Comparison of Ordinary and Penalized Power-Divergence Test Statistics for Small and Moderate Samples in Three-Way Contingency Tables via Simulation

In this article, size and power properties of the ordinary and penalized power-divergence test statistics have been compared to test a nested sequence of log-linear models for various penalization and λ values. 2 × 2 × 2 contingency tables distributed according to a multinomial distribution are considered. The simulation study results reveal that the penalized power-divergence test statistics with penalization value of one and λ = ?1.5, ?2 are the best choices for small and moderate samples, respectively.     

Testing Serial Correlation in Partially Linear Single-Index Errors-in-Variables Models

In this article, we propose two test statistics for testing the underlying serial correlation in a partially linear single-index model Y = η(Z ^τα) + X ^τβ + ? when X is measured with additive error. The proposed test statistics are shown to have asymptotic normal or chi-squared distributions under the null hypothesis of no serial correlation. Monte Carlo experiments are also conducted to illustrate the finite sample performance of the proposed test statistics. The simulation results confirm that these statistics perform satisfactorily in both estimated sizes and powers.     

Homogeneity Test of Risk Differences of Marginal and Conditional Probabilities in Several Incomplete Correlated 2 × 2 Tables

This article considers K pairs of incomplete correlated 2 × 2 tables in which the interesting measurement is the risk difference between marginal and conditional probabilities. A Wald-type statistic and a score-type statistic are presented to test the homogeneity hypothesis about risk differences across strata. Powers and sample size formulae based on the above two statistics are deduced. Figures about sample size against risk difference (or marginal probability) are given. A real example is used to illustrate the proposed methods.     

Likelihood Ratio Test Against Stochastic Order in Three Way Contingency Tables

Trend tests in dose-response have been central problems in medicine. The likelihood ratio test is often used to test hypotheses involving a stochastic order. Stratified contingency tables are common in practice. The distribution theory of likelihood ratio test has not been full developed for stratified tables and more than two stochastically ordered distributions. Under c strata of m × r tables, for testing the conditional independence against simple stochastic order alternative, this article introduces a model-free test method and gives the asymptotic distribution of the test statistic, which is a chi-bar-squared distribution. A real data set concerning an ordered stratified table will be used to show the validity of this test method.     

On the Hypothesis of No Interaction on a Linear Scale in Contingency Tables

The purpose of this paper is to review briefly the three main formulations of no Interaction hypotheses in contingency tables and to consider the formulation on a linear scale in some detail.More specifically we (i) present a situation in 2×2 tables where such a formulation may be more appropriate than others, (ii) study the geometry for this problem, (iii) give contrast-type or parametric ANOVA type formulations in the general n-dimensional tables, (iv) discuss estimation and testing procedures and (v) consider collapsibility of contingency tables in relation to the hypotheses of no interaction on a linear scale.     

Comparison of exact tests for association in unordered contingency tables using standard,mid-p,and randomized test versions

Pearson’s chi-square (Pe), likelihood ratio (LR), and Fisher (Fi)–Freeman–Halton test statistics are commonly used to test the association of an unordered r×c contingency table. Asymptotically, these test statistics follow a chi-square distribution. For small sample cases, the asymptotic chi-square approximations are unreliable. Therefore, the exact p-value is frequently computed conditional on the row- and column-sums. One drawback of the exact p-value is that it is conservative. Different adjustments have been suggested, such as Lancaster’s mid-p version and randomized tests. In this paper, we have considered 3×2, 2×3, and 3×3 tables and compared the exact power and significance level of these test’s standard, mid-p, and randomized versions. The mid-p and randomized test versions have approximately the same power and higher power than that of the standard test versions. The mid-p type-I error probability seldom exceeds the nominal level. For a given set of parameters, the power of Pe, LR, and Fi differs approximately the same way for standard, mid-p, and randomized test versions. Although there is no general ranking of these tests, in some situations, especially when averaged over the parameter space, Pe and Fi have the same power and slightly higher power than LR. When the sample sizes (i.e., the row sums) are equal, the differences are small, otherwise the observed differences can be 10% or more. In some cases, perhaps characterized by poorly balanced designs, LR has the highest power.     

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.     

An extension of the Anderson–Darling k-sample test to arbitrary sample space partition sizes

In this paper we first show that the k-sample Anderson–Darling test is basically an average of Pearson statistics in 2?×?k contingency tables that are induced by observation-based partitions of the sample space. As an extension, we construct a family of rank test statistics, indexed by c?∈??, which is based on similarly constructed c?×?k partitions. An extensive simulation study, in which we compare the new test with others, suggests that generally very high powers are obtained with the new tests. Finally we propose a decomposition of the test statistic in interpretable components.     

Wilson Confidence Intervals for the Two-Sample Log-Odds-Ratio in Stratified 2 × 2 Contingency Tables

Large-sample Wilson-type confidence intervals (CIs) are derived for a parameter of interest in many clinical trials situations: the log-odds-ratio, in a two-sample experiment comparing binomial success proportions, say between cases and controls. The methods cover several scenarios: (i) results embedded in a single 2 × 2 contingency table; (ii) a series of K 2 × 2 tables with common parameter; or (iii) K tables, where the parameter may change across tables under the influence of a covariate. The calculations of the Wilson CI require only simple numerical assistance, and for example are easily carried out using Excel. The main competitor, the exact CI, has two disadvantages: It requires burdensome search algorithms for the multi-table case and results in strong over-coverage associated with long confidence intervals. All the application cases are illustrated through a well-known example. A simulation study then investigates how the Wilson CI performs among several competing methods. The Wilson interval is shortest, except for very large odds ratios, while maintaining coverage similar to Wald-type intervals. An alternative to the Wald CI is the Agresti-Coull CI, calculated from the Wilson and Wald CIs, which has same length as the Wald CI but improved coverage.     

Minimal Basis for a Connected Markov Chain over 3 × 3 ×K Contingency Tables with Fixed Two-Dimensional Marginals

This paper considers a connected Markov chain for sampling 3 × 3 ×K contingency tables having fixed two‐dimensional marginal totals. Such sampling arises in performing various tests of the hypothesis of no three‐factor interactions. A Markov chain algorithm is a valuable tool for evaluating P‐values, especially for sparse datasets where large‐sample theory does not work well. To construct a connected Markov chain over high‐dimensional contingency tables with fixed marginals, algebraic algorithms have been proposed. These algorithms involve computations in polynomial rings using Gröbner bases. However, algorithms based on Gröbner bases do not incorporate symmetry among variables and are very time‐consuming when the contingency tables are large. We construct a minimal basis for a connected Markov chain over 3 × 3 ×K contingency tables. The minimal basis is unique. Some numerical examples illustrate the practicality of our algorithms.     

Statistical inference of risk difference in K correlated 2×2 tables with structural zero

K correlated 2×2 tables with structural zero are commonly encountered in infectious disease studies. A hypothesis test for risk difference is considered in K independent 2×2 tables with structural zero in this paper. Score statistic, likelihood ratio statistic and Wald‐type statistic are proposed to test the hypothesis on the basis of stratified data and pooled data. Sample size formulae are derived for controlling a pre‐specified power or a pre‐determined confidence interval width. Our empirical results show that score statistic and likelihood ratio statistic behave better than Wald‐type statistic in terms of type I error rate and coverage probability, sample sizes based on stratified test are smaller than those based on the pooled test in the same design. A real example is used to illustrate the proposed methodologies. Copyright © 2009 John Wiley & Sons, Ltd.     

The analysis of ratios and cross-product ratios of poisson variates with application to incidence rates

The incidence of most diseases is low enough that in. large populations the number of new cases may be considered a Poisson variate. This paper explores models and methods for analyzing such data Specific cases are the estimation and testing of ratios and the cross-product ratios, both simple and stratified* We assume the Poisson means are exponential functions of the relevant parameters. The resulting sets of sufficient statistics are partitioned into a test statistic and a vector of statistics related to the nuisance parameters . The methods derived are based on the conditional distribution of the test statistic given the other sufficient statistics. The analyses of stratified cross-product ratios are seen to be analogues of the noncentral distribution associated with theanalysis of the common odds ratio in several 2×2 tables. The various methods are illustrated in numerical examples involving incidence rates of cancer in two metropolitan areas adjusting for both age and sex.     

Permutation Tests for Linear Models

Several approximate permutation tests have been proposed for tests of partial regression coefficients in a linear model based on sample partial correlations. This paper begins with an explanation and notation for an exact test. It then compares the distributions of the test statistics under the various permutation methods proposed, and shows that the partial correlations under permutation are asymptotically jointly normal with means 0 and variances 1. The method of Freedman & Lane (1983) is found to have asymptotic correlation 1 with the exact test, and the other methods are found to have smaller correlations with this test. Under local alternatives the critical values of all the approximate permutation tests converge to the same constant, so they all have the same asymptotic power. Simulations demonstrate these theoretical results.     

Admissible Linear Estimators with Respect to Inequality Constraints

Admissibility of linear estimators is characterized in linear models E(Y)=Xβ, D(Y)=V, with an unknown multidimensional parameter (β, V) varying in the Cartesian product C × ν, where C is a subset of space and ν is a given set of non negative definite symmetric matrices. The relation between admissibility of inhomogeneous and homogeneous linear estimators is discussed, and some sufficient and necessary conditions for admissibility of an inhomogeneous linear estimator are given.     

Generalized cochran-mantel-haenszel test statistics for correlated categorical data

Three new test statistics are introduced for correlated categorical data in stratified R×C tables. They are similar in form to the standard generalized Cochran-Mantel-Haenszel statistics but modified to handle correlated outcomes. Two of these statistics are asymptotically valid in both many-strata (sparse data) and large-strata limiting models. The third one is designed specifically for the many-strata case but is valid even with a small number of strata. This latter statistic is also appropriate when strata are assumed to be random.     

Testing Serial Correlation in Semiparametric Varying-Coefficient Partially Linear Models

We propose two test statistics for testing serial correlation in semiparametric varying-coefficient partially linear models. The proposed test statistics are not only for testing zero first-order serial correlation, but also for testing higher-order serial correlations. Under the null hypothesis of no serial correlation, the test statistics are shown to have asymptotic normal or chi-square distributions. By using R, some Monte Carlo experiments are conducted to examine the finite sample performances of the proposed tests. Simulation results show that the estimated size and power of the proposed tests behave well.     

A Test of Independence Based on the Likelihood of Cut-Points

We define a test statistic C _n based on the sum of the likelihood ratio statistics for testing independence in the 2 × 2 tables defined at n sample cut-points (X _i , Y _i ). The asymptotic distribution of C _n , given the cut-points, is sum of dependent χ² variables with one degree of freedom. We use the bootstrap to obtain the distribution of C _n . We compare the performance of several tests of bivariate independence, including Pearson, Spearman, and Kendall correlations, Blum-Kiefer-Rosenblatt statistic, and C _n under several copulas and given marginal distributions.     

Bayesian Approach to Multicentre Sparse Data

In a 2 × 2 contingency table, when the sample size is small, there may be a number of cells that contain few or no observations, usually referred to as sparse data. In such cases, a common recommendation in the conventional frequentist methods is adding a small constant to every cell of the observed table to find the estimates of the unknown parameters. However, this approach is based on asymptotic properties of the estimates and may work poorly for small samples. An alternative approach would be to use Bayesian methods in order to provide better insight into the problem of sparse data coupled with fewer centers, which would otherwise be difficult to carry out the analysis. In this article, an attempt has been made to use hierarchical Bayesian model to a multicenter data on the effect of a surgical treatment with standard foot care among leprosy patients with posterior tibial nerve damage which is summarized as seven 2 × 2 tables. Monte Carlo Markov Chain (MCMC) techniques are applied in estimating the parameters of interest under sparse data setup.     

On the Expressions of Estimability in Testing General Linear Hypotheses

In testing a general linear hypothesis of the form K ′β ? ( W ′) under a general linear model, an equivalent hypothesis involving only estimable parametric functions is provided, and then an explicit test statistic in terms of the model matrices is given. The corresponding results are expanded to the case of a general linear model with a restriction and are illustrated by an example.     

Odds ratio inference from stratified samples

One of the common problems encountered in applied statistics is that of comparing two proportions from stratified samples. One approach to this problem is via inference on the corresponding odds ratio. In this paper, the various point and interval estimators of and hypothesis testing procedures for a common odds ratio from multiple 2 ×2 tables are reviewed. Based On research to date, the conditional maximum likelihood and Mantel-Haenszel estimators are recommended as the point estimators of choice. Neither confidence intervals nor hypothesis testing metthods have been studied as well as the point estimators, but there is a confidence interval method associated with the Mantel-Haenszel estimator that is a good choice.