Approximating the distribution of maximally selected McNemar's statistics

It is common in epidemiologic analyses to summarize continuous outcomes as falling above or below a threshold. With paired data and with a threshold chosen without reference to the outcomes, McNemar's test of marginal homogeneity may be applied to the resulting dichotomous pairs when testing for equality of the marginal distributions of the underlying continuous outcomes. If the threshold is chosen to maximize the test statistic, however, referring the resulting test statistic to the nominal chi 2 distribution is incorrect; instead, the p-value must be adjusted for the multiple comparisons. Here the distribution of a maximally selected McNemar's statistic is derived, and it is shown that an approximation due to Durbin (1985, Journal of Applied Probability 22, 99-122) may be used to estimate approximate p-values. The methodology is illustrated by an application to measurements of insulin-like growth factor-I (IGF-I) in matched prostate cancer cases and controls from the Physicians' Health Study. The results of simulation experiments that assess the accuracy of the approximation in moderate sample sizes are reported.     

Iso- and Anisoasymmetry Testing in Pre-post Rating Designs

It is suggested to test against anisoasymmetry rather than against marginal homogeneity if (1) a pre-post treatment matched pairs contingency table is ordinally scaled by clinical ratings and if (2) the alternative to marginal homogeneity is a shift in location from pre- to post treatment ratings. Testing against anisoasymmetry may detect treatment effects which are not manifest in marginal inhomogeneities.     

Symmetry and Asymmetry Models for Rectangular Tables

The models of complete, quasi and conditional symmetry as well as marginal homogeneity and diagonal asymmetry, applicable to square contingency tables are extended to non-square ones. Several properties of the models are investigated and estimation and test theory is presented. The utility of the new proposed models is discussed and illustrated by reanalyzing classical data sets.     

Analysis of Square Contingency Tables with Ordered Categories by a New Method of Applying the 1-Weight Modified Marginal Homogeneity Models

For square contingency tables with ordered categories, this note proposes a new method of applying Tomizawa's (1987) 1-weight modified marginal homogeneity models and applies to the 4×4 tables on unaided vision analysed by Tomizawa (1987) and by Stuart (1955). A possible explanation which is derived from fitting those models is first obtained using this new method.     

The equivalence of the generalized McNemar tests for marginal homogeneity in 2(3) and 3(2) tables.

The McNemar test for correlated proportions has been generalized to test for marginal homogeneity of either a 2c or an m2 contingency table. The tests for 2(3) and 3(2) tables turn out to be equivalent since each involves six non-diagonal elements; equal numbers of degrees of freedom in the associated chi square tests, m - 1 = c - 1 = 2; identical constraints on the fits to the non-diagonal elements; and identical measures of discrepancy between observed and fitted values to be optimized.     

A Comparison of Tests for Marginal Homogeneity in Square Contingency Tables

Several asymptotic tests were proposed for testing the null hypothesis of marginal homogeneity in square contingency tables with r categories. A simulation study was performed for comparing the power of four finite conservative conditional test procedures and of two asymptotic tests for twelve different contingency schemes for small sample sizes. While an asymptotic test proposed by STUART (1955) showed a rather satisfactory behaviour for moderate sample sizes, an asymptotic test proposed by BHAPKAR (1966) was quite anticonservative. With no a priori information the performance of (r - 1) simultaneous conditional binomial tests with a Bonferroni adjustment proved to be a quite efficient procedure. With assumptions about where to expect the deviations from the null hypothesis, other procedures favouring the larger or smaller conditional sample sizes, respectively, can have a great efficiency. The procedures are illustrated by means of a numerical example from clinical psychology.     

Simultaneous sign tests for marginal homogeneity of square contingency tables

For testing the nullhypothesis of marginal homogeneity of 1x1-contingency tables it is proposed to perform I simultaneous sign tests and to combine them according to the BONFFERRONI or the ?INáK inequality.     

Testing Quasi-Independence in Stratified Triangular Contingency Tables

A special class of incomplete contingency tables has structural zeroes in one or more cells. Some of these tables have a triangular appearance in the sense that they are square and their cells are known a priori to contain zeroes above (or below) their main diagonals. We propose methods of analyzing log-linear models and testing quasi-independence in these triangular tables. We also offer a method for combining such tables that are stratified by a concomitant categorical variable. This strategy follows the same approach used in a Cochran-Mantel-Haenszel test.     

Is the Exact Test Better than the Asymptotic Test for Testing Marginal Homogeneity in 2 × 2 Tables?

McNemar test is commonly used to test for the marginal homogeneity in 2 × 2 contingency tables. McNemar test is an asymptotic test based either on standard normal distribution or on the chi‐square distribution. When the total sample size is small, an exact version of McNemar test is available based on the binomial probabilities. The example in the paper came from a clinical study to investigate the effect of epidermal growth factor for children who had microvillus inclusion diseases. There were only six observations available. The test results differ between the exact test and the asymptotic test. It is a common belief that with this small sample size the exact test be used. However, we claim that McNemar test performs better than the exact test even when the sample size is small. In order to investigate the performances of McNemar test and the exact test, we identify the parameters that affect the test results and then perform sensitivity analysis. In addition, through Monte Carlo simulation studies we compare the empirical sizes and powers of these tests as well as other asymptotic tests such as Wald test and the likelihood ratio test.     

Test statistic and sample size for a two-sample McNemar test

McNemar's (1947, Psychometrika 12, 153-157) test of marginal homogeneity is generalized to a two-sample situation where the hypothesis of interest is that the marginal changes in each of two independently sampled tables are equal. This situation is especially applicable to two cohorts (a control and an intervention cohort), each measured at baseline and after the intervention on a binary outcome variable. Some assumptions often realistic in this situation simplify the calculation of sample size. The calculation of sample size in a study designed to increase utilization of breast cancer screening is demonstrated.     

Power and sample size for testing homogeneity of relative risks in prospective studies

Power and sample-size formulas for testing the homogeneity of relative risks using the score method are presented. The homogeneity score test (Gart, 1985, Biometrika 72, 673-677) is formally equivalent to the Pearson chi-square test, although they look different. Results of this paper may be useful in assessing the validity of the model of a common relative risk before combining several 2 x 2 tables or in designing a prospective study for detecting heterogeneity of relative risks.     

A Test Procedure of Equivalence in Ordinal Data with Matched‐Pairs

There are many epidemiologic studies or clinical trials, in which we may wish to establish an equivalence rather than to detect a difference between the distributions of responses. In this paper, we develop test procedures to detect equivalence with respect to the tail marginal distributions and the marginal proportions when the underlying data are on an ordinal scale with matched pairs. We include a numerical example concerning the unaided distance vision of two eyes over 7477 women to illustrate the practical usefulness of the proposed procedure. Finally, we include a brief discussion on the relation between the test procedures developed here and an asymptotic interval estimator proposed elsewhere for the simple difference in dichotomous data with matched‐pairs.     

Testing marginal homogeneity against stochastic order in multivariate ordinal data

Summary .  Many assessment instruments used in the evaluation of toxicity, safety, pain, or disease progression consider multiple ordinal endpoints to fully capture the presence and severity of treatment effects. Contingency tables underlying these correlated responses are often sparse and imbalanced, rendering asymptotic results unreliable or model fitting prohibitively complex without overly simplistic assumptions on the marginal and joint distribution. Instead of a modeling approach, we look at stochastic order and marginal inhomogeneity as an expression or manifestation of a treatment effect under much weaker assumptions. Often, endpoints are grouped together into physiological domains or by the body function they describe. We derive tests based on these subgroups, which might supplement or replace the individual endpoint analysis because they are more powerful. The permutation or bootstrap distribution is used throughout to obtain global, subgroup, and individual significance levels as they naturally incorporate the correlation among endpoints. We provide a theorem that establishes a connection between marginal homogeneity and the stronger exchangeability assumption under the permutation approach. Multiplicity adjustments for the individual endpoints are obtained via stepdown procedures, while subgroup significance levels are adjusted via the full closed testing procedure. The proposed methodology is illustrated using a collection of 25 correlated ordinal endpoints, grouped into six domains, to evaluate toxicity of a chemical compound.     

Testing hypotheses in case-control studies--equivalence of Mantel-Haenszel statistics and logit score tests.

The two approaches in common use for the analysis of case-control studies are cross-classification by confounding variables, and modeling the logarithm of the odds ratio as a function of exposure and confounding variables. We show here that score statistics derived from the likelihood function in the latter approach are identical to the Mantel-Haenszel test statistics appropriate for the former approach. This identity holds in the most general situation considered, testing for marginal homogeneity in mK tables. This equivalence is demonstrated by a permutational argument which leads to a general likelihood expression in which the exposure variable may be a vector of discrete and/or continuous variables and in which more than two comparison groups may be considered. This likelihood can be used in analyzing studies in which there are multiple controls for each case or in which several disease categories are being compared. The possibility of including continuous variables makes this likelihood useful in situations that cannot be treated using the Mantel-Haenszel cross-classification approach.     

Powerful Exact Unconditional Tests for Agreement between Two Raters with Binary Endpoints

Asymptotic and exact conditional approaches have often been used for testing agreement between two raters with binary outcomes. The exact conditional approach is guaranteed to respect the test size as compared to the traditionally used asymptotic approach based on the standardized Cohen''s kappa coefficient. An alternative to the conditional approach is an unconditional strategy which relaxes the restriction of fixed marginal totals as in the conditional approach. Three exact unconditional hypothesis testing procedures are considered in this article: an approach based on maximization, an approach based on the conditional p-value and maximization, and an approach based on estimation and maximization. We compared these testing procedures based on the commonly used Cohen''s kappa with regards to test size and power. We recommend the following two exact approaches for use in practice due to power advantages: the approach based on conditional p-value and maximization and the approach based on estimation and maximization.     

Notes in Case-Control Studies with Matched Pairs under Inverse Sampling

In case-control studies with matched pairs, the traditional point estimator of odds ratio (OR) is well-known to be biased with no exact finite variance under binomial sampling. In this paper, we consider use of inverse sampling in which we continue to sample subjects to form matched pairs until we obtain a pre-determined number (>0) of index pairs with the case unexposed but the control exposed. In contrast to use of binomial sampling, we show that the uniformly minimum variance unbiased estimator (UMVUE) of OR does exist under inverse sampling. We further derive an exact confidence interval of OR in closed form. Finally, we develop an exact test and an asymptotic test for testing the null hypothesis H₀: OR = 1, as well as discuss sample size determination on the minimum required number of index pairs for a desired power at α-level.     

Unconditional versions of several tests commonly used in the analysis of contingency tables

By focusing on a confidence interval for a nuisance parameter, Berger and Boos (1994, Journal of the American Statistical Association 89, 1012-1016) proposed new unconditional tests. In particular, they showed that, for a 2 x 2 table, this procedure generally was more powerful than Fisher's exact test. This paper utilizes and extends their approach to obtain unconditional tests for combining several 2 x 2 tables and testing for trend and homogeneity in a 2 x K table. The unconditional procedures are compared to the conditional ones by reanalyzing some published biomedical data.     

Notes on Testing Equality in Dichotomous Data with Matched Pairs

In attempting to improve the efficiency of McNemar's test statistic, we develop two test procedures that account for the information on both the discordant and concordant pairs for testing equality between two comparison groups in dichotomous data with matched pairs. Furthermore, we derive a test procedure derived from one of the most commonly‐used interval estimators for odds ratio. We compare these procedures with those using McNemar's test, McNemar's test with the continuity correction, and the exact test with respect to type I error and power in a variety of situations. We note that the test procedures using McNemar's test with the continuity correction and the exact test can be quite conservative and hence lose much efficiency, while the test procedure using McNemar's test can actually perform well even when the expected number of discordant pairs is small. We also find that the two test procedures, which incorporate the information on all matched pairs into hypothesis testing, may slightly improve the power of using McNemar's test without essentially losing the precision of type I error. On the other hand, the test procedure derived from an interval estimator of adds ratio with use of the logarithmic transformation may have type I error much larger than the nominal α‐level when the expected number of discordant pairs is not large and therefore, is not recommended for general use.     

Systematic detection of epistatic interactions based on allele pair frequencies

Epistatic genetic interactions are key for understanding the genetic contribution to complex traits. Epistasis is always defined with respect to some trait such as growth rate or fitness. Whereas most existing epistasis screens explicitly test for a trait, it is also possible to implicitly test for fitness traits by searching for the over- or under-representation of allele pairs in a given population. Such analysis of imbalanced allele pair frequencies of distant loci has not been exploited yet on a genome-wide scale, mostly due to statistical difficulties such as the multiple testing problem. We propose a new approach called Imbalanced Allele Pair frequencies (ImAP) for inferring epistatic interactions that is exclusively based on DNA sequence information. Our approach is based on genome-wide SNP data sampled from a population with known family structure. We make use of genotype information of parent-child trios and inspect 3×3 contingency tables for detecting pairs of alleles from different genomic positions that are over- or under-represented in the population. We also developed a simulation setup which mimics the pedigree structure by simultaneously assuming independence of the markers. When applied to mouse SNP data, our method detected 168 imbalanced allele pairs, which is substantially more than in simulations assuming no interactions. We could validate a significant number of the interactions with external data, and we found that interacting loci are enriched for genes involved in developmental processes.     

亲代传递的独立性与传递不平衡的边缘齐性检验

传递不平衡检验其实质是边缘齐性检验,需同时考察家系中两个亲代向受累子代的传递情形．但是,其零假设成立时,同一家系中父亲的传递与母亲的传递并不一定相互独立．本文探讨了父母传递的独立性条件对检验统计量的影响．结果表明,即使父母传递不独立时,边缘齐性检验仍然适合．