The analysis of stratified 2 x 2 contingency tables

We consider the problem of testing for independence against the consistent superiority of one treatment over another when the response variable is binary and is compared across two treatments in each of several strata. Specifically, we consider the randomized clinical trial setting. A number of issues arise in this context. First, should tables be combined if there are small or zero margins? Second, should one assume a common odds ratio across strata? Third, if the odds ratios differ across strata, then how does the standard test (based on a common odds ratio) perform? Fourth, are there other analyzes that are more appropriate for handling a situation in which the odds ratios may differ across strata? In addressing these issues we find that the frequently used Cochran–Mantel–Haenszel test may have a poor power profile, despite being optimal when the odds ratios are common. We develop novel tests that are analogous to the Smirnov, modified Smirnov, convex hull, and adaptive tests that have been proposed for ordered categorical data. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Describing Ordinal Odds Ratios for Stratified r × c Tables

For an r × ctable with ordinal responses, odds ratios are commonly used to describe the relationship between the row and column variables. This article shows two types of ordinal odds ratios where local‐global odds ratios are used to compare several groups on a c‐category ordinal response and a global odds ratio is used to measure the global association between a pair of ordinal responses. When there is a stratification factor, we consider Mantel‐Haenszel (MH) type estimators of these odds ratios to summarize the association from several strata. Like the ordinary MH estimator of the common odds ratio for several 2 × 2 contingency tables, the estimators are used when the association is not expected to vary drastically among the strata. Also, the estimators are consistent under the ordinary asymptotic framework in which the number of strata is fixed and also under sparse asymptotics in which the number of strata grows with the sample size. Compared to the maximum likelihood estimators, simulations find that the MH type estimators perform better especially when each stratum has few observations. This article provides variances and covariances formulae for the local‐global odds ratios estimators and applies the bootstrap method to obtain a standard error for the global odds ratio estimator. At the end, we discuss possible ways of testing the homogeneity assumption.     

A Modified Exact Test for 2 × 2 Contingency Tables

A modified exact test is proposed for 2×2 contingency tables. This test, which is based on a less conservative definition of the concept of significance (STONE, 1969) is compared with a modified form of Pearson's X² test and with Tocher's randomized exact (UMPU) test. The sizes of the new test lie near the nominal 0.05 levels while those of the X² test usually exceed the nominal level, sometimes by a factor of 2 or more. The power of the modified test is usually close to that of the UMPU test.     

Effects of aerobic exercise on metabolic syndrome improvement in response to weight reduction

Objective: The objective was to test effects of aerobic exercise training on metabolic syndrome (MetSyn) improvement in response to weight reduction. Research Methods and Procedures: A total of 459 overweight and obese women (age, 49 ± 9 years; BMI, 28 ± 3 kg/m²) were recruited for a baseline examination to test the relationship between cardiorespiratory fitness and metabolic syndrome prevalence; among these, 67 subjects with MetSyn were treated with 14‐week weight‐loss programs, which included low‐calorie diet and aerobic exercise. The MetSyn was defined according to the Examination Committee of Criteria for “Metabolic Syndrome” in Japan. Maximal oxygen uptake (V?o _2max) during a maximal cycling test was measured as an index of cardiorespiratory fitness at baseline and after the intervention. Results: In the baseline examination, age‐ and BMI‐adjusted odds ratios for MetSyn prevalence in the low, middle, and upper thirds of V?o _2max were 1.0 (referent), 0.50 (95% confidence interval, 0.26 to 0.95), and 0.39 (95% confidence interval, 0.14 to 0.96), respectively (linear trend, p = 0.02). The adjusted odds ratios for MetSyn improvement in the two interventions with diet alone and diet plus exercise were 1.0 and 3.68 (95% confidence interval, 1.02 to 17.6; p = 0.04), respectively. Discussion: These results suggest that adding aerobic exercise training to a dietary weight‐reduction program further improves MetSyn (adjusted odds ratio, 3.68) in obese women, compared with diet alone. Further studies on an association between V?o _2max change and MetSyn improvement are needed.     

Small Sample Properties of the Mantel-Haenszel Test Statistic for Density Follow-Up Studies

For the analysis of combinations of 2×2 non-contingency tables as obtained from density follow-up studies (relating a number of events to a number of person-years of follow-up) an analogue of the Mantel-Haenszel test for 2×2 contingency tables is widely used. In this paper the small sample properties of this test, both with and without continuity correction, are evaluated. Also the improvement of the test-statistic by using the first four cumulants via the Edgeworth expansion was studied. Results on continuity correction agree with similar studies on the Mantel-Haenszel statistic for 2×2 contingency tables: Continuity correction gives a p-value which approximates the exact p-value better than the p-value obtained without this correction; both the exact test and its approximations show considerable conservatism in small samples; the uncorrected Mantel-Haenszel test statistic gives a p-value that agrees more with the nominal significance level, but can be anti-conservative. The p-value based on the first four cumulants gives a better approximation of the exact p-value than the continuity corrected test, especially when the distribution has marked skewness.     

Conditional Rejection Probabilities of Student's t‐test and Design Adaptations

For clinical trials with interim analyses conditional rejection probabilities play an important role when stochastic curtailment or design adaptations are performed. The conditional rejection probability gives the conditional probability to finally reject the null hypothesis given the interim data. It is computed either under the null or the alternative hypothesis. We investigate the properties of the conditional rejection probability for the one sided, one sample t‐test and show that it can be non monotone in the interim mean of the data and non monotone in the non‐centrality parameter for the alternative. We give several proposals how to implement design adaptations (that are based on the conditional rejection probability) for the t‐test and give a numerical example. Additionally, the conditional rejection probability given the interim t‐statistic is investigated. It does not depend on the unknown σ and can be used in stochastic curtailment procedures. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Association between FABP2 Ala54Thr polymorphisms and type 2 diabetes mellitus risk: a HuGE Review and Meta‐Analysis

Many studies have examined the association between the FABP2 (rs1799883) Ala54Thr gene polymorphism and type 2 diabetes mellitus risk (T2DM) in various populations, but their results have been inconsistent. To assess this relationship more precisely, A HuGE review and meta‐analysis were performed. The PubMed and CNKI database was searched for case‐control studies published up to April 2014. Data were extracted and pooled odds ratios (OR) with 95% confidence intervals (CI) were calculated. Ultimately, 13 studies, comprising 2020 T2DM cases and 2910 controls were included. Overall, for the Thr carriers (Ala/Thr and Thr/Thr) versus the wild‐type homozygotes (Ala/Ala), the pooled OR was 1.18 (95% CI = 1.04–1.34, P = 0.062 for heterogeneity), for Thr/Thr versus Ala/Ala the pooled OR was 1.17 (95% CI = 1.05–1.41 P = 0.087 for heterogeneity). In the stratified analysis by ethnicity, the significantly risks were found among Asians but not Caucasians. This meta‐analysis suggests that the FABP2 (rs1799883) Ala54Thr polymorphisms are associated with increased susceptibility to T2DM risk among Asians but not Caucasians.     

On the Analysis of Special Incomplete Three-dimensional Contingency Tables

Incomplete contingency tables, i.e. tables with structurally caused empty cells, are analysed by means of so-called quasilog-linear models. In general the expected values can be calculated by means of iterative cyclic adaption to corresponding marginals of the empirical contingency tables (in the same way as in complete tables) under different hierarchical hypotheses concerning the parameters of the models. For important cases of 2-dimensional contingency tables it is possible to demonstrate that expected values and test statistics are to find in a closed form. If all 2-dimensional sub or partial tables of a 3-dimensional table can be assigned to such cases then the hypotheses of classes (AB×C) (??), (B×C)/A(??), (A??B)/A(??) etc. are testable in closed form. But the expected values to (A×B×C) (×) have to be calculated iteratively. An example shows that some definite additive decompositions of the test statistic 2 I are no longer valid while some others remain valid in spite of incompleteness of the tables.     

Stratified or Multicentre Analysis with Extended Mantel–Haenszel Statistics

Since its introduction in 1959 the ability of the classical Mantel-Haenszel (M–H) procedure for combining the odds ratios of a set of I 2 × 2 tables has led to its use also in stratified or multicentre type clinical trials. A familiar application is the M–H logrank test in survival analysis. An extension of the M–H procedure covering the case of 2 × K contingency tables (MANTEL , 1963) with ordered levels retains the essential property of pooling the results of I homogeneous tables (i.e. in absence of qualitative interactions). The assignment of some score for the K columns of a table is essential for the use of the method (in comparing 2 treatments). Some possibilities of score assignment are discussed: for clinical outcome variables such as the degree of severity of a disease, pain and so on, the score is at hand in a natural way. A less well-known type of scoring consists in ranking the observations of a continuous variable, leading to cell sizes of 1 or 0. In this case, however, if equidistant ranking was used, the E–M–H procedure appears as an extension of Wilcoxon's rank sum test and represents a powerful non-parametric approach in stratified or multicentre type designs with non normally distributed outcome variables. The results of some Monte-Carlo simulations for 2 possible equidistant ranking procedures are presented, which indicate only a moderate gain in power as compared to Wilcoxon's rank sum test under the common situation of centre effects not exceeding treatment effects. Use of the E–M–H pro?edure is also recommended as a simple method to overcome the potential bias due to unequally distributed prognostic factors among treatment groups.     

Testing the rate isomorphy hypothesis using five statistical methods

Abstract Organisms are said to be in developmental rate isomorphy when the proportions of developmental stage durations are unaffected by temperature. Comprehensive stage‐specific developmental data were generated on the cabbage beetle, Colaphellus bowringi Baly (Coleoptera: Chrysomelidae), at eight temperatures ranging from 16°C to 30°C (in 2°C increments) and five analytical methods were used to test the rate isomorphy hypothesis, including: (i) direct comparison of lower developmental thresholds with standard errors based on the traditional linear equation describing developmental rate as the linear function of temperature; (ii) analysis of covariance to compare the lower developmental thresholds of different stages based on the Ikemoto‐Takai linear equation; (iii) testing the significance of the slope item in the regression line of versus temperature, where p is the ratio of the developmental duration of a particular developmental stage to the entire pre‐imaginal developmental duration for one insect or mite species; (iv) analysis of variance to test for significant differences between the ratios of developmental stage durations to that of pre‐imaginal development; and (v) checking whether there is an element less than a given level of significance in the p‐value matrix of rotating regression line. The results revealed no significant difference among the lower developmental thresholds or among the aforementioned ratios, and thus convincingly confirmed the rate isomorphy hypothesis.     

Stratified Fisher's exact test and its sample size calculation

Chi‐squared test has been a popular approach to the analysis of a 2 × 2 table when the sample sizes for the four cells are large. When the large sample assumption does not hold, however, we need an exact testing method such as Fisher's test. When the study population is heterogeneous, we often partition the subjects into multiple strata, so that each stratum consists of homogeneous subjects and hence the stratified analysis has an improved testing power. While Mantel–Haenszel test has been widely used as an extension of the chi‐squared test to test on stratified 2 × 2 tables with a large‐sample approximation, we have been lacking an extension of Fisher's test for stratified exact testing. In this paper, we discuss an exact testing method for stratified 2 × 2 tables that is simplified to the standard Fisher's test in single 2 × 2 table cases, and propose its sample size calculation method that can be useful for designing a study with rare cell frequencies.     

Confidence Interval for Rate Ratio in a 2 × 2 Table with Structural Zero: An Application in Assessing False-Negative Rate Ratio When Combining Two Diagnostic Tests

Summary . In this article, we consider problems with correlated data that can be summarized in a 2 × 2 table with structural zero in one of the off‐diagonal cells. Data of this kind sometimes appear in infectious disease studies and two‐step procedure studies. Lui (1998, Biometrics 54, 706–711) considered confidence interval estimation of rate ratio based on Fieller‐type, Wald‐type, and logarithmic transformation statistics. We reexamine the same problem under the context of confidence interval construction on false‐negative rate ratio in diagnostic performance when combining two diagnostic tests. We propose a score statistic for testing the null hypothesis of nonunity false‐negative rate ratio. Score test–based confidence interval construction for false‐negative rate ratio will also be discussed. Simulation studies are conducted to compare the performance of the new derived score test statistic and existing statistics for small to moderate sample sizes. In terms of confidence interval construction, our asymptotic score test–based confidence interval estimator possesses significantly shorter expected width with coverage probability being close to the anticipated confidence level. In terms of hypothesis testing, our asymptotic score test procedure has actual type I error rate close to the pre‐assigned nominal level. We illustrate our methodologies with real examples from a clinical laboratory study and a cancer study.     

Normal Approximation of a Discrete Distribution on Exact Test of Uniform Association in a 2 × 3 Table

For the analysis of 2 × 3 tables, TOMIZAWA (1993) considered an exact test of uniform association, which is an extension of independence, and then derived a discrete distribution. This paper gives a normal approximation of the discrete distribution and describes that the normalized statistic can test a one-sided hypothesis on the uniform association. Also it points out that the square of the normalized test statistic is equal to the Pearson's chi-squared statistic for testing the uniform association.     

Testing for Treatment Effects on Subsets of Endpoints

Multiple endpoints are tested to assess an overall treatment effect and also to identify which endpoints or subsets of endpoints contributed to treatment differences. The conventional p‐value adjustment methods, such as single‐step, step‐up, or step‐down procedures, sequentially identify each significant individual endpoint. Closed test procedures can also detect individual endpoints that have effects via a step‐by‐step closed strategy. This paper proposes a global‐based statistic for testing an a priori number, say, r of the k endpoints, as opposed to the conventional approach of testing one (r = 1) endpoint. The proposed test statistic is an extension of the single‐step p‐value‐based statistic based on the distribution of the smallest p‐value. The test maintains strong control of the FamilyWise Error (FWE) rate under the null hypothesis of no difference in any (sub)set of r endpoints among all possible combinations of the k endpoints. After rejecting the null hypothesis, the individual endpoints in the sets that are rejected can be tested further, using a univariate test statistic in a second step, if desired. However, the second step test only weakly controls the FWE. The proposed method is illustrated by application to a psychosis data set.     

Finger length ratios (2D:4D) in anthropoids implicate reduced prenatal androgens in social bonding

The second‐to‐fourth digit ratio (2D:4D) has been proposed as a biomarker reflecting prenatal androgen effects (PAE), such that individuals with lower ratios have experienced higher PAE than those with higher ratios. 2D:4D has been correlated with a number of sex‐linked traits in humans such as aggression, promiscuity, and competitiveness. In addition, polygynous societies reportedly have lower 2D:4D (higher PAE) than more monogamous populations. This evidence suggests that PAE may be implicated in the development of sexually selected behaviors in humans. To place 2D:4D research into a broader context, we test the relationship between digit ratios and behavior across nonhuman anthropoids; polygynous species, with higher levels of intrasexual competition, should have more pronounced markers of PAE (lower 2D:4D) than pair‐bonded species. Our results accord with those found in humans: 2D:4D is lower in polygynous species and higher (lower PAE) in pair‐bonded species. Old World monkeys have low, and relatively invariant 2D:4D (high PAE), which is coupled with high levels of intrasexual competition. This contrasts with higher and more variable ratios in both great apes and New World monkeys. In addition, both male and female ratios decrease with increasing levels of intrasexual competition. Human ratios are intermediate between pair‐bonded and more promiscuous hominoids. We propose that PAE may be involved in promoting species characteristic social behavior in anthropoids. Am J Phys Anthropol, 2010. © 2009 Wiley‐Liss, Inc.     

Anthropometric measures, body composition, body fat distribution, and knee osteoarthritis in women

Objective: Increased BMI is a well‐recognized risk factor for radiographic knee osteoarthritis (rKOA); however, the contributions of the components of body composition, body fat distribution, and height to this association are not clear. Research Methods and Procedures: We examined 779 women ≥45 years of age from the Johnston County Osteoarthritis Project. Body composition was assessed using DXA, and rKOA was defined as Kellgren‐Lawrence grade ≥2. Logistic regression models examined the association between rKOA and the fourth compared with the first quartiles of anthropometric, body composition, and fat distribution measures adjusting for age, ethnicity, and prior knee injury. Results: The adjusted odds ratios and 95% confidence interval of BMI and weight were 5.27 (3.05, 9.13) and 5.28 (3.05, 9.16), respectively. In separate models, higher odds of rKOA were also found for fat mass [4.54 (2.68, 7.69)], percent fat mass [3.84 (2.26, 6.54)], lean mass [3.94 (2.22, 6.97)], and waist circumference [4.15 (2.45, 7.02)]. Waist‐to‐hip ratio was not associated with rKOA [1.45 (0.86, 2.43)], and percent lean mass was associated with lower odds [0.20 (0.11, 0.35)]. Taller women had higher odds of rKOA after adjustment for BMI [1.77 (1.05, 3.00)]. Discussion: This study confirms that BMI and weight are strongly associated with rKOA in women and suggests that precise measurements of body composition and measures of fat distribution may offer no advantage over the more simple measures of BMI or weight in assessment of risk of rKOA.     

A Two-Tailed P-Value for Fisher's Exact Test

A two-tailed P-value is presented for a significance test in two by two contingency tables. There is no extraneous quasi-observation such as is needed in the exact randomized uniformly most powerful unbiased (UMPU) test of the hypothesis of independence. The proposed P-value can never exceed unity and is always two-tailed, unlike other P-values proposed in the literature     

The pigment composition of Phaeocystis antarctica (Haptophyceae) under various conditions of light,temperature, salinity,and iron

The pigment composition of Phaeocystis antarctica was monitored under various conditions of light, temperature, salinity, and iron. 19′‐Hexanoyloxyfucoxanthin (Hex‐fuco) always constituted the major light‐harvesting pigment, with remarkably stable ratios of Hex‐fuco‐to‐chl a under the various environmental conditions. Increased pigment‐to‐chl a ratios at low irradiance confirmed the light‐harvesting function of Fucoxanthin (Fuco), 19′‐Hexanoyloxy‐4‐ketofucoxanthin (Hex‐kfuco), 19′‐butanoyloxyfucoxanthin (But‐fuco), and chl c2 and c3. Increased pigment‐to‐chl a ratios at high irradiance, low iron concentrations, and to a lesser extent at high salinity confirmed the photoprotective function of diadinoxanthin, diatoxanthin, and ß,ß‐carotene. Pigment ratios were not always according to expectations. The consistent increase in But‐fuco/chl at high temperature, high salinity, and low iron suggests a role in photoprotection rather than in light harvesting. Low Hex‐kfuco/chl ratios at high salinity were consistent with a role as light harvester, but the high ratios at high temperature were not, leaving the function of Hex‐kfuco enigmatic. Dedicated experiments were performed to test whether or not the light‐harvesting pigment Fuco could be converted into its structural relative Hex‐fuco, and vice versa, in response to exposure to light shifts. Rapid conversions could not be confirmed, but long‐term conversions cannot be excluded. New pigment ratios are proposed for chemotaxonomic applications. The ratios will improve pigment‐based diagnosis of algal species in waters dominated by P. antarctica.     

Exact Calculation of Permutational Distributions for Two Independent Samples

A simple Fortran Subroutine is given for the exact calculation of permutational distributions for two independent samples. Important special cases are the Fisher-Pitman randomization test, the Mann-Whitney U-test, the Mantel-Haenszel-test and the exact test for 2×k contingency tables. The algorithm needs polynomial time and is of use even for personal- or microcomputers.     

Bootstrap and Second‐Order Tests of Risk Difference

Summary Clinical trials data often come in the form of low‐dimensional tables of small counts. Standard approximate tests such as score and likelihood ratio tests are imperfect in several respects. First, they can give quite different answers from the same data. Second, the actual type‐1 error can differ significantly from nominal, even for quite large sample sizes. Third, exact inferences based on these can be strongly nonmonotonic functions of the null parameter and lead to confidence sets that are discontiguous. There are two modern approaches to small sample inference. One is to use so‐called higher order asymptotics ( Reid, 2003 , Annal of Statistics 31 , 1695–1731) to provide an explicit adjustment to the likelihood ratio statistic. The theory for this is complex but the statistic is quick to compute. The second approach is to perform an exact calculation of significance assuming the nuisance parameters equal their null estimate ( Lee and Young, 2005 , Statistic and Probability Letters 71 , 143–153), which is a kind of parametric bootstrap. The purpose of this article is to explain and evaluate these two methods, for testing whether a difference in probabilities p₂? p₁ exceeds a prechosen noninferiority margin δ₀ . On the basis of an extensive numerical study, we recommend bootstrap P‐values as superior to all other alternatives. First, they produce practically identical answers regardless of the basic test statistic chosen. Second, they have excellent size accuracy and higher power. Third, they vary much less erratically with the null parameter value δ₀ .