Bayesian Analysis of Contingency Tables

ABSTRACT

The display of the data by means of contingency tables is used in different approaches to statistical inference, for example, to broach the test of homogeneity of independent multinomial distributions. We develop a Bayesian procedure to test simple null hypotheses versus bilateral alternatives in contingency tables. Given independent samples of two binomial distributions and taking a mixed prior distribution, we calculate the posterior probability that the proportion of successes in the first population is the same as in the second. This posterior probability is compared with the p-value of the classical method, obtaining a reconciliation between both results, classical and Bayesian. The obtained results are generalized for r × s tables.     

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.     

Analysis of Contingency Tables with Clustered Observations

A new approach is presented for testing independence in contingency tables with clustered observations. The approach is based on the framework of generalized linear mixed models. Under the multinomial logistic link function, the category counts are modelled with random cluster effects and a modified likelihood ratio statistic is used for testing independence. The method is applicable to multi-way tables, and can accommodate multiple levels of clustering. It is illustrated using a benchmark dataset.     

Cumulative Correspondence Analysis of Two-Way Ordinal Contingency Tables

A suitable measure of association for two ordered variables is the doubly cumulative chi-squared statistic (Hirotsu, 1994 Hirotsu , C. ( 1994 ). Modelling and analysing the generalized interaction . Proc. Third IEEE Conf. Control Applic. 2 : 1283 – 1288 .[Crossref] , [Google Scholar]). This statistic is obtained by considering the cumulative sum of cell frequencies across the variables. In this article, we explore the development of correspondence analysis which takes into account the presence of two ordered variables by partitioning the doubly cumulative chi-squared statistic.     

属性统计与列联表分析（Ⅱ）

四、列联表的分割§4-1 X~2的分解与列联表分割 如果一个r×c列联表的变量独立性的假设经检验遭到拒绝,说明所涉及的两个变量之间存在一定的关联。在2×k表情形,这也等价于k个率之间有显著性差异。但以上结论是指表的整体而言的,并不意味着两个变量的所有分类都存在着关联或任意两个率之间都存在显著性差异。完全有可能变量之间的关联性仅存在于某些类别(或某些率)之间。这正如在一     

Marginal Nested Interactions for Contingency Tables

We introduce a new definition of generalized marginal interactions, called marginal nested interactions, which includes baseline, local, continuation and reverse continuation logits and odds ratios as special cases. The significant aspect of this definition is the inclusion of new types of logits and odds ratios that can handle non-ordinal, ordinal and partially ordered categorical variables in a flexible and appropriate way. It is proved also that the marginal nested interactions define a saturated model of a multi-way contingency table.     

属性统计与列联表分析（Ⅰ）

一、引 言§1—1 若干基本概念 属性就是用以说明总体中每个个体的一种(定性的)特性。设总体中的所有个体可以按一个或多个属性分类。例如,若将某市全体成年居民作为总体,则这个总体中的每个个体—每个成年人,可按性别(男或女)、年龄档次(青年、中年或老年)、文化程度(文盲、小学、初中、高中及大专以上)、婚姻状况(未婚、已婚、离婚或丧偶)等属性分类。再如若总体是某省所有基层商业单位,则可考虑按所有制性质(全民、集体或个体)分类;也可以按其     

Independence in Contingency Tables Using Simplicial Geometry

Frequently, contingency tables are generated in a multinomial sampling. Multinomial probabilities are then organized in a table assigning probabilities to each cell. A probability table can be viewed as an element in the simplex. The Aitchison geometry of the simplex identifies independent probability tables as a linear subspace. An important consequence is that, given a probability table, the nearest independent table is obtained by orthogonal projection onto the independent subspace. The nearest independent table is identified as that obtained by the product of geometric marginals, which do not coincide with the standard marginals, except in the independent case. The original probability table is decomposed into orthogonal tables, the independent and the interaction tables. The underlying model is log-linear, and a procedure to test independence of a contingency table, based on a multinomial simulation, is developed. Its performance is studied on an illustrative example.     

Generalized Least Squares Estimation in Contingency Tables Analysis: Asymptotic Properties and Applications

Different sorts of bilinear models (models with bilinear interaction terms) are currently used when analyzing contingency tables: association models, correlation models... All these can be included in a general family of bilinear models: power models. In this framework, Maximum Likelihood (ML) estimation is not always possible, as explained in an introductory example. Thus, Generalized Least Squares (GLS) estimation is sometimes needed in order to estimate parameters. A subclass of power models is then considered in this paper: separable reduced-rank (SRR) models. They allow an optimal choice of weights for GLS estimation and simplifications in asymptotic studies concerning GLS estimators. Power 2 models belong to the subclass of SRR models and the asymptotic properties of GLS estimators are established. Similar results are also established for association models which are not SRR models. However, these results are more difficult to prove. Finally, 2 examples are considered to illustrate our results.     

A Note on the Canonical Correlations From Contingency Tables

Explicit formulae are obtained for the asymptotic variances and covariances of canonical correlations which correspond to non-zero theoretical correlations in (p+ 1) x (q+1) contingency tables, with p≤q. The moments of the roots of a central Wishart matrix distributed as Wp(q; I ) are also given in general, with means, variances and covariances tabulated for p= 2, 3, 4: these may apply to canonical correlations corresponding to zeros.     

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.     

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.     

Measure of Asymmetry for Square Contingency Tables Having Ordered Categories

For the analysis of square contingency tables with nominal categories, Tomizawa and coworkers have considered measures that represent the degree of departure from symmetry. This paper proposes a measure that represents the degree of asymmetry for square contingency tables with ordered categories (instead of those with nominal categories). The measure proposed is expressed using the Cressie–Read power-divergence or Patil–Taillie diversity index, defined for the cumulative probabilities that an observation falls in row (column) category i or below and column (row) category j (> i ) or above. The measure depends on the order of listing the categories. It should be useful for comparing the degree of asymmetry in several tables with ordered categories. The relationship between the measure and the normal distribution is shown.     

Jackknifing the Mantel–Haenszel Estimator in Sparse Contingency Tables

This article investigates the performance of two jackknife techniques under an asymptotic model in which the number of 2 × 2 tables increases but the possible marginal configurations remain fixed. These approaches are applied to the Mantel–Haenszel estimator, or transformed versions of this estimator, respectively. The resulting jackknife estimators are shown to be consistent for the common odds ratio. Their asymptotic distributions are derived; they can be used for constructing appropriate sparse-data confidence intervals.     

A Perturbation Approach to Outlier Detection in Two-Way Contingency Tables

In order to identify outliers in contingency tables, we evaluate the derivatives of the perturbation-formed surface of the Pearson goodness-of-fit statistic. The resulting diagnostics are shown to be less susceptible to masking and swamping problems than residual-based measures. A Monte Carlo study further confirms the effectiveness of the proposed diagnostics.     

Partitioning Pearson's Chi-squared Statistic for Singly Ordered Two-way Contingency Tables

This paper presents a partition of Pearson's chi-squared statistic for singly ordered two-way contingency tables. The partition involves using orthogonal polynomials for the ordinal variable while generalized basic vectors are used for the non-ordinal variable. The benefit of this partition is that important information about the structure of the ordered variable can be identified in terms of locations, dispersion and higher order components. For the non-ordinal variable, it is shown that the squared singular values from the singular value decomposition of the transformed dataset can be partitioned into location, dispersion and higher order components. The paper also uses the chi-squared partition to present an alternative to the maximum likelihood technique of parameter estimation for the log-linear analysis of the contingency table.     

Tests of Symmetry in Three-dimensional Contingency Tables Based on Phi-divergence Statistics

In this paper we introduce a family of test statistics for testing complete symmetry in three-dimensional contingency tables based on phi- divergence families. These test statistics yield the likelihood ratio test and the Pearson test statistics as special cases. Asymptotic distribution for the new test statistics are derived under both the null and the alternative hypotheses. A simulation study is presented to show that some new statistics offer an attractive alternative to the classical Pearson and likelihood ratio test statistics for this problem of complete symmetry.     

New Tests of Spatial Segregation Based on Nearest Neighbour Contingency Tables

Abstract.  The spatial clustering of points from two or more classes (or species) has important implications in many fields and may cause segregation or association, which are two major types of spatial patterns between the classes. These patterns can be studied using a nearest neighbour contingency table (NNCT) which is constructed using the frequencies of nearest neighbour types. Three new multivariate clustering tests are proposed based on NNCTs using the appropriate sampling distribution of the cell counts in a NNCT. The null patterns considered are random labelling (RL) and complete spatial randomness (CSR) of points from two or more classes. The finite sample performance of these tests are compared with other tests in terms of empirical size and power. It is demonstrated that the newly proposed NNCT tests perform relatively well compared with their competitors and the tests are illustrated using two example data sets.     

中国第一批统计表的创制

统计表是表现和分析统计资料的重要工具。在统计资料的搜集、整理、分析、储存和传输中,都要应用统计表。统计表的发明为利用大量数据进行科学研究和经济管理提供了有效的途径。 统计表创制于何时何地?创始者是谁?有许多不同的说法。苏联学者M·V普图哈说:“I·K基洛夫于1726-1927年发明了国势学派的统计表式”,更多的学者认为统计表是丹麦历史学家H·P安彻逊创始的。在安彻逊1741年出版的《文明国家一览表》一书中,有一个包括15个欧洲国家的简表,表的主栏列有人口、领土、宗教、军事、行政组织、货币、度量衡等项目。     

Outliers and Patterns of Outliers in Contingency Tables with Algebraic Statistics

Abstract. In this paper, we provide a definition of pattern of outliers in contingency tables within a model‐based framework. In particular, we make use of log‐linear models and exact goodness‐of‐fit tests to specify the notions of outlier and pattern of outliers. The language and some techniques from Algebraic Statistics are essential tools to make the definition clear and easily applicable. We also analyse several numerical examples to show how to use our definitions.