Large sample results for tests of association ii. multinomial and stratified sampling

This paper deals with the asymptotics of a class of tests for association in 2-way contingency tables based on square forms in cell frequencies, given the total number of observations (multinomial sampling) or one set of marginal totals (stratified sampling). The case when both row and column marginal totals are fixed (hypergeometric sampling) was studied in Kulinskaya (1994), The class of tests under consideration includes a number of classical measures for association, Its two subclasses are the tests based on statistics using centralized cell frequencies (asymptotically distributed as weighted sums of central chi-squares) and those using the non-centralized cell frequencies (asymptotically normal). The parameters of asymptotic distributions depend on the sampling model and on true marginal probabilities. Maximum efficiency for asymptotically normal statistics is achieved under hypergeometric sampling, If the cell frequencies or the statistic as a whole are centralized using marginal proportions as estimates for marginal probabilities, the asymptotic distribution does not differ much between models and it is equivalent to that under hypergeometric sampling. These findings give an extra justification for the use of permutation tests for association (which are based on hypergeometric sampling). As an application, several well known measures of association are analysed.     

Point-symmetry models and decomposition for collapsed square contingency tables

For square contingency tables with ordered categories, there may be some cases that one wants to analyze them by considering collapsed tables with some adjacent categories combined in the original table. This paper proposes three kinds of new models which have the structure of point-symmetry (PS), quasi point-symmetry and marginal point-symmetry for collapsed square tables. This paper also gives a decomposition of the PS model for collapsed square tables. The father's and his daughter's occupational mobility data are analyzed using new models.     

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.     

Minimum chi-square estimator and test for incomplete multidimensional contingency tables

A modified transformed chi-square statistic is defined for testing hypotheses of quasi-independence in the incomplete multi-dimensional contingency table and a simple method for determining degrees of freedom is given. A modified transformed chi-squareestimator of the expected cell frequencies is given in closed form for a general class of exact linear constraints. The co-variance matrix of estimated cell frequencies is derived under the assumption of a conditional Poisson distribution.     

Data augmentation in multi-way contingency tables with fixed marginal totals

We describe and illustrate approaches to data augmentation in multi-way contingency tables for which partial information, in the form of subsets of marginal totals, is available. In such problems, interest lies in questions of inference about the parameters of models underlying the table together with imputation for the individual cell entries. We discuss questions of structure related to the implications for inference on cell counts arising from assumptions about log-linear model forms, and a class of simple and useful prior distributions on the parameters of log-linear models. We then discuss “local move” and “global move” Metropolis–Hastings simulation methods for exploring the posterior distributions for parameters and cell counts, focusing particularly on higher-dimensional problems. As a by-product, we note potential uses of the “global move” approach for inference about numbers of tables consistent with a prescribed subset of marginal counts. Illustration and comparison of MCMC approaches is given, and we conclude with discussion of areas for further developments and current open issues.     

Improving Path Trimming in a Network Algorithm for Fisher's Exact Test in Two-way Contingency Tables

This article provides an improvement of the network algorithm for calculating the exact p value of the generalized Fisher's exact test in two-way contingency tables. We give a new exact upper bound and an approximate upper bound for the maximization problems encountered in the network algorithm. The approximate bound has some very desirable computational properties and the meaning is elucidated from a viewpoint of differential geometry. Our proposed procedure performs well regardless of the pattern of marginal totals of data.     

Exact confidence regions for species assignment based on DNA markers

Assignment of individuals to correct species or population of origin based on a comparison of allele profiles has in recent years become more accurate due to improvements in DNA marker technology. A method of assessing the error in such assignment problems is présentés. The method is based on the exact hypergeometric distributions of contingency tables conditioned on marginal totals. The result is a confidence region of fixed confidence level. This confidence level is calculable exactly in principle, and estimable very accurately by simulation, without knowledge of the true population allele frequencies. Various properties of these techniques are examined through application to several examples of actual DNA marker data and through simulation studies. Methods which may reduce computation time are discussed and illustrated.     

Maximum likelihood estimation of the parameters of the bivariate binomial distribution

We consider the problem of maximum likelihood estimation of the parameters of the bxvariate binomial distribution, In the statistical literature, this problem is solved when the observed sample is available in the form of a 2x2 contingency table, that is, with all four cell fre quencies given,, The present paper provides a solution for this problem when only the marginal totals of the 2x2 table are observed, which is the natural set-up in a bivariate sampling situation.. Thus, based on a sample [(X_i,Y_i:), i = 1, …, k] from a bivariate binomial population, we derive maximum likelihood (ML) estimators for the two marginal parameters p₁,p₂: and the covariance parameter p₁₁: It. turns out that the ML estimators for P₁: and P₂: are expressed explicitly in terms of the sample values, whereas the ML estimator for p₁₁: can only be obtained numerically by iterative methods Two nu merical illustrations are also presented     

Measure of departure from symmetry for the analysis of collapsed square contingency tables with ordered categories

For square contingency tables with ordered categories, there may be some cases that one wants to analyze them by considering collapsed tables with some adjacent categories combined in the original table. This paper considers the symmetry model for collapsed square contingency tables and proposes a measure to represent the degree of departure from symmetry. The proposed measure is defined as the arithmetic mean of submeasures each of which represents the degree of departure from symmetry for each collapsed 3×3 table. Each submeasure also represents the mean of power-divergence or diversity index for each collapsed table. Examples are given.     

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.     

Synthetic two-way contingency tables that preserve conditional frequencies

In the area of statistical limitation, releasing synthetic data sets has become a popular method for limiting the risks of disclosure of sensitive information and at the same time maintaining analytic utility of data. However, less work has been done on how to create synthetic contingency tables that preserve some summary statistics of the original table. Studies in this area have primarily focused on generating replacement tables that preserve the margins of the original table since the latter support statistical inferences for a large set of parametric tests and models. Yet, not all synthetic tables that preserve a set of margins yield consistent results. In this paper, we propose alternative synthetic table releases. We describe how to generate complete two-way contingency tables that have the same set of observed conditional frequencies by using tools from computational algebra. We study both the disclosure risk and the data utility associated with such synthetic tabular data releases, and compare them to the traditionally released synthetic tables.     

Use of the 21% rulf in experimental design

When the probabilities underlying a contingency table are assumed to be symmetric, it is advantageous to combine the frequencies of equiprobable cells (Haber, 1981, 1982).The present work introduces models, as well as exact and asymptotic methods, for analyzing such ‘intraclass’ contingency tables in several dimensions.     

Editor's Report

There are two common methods for statistical inference on 2 × 2 contingency tables. One is the widely taught Pearson chi-square test, which uses the well-known χ²statistic. The chi-square test is appropriate for large sample inference, and it is equivalent to the Z-test that uses the difference between the two sample proportions for the 2 × 2 case. Another method is Fisher’s exact test, which evaluates the likelihood of each table with the same marginal totals. This article mathematically justifies that these two methods for determining extreme do not completely agree with each other. Our analysis obtains one-sided and two-sided conditions under which a disagreement in determining extreme between the two tests could occur. We also address the question whether or not their discrepancy in determining extreme would make them draw different conclusions when testing homogeneity or independence. Our examination of the two tests casts light on which test should be trusted when the two tests draw different conclusions.     

On tests of symmetry,marginal homogeneity and quasi-symmetry in two-way contingency tables based on minimum φ-divergence estimator with constraints

The restricted minimum φ-divergence estimator, [Pardo, J.A., Pardo, L. and Zografos, K., 2002, Minimum φ-divergence estimators with constraints in multinomial populations. Journal of Statistical Planning and Inference, 104, 221–237], is employed to obtain estimates of the cell frequencies of an I×I contingency table under hypotheses of symmetry, marginal homogeneity or quasi-symmetry. The associated φ-divergence statistics are distributed asymptotically as chi-squared distributions under the null hypothesis. The new estimators and test statistics contain, as particular cases, the classical estimators and test statistics previously presented in the literature for the cited problems. A simulation study is presented, for the symmetry problem, to choose the best function φ₂ for estimation and the best function φ₁ for testing.     

Contingency tables with prescribed marginals

For an adjustment of contingency tables to prescribed marginal frequencies Deming and Stephan (1940) minimize a Chi-square expression. Asymptotically equivalently, Ireland and Kullback (1968) minimize a Leibler-Kullback divergence, where the probabilistical arguments for both methods remain vague. Here we deduce a probabilistical model based on observed contingency tables. It shows that the two above methods and the maximum likelihood approach in Smith (1947) yield asymptotically the ‘most probable’ adjustment under prescribed marginal frequencies. The fundamental hypothesis of statistical mechanics relates observations to ‘most probable’ realizations. ‘Most probable’ is going to be used in the sense of so-called large deviations. The proposed adjustment has a significant product form and will be generalized to contingency tables with infinitely many cells.     

A NOTE ON MULTIVARIATE LOGISTIC MODELS FOR CONTINGENCY TABLES

The log-linear model is a tool widely accepted for modelling discrete data given in a contingency table. Although its parameters reflect the interaction structure in the joint distribution of all variables, it does not give information about structures appearing in the margins of the table. This is in contrast to multivariate logistic parameters, recently introduced by Glonek & McCullagh (1995), which have as parameters the highest order log odds ratios derived from the joint table and from each marginal table. Glonek & McCullagh give the link between the cell probabilities and the multivariate logistic parameters, in an algebraic fashion. The present paper focuses on this link, showing that it is derived by general parameter transformations in exponential families. In particular, the connection between the natural, the expectation and the mixed parameterization in exponential families (Barndorff-Nielsen, 1978) is used; this also yields the derivatives of the likelihood equation and shows properties of the Fisher matrix. The paper emphasises the analysis of independence hypotheses in margins of a contingency table.     

Estimation under generalized sampling of cell proportions for contingency tables subject to marginal constraints

Under simple random (multinomial) sampling the problem of estimating cell proportions for a contingency table subject to marginal constraints has been well explored. We briefly review methods that have been considered; then we develop a general method, for more complicated sampling, which reflects the variance structure of the estimated cell proportions. For stratified and cluster sampling we compare our method against earlier methods for the 2×2 table and find it potentially advantageous.     

The iterative proportional fitting algorithm for adjusted agreement in a non‐inferiority diagnostic clinical trial

In this paper, we describe a method of comparing agreement between two diagnostic contingency tables after adjustment to more clinically relevant marginal distributions using the iterative proportional fitting algorithm. When the categories of a contingency table represent mild, moderate, and severe outcomes, the majority of patients often are in the mild category. Because it is often of more interest to evaluate agreement when patients are uniformly distributed among categories, we present the primary results of two clinical trials with adjustment to this structure. We also describe the relationship between the sponsor's pre‐specified agreement measure for the observed contingency table and kappa for the adjusted table; and by either criterion, we then show that the agreement of the new diagnostic tool with the standard diagnostic tool is comparably non‐inferior to the agreement of the standard diagnostic tool with itself. Copyright © 2014 John Wiley & Sons, Ltd.     

Double linear diagonals-parameter symmetry and decomposition of double symmetry for square tables

For square contingency tables with ordered category, the present paper proposes the double linear diagonals-parameter symmetry (D-LDPS) model which implies the structure of both asymmetry with respect to the main diagonal and with respect to the reverse diagonal in the table. The D-LDPS model may be appropriate for a square ordinal table if it is reasonable to assume an underlying bivariate normal distribution with equal marginal variances. The present paper also gives the orthogonal decomposition of the double symmetry model into the D-LDPS model and the double marginal mean equality model. An example is given.     

Effective directed tests for models with ordered categorical data

This paper offers a new method for testing one‐sided hypotheses in discrete multivariate data models. One‐sided alternatives mean that there are restrictions on the multidimensional parameter space. The focus is on models dealing with ordered categorical data. In particular, applications are concerned with R×C contingency tables. The method has advantages over other general approaches. All tests are exact in the sense that no large sample theory or large sample distribution theory is required. Testing is unconditional although its execution is done conditionally, section by section, where a section is determined by marginal totals. This eliminates any potential nuisance parameter issues. The power of the tests is more robust than the power of the typical linear tests often recommended. Furthermore, computer programs are available to carry out the tests efficiently regardless of the sample sizes or the order of the contingency tables. Both censored data and uncensored data models are discussed.