarchical model, while preserving the localization property of the inference in the hierarchical model. This approach also enables us to evaluate the localization property of a given log-affine model. We discuss estimation and exact tests of the proposed model and illustrate the advantage of the proposed modeling with some data sets.     

From Archimedean to Liouville copulas

We use a recent characterization of the d-dimensional Archimedean copulas as the survival copulas of d-dimensional simplex distributions (McNeil and Nešlehová (2009) [1]) to construct new Archimedean copula families, and to examine the relationship between their dependence properties and the radial parts of the corresponding simplex distributions. In particular, a new formula for Kendall’s tau is derived and a new dependence ordering for non-negative random variables is introduced which generalises the Laplace transform order. We then generalise the Archimedean copulas to obtain Liouville copulas, which are the survival copulas of Liouville distributions and which are non-exchangeable in general. We derive a formula for Kendall’s tau of Liouville copulas in terms of the radial parts of the corresponding Liouville distributions.     

Mutual information,phi-squared and model-based co-clustering for contingency tables

Many of the datasets encountered in statistics are two-dimensional in nature and can be represented by a matrix. Classical clustering procedures seek to construct separately an optimal partition of rows or, sometimes, of columns. In contrast, co-clustering methods cluster the rows and the columns simultaneously and organize the data into homogeneous blocks (after suitable permutations). Methods of this kind have practical importance in a wide variety of applications such as document clustering, where data are typically organized in two-way contingency tables. Our goal is to offer coherent frameworks for understanding some existing criteria and algorithms for co-clustering contingency tables, and to propose new ones. We look at two different frameworks for the problem of co-clustering. The first involves minimizing an objective function based on measures of association and in particular on phi-squared and mutual information. The second uses a model-based co-clustering approach, and we consider two models: the block model and the latent block model. We establish connections between different approaches, criteria and algorithms, and we highlight a number of implicit assumptions in some commonly used algorithms. Our contribution is illustrated by numerical experiments on simulated and real-case datasets that show the relevance of the presented methods in the document clustering field.     

Probability matrices, non-negative rank, and parameterization of mixture models

In this paper, we parameterize non-negative matrices of sum one and rank at most two using the least possible number of parameters. We also show how this parameterization relates to a class of statistical models, known in Probability and Statistics as mixture models for contingency tables. In particular, we show how to use this parameterization to make some optimization problems computationally easier.     

A parametric approach to correspondence analysis

We compare correspondence analysis (CA) and the alternative approach using Hellinger distance (HD), for representing categorical data in a contingency table. As both methods may be appropriate, we introduce a parameter and define a generalized version of correspondence analysis (GCA) which contains CA and HD as particular cases. Comparison with alternative approaches are performed. We propose a coefficient which globally measures the similarity between CA and GCA, which can be decomposed into several components, one component for each principal dimension, indicating the contribution of the dimensions on the difference between both representations. Two criteria for choosing the best value of the parameter are proposed.     

Estimating the number of zero-one multi-way tables via sequential importance sampling

In 2005, Chen et al. introduced a sequential importance sampling (SIS) procedure to analyze zero-one two-way tables with given fixed marginal sums (row and column sums) via the conditional Poisson (CP) distribution. They showed that compared with Monte Carlo Markov chain (MCMC)-based approaches, their importance sampling method is more efficient in terms of running time and also provides an easy and accurate estimate of the total number of contingency tables with fixed marginal sums. In this paper, we extend their result to zero-one multi-way ( $d$ -way, $d \ge 2$ ) contingency tables under the no $d$ -way interaction model, i.e., with fixed $d-1$ marginal sums. Also, we show by simulations that the SIS procedure with CP distribution to estimate the number of zero-one three-way tables under the no three-way interaction model given marginal sums works very well even with some rejections. We also applied our method to Samson’s monks data set.     

Nonparametric tests for conditional independence in two-way contingency tables

Testing for the independence between two categorical variables R and S forming a contingency table is a well-known problem: the classical chi-square and likelihood ratio tests are used. Suppose now that for each individual a set of p characteristics is also observed. Those explanatory variables, likely to be associated with R and S, can play a major role in their possible association, and it can therefore be interesting to test the independence between R and S conditionally on them. In this paper, we propose two nonparametric tests which generalise the chi-square and the likelihood ratio ideas to this case. The procedure is based on a kernel estimator of the conditional probabilities. The asymptotic law of the proposed test statistics under the conditional independence hypothesis is derived; the finite sample behaviour of the procedure is analysed through some Monte Carlo experiments and the approach is illustrated with a real data example.     

A likelihood ratio test for separability of covariances

We propose a formal test of separability of covariance models based on a likelihood ratio statistic. The test is developed in the context of multivariate repeated measures (for example, several variables measured at multiple times on many subjects), but can also apply to a replicated spatio-temporal process and to problems in meteorology, where horizontal and vertical covariances are often assumed to be separable. Separable models are a common way to model spatio-temporal covariances because of the computational benefits resulting from the joint space-time covariance being factored into the product of a covariance function that depends only on space and a covariance function that depends only on time. We show that when the null hypothesis of separability holds, the distribution of the test statistic does not depend on the type of separable model. Thus, it is possible to develop reference distributions of the test statistic under the null hypothesis. These distributions are used to evaluate the power of the test for certain nonseparable models. The test does not require second-order stationarity, isotropy, or specification of a covariance model. We apply the test to a multivariate repeated measures problem.     

Singular value decomposition of large random matrices (for two-way classification of microarrays)

Asymptotic behavior of the singular value decomposition (SVD) of blown up matrices and normalized blown up contingency tables exposed to random noise is investigated. It is proved that such an m×n random matrix almost surely has a constant number of large singular values (of order ), while the rest of the singular values are of order as m,n→∞. We prove almost sure properties for the corresponding isotropic subspaces and for noisy correspondence matrices. An algorithm, applicable to two-way classification of microarrays, is also given that finds the underlying block structure.     

Relational models for contingency tables

The paper considers general multiplicative models for complete and incomplete contingency tables that generalize log-linear and several other models and are entirely coordinate free. Sufficient conditions for the existence of maximum likelihood estimates under these models are given, and it is shown that the usual equivalence between multinomial and Poisson likelihoods holds if and only if an overall effect is present in the model. If such an effect is not assumed, the model becomes a curved exponential family and a related mixed parameterization is given that relies on non-homogeneous odds ratios. Several examples are presented to illustrate the properties and use of such models.     

Markov bases and subbases for bounded contingency tables

In this paper we study the computation of Markov bases for contingency tables whose cell entries have an upper bound. It is known that in this case one has to compute universal Gröbner bases, and this is often infeasible also in small- and medium-sized problems. Here we focus on bounded two-way contingency tables under independence model. We show that when these bounds on cells are positive the set of basic moves of all 2 × 2 minors connects all tables with given margins. We also give some results about bounded incomplete table and we conclude with an open problem on the necessary and sufficient condition on the set of structural zeros so that the set of basic moves of all 2 × 2 minors connects all incomplete contingency tables with given positive margins.     

Conditional Inference on Tables With Structural Zeros

We develop a set of sequential importance sampling (SIS) strategies for sampling nearly uniformly from two-way zero-one or contingency tables with fixed marginal sums and a given set of structural zeros. The SIS procedure samples tables column by column or cell by cell by using appropriate proposal distributions, and enables us to approximate closely the null distributions of a number of test statistics involved in such tables. When structural zeros are on the diagonal or follow certain patterns, more efficient SIS algorithms are developed which guarantee that every generated table is valid. Examples show that our methods can be applied to make conditional inference on zero-one and contingency tables, and are more efficient than other existing Monte Carlo algorithms.     

Concordance measures for multivariate non-continuous random vectors

A notion of multivariate concordance suitable for non-continuous random variables is defined and many of its properties are established. This allows the definition of multivariate, non-continuous versions of Kendall’s tau, Spearman’s rho and Spearman’s footrule, which are concordance measures. Since the maximum values of these association measures are not +1 in general, a special attention is given to the computation of upper bounds. The latter turn out to be multivariate generalizations of earlier findings made by Nešlehová (2007) [9] and Denuit and Lambert (2005) [2]. They are easy to compute and can be estimated from a data set of (possibly) discontinuous random vectors. Corrected versions are considered as well.     

Latent models for cross-covariance

We consider models for the covariance between two blocks of variables. Such models are often used in situations where latent variables are believed to present. In this paper we characterize exactly the set of distributions given by a class of models with one-dimensional latent variables. These models relate two blocks of observed variables, modeling only the cross-covariance matrix. We describe the relation of this model to the singular value decomposition of the cross-covariance matrix. We show that, although the model is underidentified, useful information may be extracted. We further consider an alternative parameterization in which one latent variable is associated with each block, and we extend the result to models with r-dimensional latent variables.     

Existence of MLEs for discrete linear exponential models

Necessary and sufficient conditions are given for the existence of an MLE for log-linear and regression models for contingency tables. A partial compactification of the parameter space is used to elucidate a more abstract compactification given by Lauritzen [14]. A modification of the Newton-Raphson approximation yields MLEs in the partial compactification. This work was done while the author was Assistant Professor, Department of Mathematics, Arizona State University, Tempe 85281.     

A VIEW OF THE CHI-SQUARE STATISTIC

Statistical analysis of contingency tables is essentially a discrete multivariate problem. To test independence in a two-way contingency table, one can use different principles such as general class of distance or Mahalanobis distance to derive test statistic. Most of them result in the chi-square statistic, which is a simple test statistic. The disadvantage of the chi-square test is discussed from a multivariate point of view. A new test statistic is proposed for testing independence. This statistic is more sensitive to dependence than the chi-square statistic.     

A Non-Iterative Alternative to Ordinal Log-Linear Models

Log-linear modeling is a popular statistical tool for analysing a contingency table. This presentation focuses on an alternative approach to modeling ordinal categorical data. The technique, based on orthogonal polynomials, provides a much simpler method of model fitting than the conventional approach of maximum likelihood estimation, as it does not require iterative calculations nor the fitting and refitting to search for the best model. Another advantage is that quadratic and higher order effects can readily be included, in contrast to conventional log-linear models which incorporate linear terms only.

The focus of the discussion is the application of the new parameter estimation technique to multi-way contingency tables with at least one ordered variable. This will also be done by considering singly and doubly ordered two-way contingency tables. It will be shown by example that the resulting parameter estimates are numerically similar to corresponding maximum likelihood estimates for ordinal log-linear models.     

An estimation method for the Neyman chi-square divergence with application to test of hypotheses

We propose a new definition of the Neyman chi-square divergence between distributions. Based on convexity properties and duality, this version of the χ² is well suited both for the classical applications of the χ² for the analysis of contingency tables and for the statistical tests in parametric models, for which it is advocated to be robust against outliers.We present two applications in testing. In the first one, we deal with goodness-of-fit tests for finite and infinite numbers of linear constraints; in the second one, we apply χ²-methodology to parametric testing against contamination.     

Characterizations of multivariate life distributions

Characterizations of multivariate distributions has been a topic of great interest in applied statistics literature for the last three decades. In this paper, we develop characterizations of multivariate lifetime distributions by relationship between multivariate failure rates (reversed failure rates) and the left (right) truncated expectations of functions of random variables. We, then, discuss the application of the results to derive a multivariate Stein type identity.