Fractional imputation for incomplete two-way contingency tables

Imputation procedures such as fully efficient fractional imputation (FEFI) or multiple imputation (MI) create multiple versions of the missing observations, thereby reflecting uncertainty about their true values. Multiple imputation generates a finite set of imputations through a posterior predictive distribution. Fractional imputation assigns weights to the observed data. The focus of this article is the development of FEFI for partially classified two-way contingency tables. Point estimators and variances of FEFI estimators of population proportions are derived. Simulation results, when data are missing completely at random or missing at random, show that FEFI is comparable in performance to maximum likelihood estimation and multiple imputation and superior to simple stochastic imputation and complete case anlaysis. Methods are illustrated with four data sets.     

Some model-restricted shrinkage estimators for contingency tables with missing data

For contingency tables with extensive missing data, the unrestricted MLE under the saturated model, computed by the EM algorithm, is generally unsatisfactory. In this case, it may be better to fit a simpler model by imposing some restrictions on the parameter space. Perlman and Wu (1999) propose lattice conditional independence (LCI) models for contingency tables with arbitrary missing data patterns. When this LCI model fits well, the restricted MLE under the LCI model is more accurate than the unrestricted MLE under the saturated model, but not in general. Here we propose certain empirical Bayes (EB) estimators that adaptively combine the best features of the restricted and unrestricted MLEs. These EB estimators appear to be especially useful when the observed data is sparse, even in cases where the suitability of the LCI model is uncertain. We also study a restricted EM algorithm (called the ER algorithm) with similar desirable features. Received: July 1999     

Conditional independence and log linear models for multi-dimensional contingency tables

Interference about conditional independence in relation to log linear models are discussed for contingency tables. The parameters and likelihood ratios for a log linear model with a dependent variable are shown to be identical to those for a multivariate model. An approximaate method of calculating log likelihood ratios, even when all dimensions of the table have more than two levels (no binary variables) is derived. The implications for sociological “causal” models are discussed.     

Comparisons of Bayesian estimation procedures for two-way contingency tables without interaction

Bayesian and empirical Bayesian estimation methods are reviewed and proposed for the row and column parameters in two-way Contingency tables without interaction. Rasch's multiplicative Poisson model for misreadings is discussed in an example. The case is treated where assumptions of exchangeability are reasonable a priori for the unknown parameters. Two different types of prior distributions are compared, It appears that gamma priors yield more tractable results than lognormal priors.     

Testing conditional symmetry against one-sided alternatives in square contingency tables

For a (k×k) square contingency table with ordered categories, letX(Y) denote the row (column) number. The conditional symmetry model is given byP(X=i, Y=j|X<Y)=P(X=j, Y=i |X>Y), ∀i<j. In this paper, we study the likelihood ratio tests of conditional symmetry in a square contingency table against two particular classes of one-sided alternatives. We obtain the maximum likelihood estimators under each alternative. The asymptotic null distributions of the likelihood ratio statistics are shown to have chi-bar square type distributions. A simulation study is performed by comparing the powers of different tests. The theory developed is illustrated by using the famous eye vision data from Stuart (1953).     

A new approach to distribution free tests in contingency tables

We suggest an extremely wide class of asymptotically distribution free goodness of fit tests for testing independence in two-way contingency tables, or equivalently, independence of two discrete random variables. The nature of these tests is that the test statistics can be viewed as definite functions of the transformation of \(\widehat{T}_n = (\widehat{T}_{ij})=\Big (\frac{\nu _{ij}- n\hat{a}_i\hat{b}_j}{\sqrt{n\hat{a}_i\hat{b}_j}}\Big )\) where \(\nu _{ij}\) are frequencies and \(\hat{a}_i, \hat{b}_j\) are estimated marginal distributions. Our method is also applicable for testing independence of two discrete random vectors. We make some comparisons on statistical powers of the new tests with the conventional chi-square test and suggest some cases in which this class is significantly more powerful.     

Alternative interaction structures in square contingency tables having ordered classificatory variables

Goodman (1972) proposed several models for the analysis of the general I x I square tables with particular emphasis on social mobility data. We demonstrate in this paper, that most of his models can be reproduced by combinations of both new models proposed here and the various well known models that have received considerable attention in the literature. Our presentation here is both concise and simple to comprehend. The various models considered in this study are fitted to ten data sets that include the much analyzed 5×5 Danish and British Social mobility data sets. Results suggest that in some cases more parsimonious models than those considered earlier by various authors are possible for the explanations of the variations in the data analyzed in this study.     

Class-specific tests of spatial segregation based on nearest neighbor contingency tables

The spatial interaction between two or more classes might cause multivariate clustering patterns such as segregation or association, which can be tested using a nearest neighbor contingency table (NNCT). The null hypothesis is randomness in the nearest neighbor structure, which may result from random labeling (RL) or complete spatial randomness of points from two or more classes (which is henceforth called CSR independence ). We consider Dixon's class-specific segregation test and introduce a new class-specific test, which is a new decomposition of Dixon's overall chi-squared segregation statistic. We analyze the distributional properties and compare the empirical significant levels and power estimates of the tests using extensive Monte Carlo simulations. We demonstrate that the new class-specific tests have comparable performance with the currently available tests based on NNCTs. For illustrative purposes, we use three example data sets and provide guidelines for using these tests.     

Extensions of linear diagonal-parameter symmetry and quasi-symmetry models for cumulative probabilities in square contingency tables

For square contingency tables with ordered categories, CAUSSINUS [Annales de la Faculté des Sciences de l'Université de Toulouse (1965) Vol. 29, pp. 77–182] and AGRESTI [Statistics and Probability Letters (1983) Vol. 1, pp. 313–316] considered the quasi-symmetry and the linear diagonal-parameter symmetry models, respectively, which have multiplicative forms for cell probabilities. This paper proposes two kinds of models that have the similar multiplicative forms for cumulative probabilities that an observation will fall in row (column) category i or below and column (row) category j (> i ) or above. The endometrial cancer data are analyzed using these models.     

The causal analysis of contingency tables: A return to first principles

Given the undoubtedly major advances in the analysis of contingency tables which have been achieved over the past ten years or so (see for example, Fienberg, 1980; Upton, 1978; Everitt, 1977; Haberman, 1978, 1979), it might seem rather unnecessary to want to return to first princples again. However, the need arises precisely because of these advances; for progress in the specifically causal analysis of contingency tables has not matched the other advances at all. Whilst Fienberg devoted a chapter to causal analysis, he made it clear that he views “the assignment of numerical values [to the arrows in a path diagram] as problematic, and [he] would limit [the analysis] to an indication of sign for causal relationships, in a fashion similar to that described by Blalock (1964)” (Fienberg, 1980, pp. 91–92). Considering how far quantitative-variable causal analysis has developed since Blalock (1964), it becomes clear that the causal analysis of qualitative data is still at a rather primitive stage. Indeed, Haberman (1978, 1979), in his two-volume survey of the analysis of qualitative data, does not mention it at all. The problem, I believe, is that log-linear and logit methodology are not particularly suited to the logic of causality in contingency tables. In order to derive a suitable method, it is necessary to uncover the logic underlying causality when applied to qualitative variables. A few others have taken seriously the idea that a direct analysis of the form of a contingency table can lead to fruitful results (see, especially, for example, Boudon, 1967), but their work has been overshadowed by the statistically more profound advances made in log-linear methods. This article is an attempt to provide a statistically rigorous analysis based on the direct interpretation of causality embodied in a contingency table.     

A class of statistical models to weaken independence in two-way contingency tables

In this paper we study a new class of statistical models for contingency tables. We define this class of models through a subset of the binomial equations of the classical independence model. We prove that they are log-linear and we use some notions from Algebraic Statistics to compute their sufficient statistic and their parametric representation. Moreover, we show how to compute maximum likelihood estimates and to perform exact inference through the Diaconis-Sturmfels algorithm. Examples show that these models can be useful in a wide range of applications.     

On X2- tests for contingency tables of negative multinomial types

Om te toets of die twee kenmerke waarvolgens die frekwensies in 'n gebeurlikseidstabel opgestel is onafhanklik is, word gewoonlik 'n X² toets gebruik. In die opstelling van so 'n toets word die totale frekwensie as vas beskou. In hierdie artikel word aangetoon hoe 'n X²-toets nog gebruik kan word wanneer die steekproefneming voortgesit word totdat 'n vooraf vasgestelde frekwensie in een selbereik word. Die totale frekwensie is dan 'n stogastiese veranderlike. Die eienskappe van die randverdelings van 'n negatiewe multinomiaalverdeling word gebruik om die parameters wat in die X²-verdeling optree te skat.     

A generalization of Cochran's procedure for the combining of r×c contingency tables*

Abstract Cochran [3] derives a test of association when k 2 × 2 contingency tables are combined. We show in this paper how to extend Cochran's test to the combining of k r×c contingency tables using a multiple comparison technique similar to the one presented by Dunn [4]. An example is included.     

Double symmetry model and its decompositions in square contingency tables having ordered categories

In this paper, we have employed the non-standard log-linear models to fit the double symmetry models and some of its decompositions to square contingency tables having ordered categories. SAS PROC GENMOD was employed to fit these models although we could similarly have used GENLOG in SPSS or GLM in STATA. A SAS macro generates the factor or scalar variables required to fit these models. Two sets of \(4 \times 4\) unaided distance vision data that have been previously analyzed in (Tahata and Tomizawa, Journal of the Japan Statistical Society 36:91–106, 2006) were employed for verification of results. We also extend the approach to the Danish \(5 \times 5\) Mobility data as well as to the \(3 \times 3\) Danish longitudinal study data of subjective health, firstly reported in (Andersen, The Statistical Analysis of Categorical Data, Springer:Berlin, 1994) and analyzed in (Tahata and Tomizawa, Statistical Methods and Applications 19:307–318, 2010). Results obtained agree with those published in previous literature on the subject. The approaches suggest here eliminate any programming that might be required in order to apply these class of models to square contingency tables.     

The large-sample power of the 195-1195-1195-1test for multidimensional contingency tables

Summary The asymptotic power of the frequency ² test depends on a noncentrality parameter, ,Mitra [1958] offered a general expression for , which is rather difficult to apply. This work provides simplified formulae for in various models associated with multidimensional contingency tables.