Testing for or against a trend in odds ratios

Consider data arranged into k × 2 × 2 contingency tables. The principal result of this paper is the derivation of the likelihood ratio test and its asymptotic distribution for testing for or against an order restriction placed upon the odds ratios. We will show that the limiting distributions are of chi-bar square type and give the expression of the weighting values.     

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.     

Testing for and against a set of linear inequality constraints in a multinomial setting

There are numerous situations in categorical data analysis where one wishes to test hypotheses involving a set of linear inequality constraints placed upon the cell probabilities. For example, it may be of interest to test for symmetry in k × k contingency tables against one-sided alternatives. In this case, the null hypothesis imposes a set of linear equalities on the cell probabilities (namely p_ij = P_ji ×i > j), whereas the alternative specifies directional inequalities. Another important application (Robertson, Wright, and Dykstra 1988) is testing for or against stochastic ordering between the marginals of a k × k contingency table when the variables are ordinal and independence holds. Here we extend existing likelihood-ratio results to cover more general situations. To be specific, we consider testing H_t,0 against H₁ - H₀ and H₁ against H₂ - H ₁ when H₀:k × i=1 p_ix_ji = 0, j = 1,…, s, H₁:k × i=1 p_ix_ji × 0, j = 1,…, s, and does not impose any restrictions on p. The x_ji's are known constants, and s × k - 1. We show that the asymptotic distributions of the likelihood-ratio tests are of chi-bar-square type, and provide expressions for the weighting values.     

Power-Divergence Test Statistics for Testing Linear by Linear Association

Power-divergence test statistics have been considered to test linear by linear association for two-way contingency tables. These test statistics have been compared based on designed simulation study and asymptotic results for 2 × 2, 2 × 3, and 3 × 3 tables. According to the results, there are test statistics with better properties than the well-known likelihood ratio test statistic for small and moderate samples.     

Comparison of dependent two-way contingency tables

Comparison of two-way contingency tables using measures of association is considered. Multiple comparison procedures for dependent tables are proposed, enabling us to compare tables that are faces from larger multl-dimensional tables. An example

is given to Illustrate the analysis of two 2 × 2-tables formed

from a 2⁴-table.     

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.     

Bayesian reconstruction of contingency tables with partially categorized data

This paper suggests a Bayesian approach to the reconstruction of a 2 × 2 contingency table where some of the observations are only partially categorized and others are fully categorized. In contrast, most previous Bayesian and non-Bayesian analyses of the partially categorized data problem have been concerned with estimation of the parameters that generated the data. We show in an example that estimates may not be extremely sensitive to the weight placed on prior information relative to the sample data.     

On testing and estimating the interaction between treatments and environmental conditions in binomial experiments: The case of two stations

Estimating confidence intervals for the interaction between treatments and environmental conditions in binomial experiments is analyzed. Testing the interaction is studied also. The problem is reduced to that of estimating or testing the interaction parameter in 2 × 2 × 2 contingency tables with given marginals. Programs for determining the exact conditional tests and their power functions are provided for sample of size not exceeding 100. Large sample approximations based on maximum likelihood (ML) and on the arcsin transformation for proportions are studied.     

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.     

On the small sample estimation of the log odds ratio in 2×2 contingency tables with one set of fixed margins

For the analysis of 2 × 2 contingency tables with one set of fixed margins, a number of authors (e.g. Wolf, 1955; Cox, 1970) have proposed the use of various modified estimators based upon the empirical logistic transform. In this paper the moments of such estimators are considered and their small sample properties are investigated numerically.     

Measures of Association and Visualization of Log Odds Ratio Structure for a Two Way Contingency Table

The odds ratio (OR) is a measure of association used for analysing an I × J contingency table. The total number of ORs to check grows with I and J. Several statistical methods have been developed for summarising them. These methods begin from two different starting points, the I × J contingency table and the two‐way table composed by the ORs. In this paper we focus our attention on the relationship between these methods and point out that, for an exhaustive analysis of association through log ORs, it is necessary to consider all the outcomes of these methods. We also introduce some new methodological and graphical features. In order to illustrate previously used methodologies, we consider a data table of the cross‐classification of the colour of eyes and hair of 5387 children from Scotland. We point out how, through the log OR analysis, it is possible to extract useful information about the association between variables.     

On the Comparison of Two Ordinal Responses

Responses of two groups, measured on the same ordinal scale, are compared through the column effect association model, applied on the corresponding 2 × J contingency table. Monotonic or umbrella shaped ordering for the scores of the model are related to stochastic or umbrella ordering of the underlying response distributions, respectively. An algorithm for testing all possible hypotheses of stochastic ordering and deciding on an appropriate one is proposed.     

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.     

Inference with overlapping 2×2 tables

Exact probabilities (under a Markov-like assumption) are calculated for overlapping 2×2 contingency tables. These contingency tables often arise in biological and legal situations that yield dichotomous responses, including the evaluation of clinical trials and the determination of prima facie evidence of employer discrimination. In this note, probabilities, means and variances are derived and comparisons are made with results when assuming no overlap exists.     

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.     

ON THE USE OF X2 AS A TEST OF INDEPENDENCE IN CONTINGENCY TABLES WITH SMALL CELL EXPECTATIONS

When an r×c contingency table has many cells having very small expectations, the usual χ² approximation to the upper tail of the Pearson χ² goodness-of-fit statistic becomes very conservative. The alternatives considered in this paper are to use either a lognormal approximation, or to scale the usual χ² approximation. The study involves thousands of tables with various sample sizes, and with tables whose sizes range from 2×2 through 2×10×10. Subject to certain restrictions the new scaled χ² approximations are recommended for use with tables having an average cell expectation as small as 0·5.     

The Tale of Cochran's Rule: My Contingency Table has so Many Expected Values Smaller than 5, What Am I to Do?

In an informal way, some dilemmas in connection with hypothesis testing in contingency tables are discussed. The body of the article concerns the numerical evaluation of Cochran's Rule about the minimum expected value in r × c contingency tables with fixed margins when testing independence with Pearson's X² statistic using the χ² distribution.     

A new C(α) test for 2×2 tables

Two by two contingency tables which arise from experiments in which each subject serves as a self control are examined, Two types of experimental design are discussed: the confounded design and the counterbalanced design. In the case of the counterbalanced design, the locally asymptotically optimal C(α) test is compared with the binomial test for symmetry in a 2×2 contingency table. The binomial test is found to be less efficient than the C(α) test.     

Perfect sampling algorithm for small <Emphasis Type="Italic">m</Emphasis>×<Emphasis Type="Italic">n</Emphasis> contingency tables

A Markov chain is proposed that uses coupling from the past sampling algorithm for sampling m×n contingency tables. This method is an extension of the one proposed by Kijima and Matsui (Rand. Struct. Alg., 29:243–256, 2006). It is not polynomial, as it is based upon a recursion, and includes a rejection phase but can be used for practical purposes on small contingency tables as illustrated in a classical 4×4 example.     

Odds ratio for a single 2 × 2 table with correlated binomials for two margins

While analyzing 2 × 2 contingency tables, the log odds ratio for measuring the strength of association is often approximated by a normal distribution with some variance. We show that the expression of that variance needs to be modified in the presence of correlation between two binomial distributions of the contingency table. In the present paper, we derive a correlation-adjusted variance of the limiting normal distribution of log odds ratio. We also propose a correlation adjusted test based on the standard odds ratio for analyzing matched-pair studies and any other study settings that induce correlated binary outcomes. We demonstrate that our proposed test outperforms the classical McNemar’s test. Simulation studies show the gains in power are especially manifest when sample size is small and strong correlation is present. Two examples of real data sets are used to demonstrate that the proposed method may lead to conclusions significantly different from those reached using McNemar’s test.