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.     

塔克尔莫乎尔沙漠银沙槐群落主要植物的种间关系

根据野外调查的样地数据资料,运用2×2列联表的Fisher精确检验、Pearson相关系数和Spearman秩相关系数检验,并结合DCA排序,分析塔克尔莫乎尔沙漠银沙槐群落主要植物种群之间的种间关系。结果表明:Fisher精确检验有3个种对呈显著正关联,3个种对呈显著负关联;Pearson相关系数检验有1个种对呈显著正相关,12个种对呈显著负相关;Spearman秩相关系数检验,有14个种对存在显著相关,其中负相关13对,占所有种对数的14.29%,正相关1对,占所有种对数的1.1%。银沙槐群落种间关系相对松散,反映出植物群落随着环境变化其物种组成也发生相应变化,正向着相对稳定独立的无关联方向发展。     

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.     

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.     

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.     

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.     

Statistical Inference For Risk Difference in an Incomplete Correlated 2 × 2 Table

In some infectious disease studies and 2‐step treatment studies, 2 × 2 table with structural zero could arise in situations where it is theoretically impossible for a particular cell to contain observations or structural void is introduced by design. In this article, we propose a score test of hypotheses pertaining to the marginal and conditional probabilities in a 2 × 2 table with structural zero via the risk/rate difference measure. Score test‐based confidence interval will also be outlined. We evaluate the performance of the score test and the existing likelihood ratio test. Our empirical results evince the similar and satisfactory performance of the two tests (with appropriate adjustments) in terms of coverage probability and expected interval width. Both tests consistently perform well from small‐ to moderate‐sample designs. The score test however has the advantage that it is only undefined in one scenario while the likelihood ratio test can be undefined in many scenarios. We illustrate our method by a real example from a two‐step tuberculosis skin test study.     

山西崦山自然保护区侧柏群落优势种群种间关系分析

在样方调查的基础上,采用方差比率法、Fisher精确检验、Jaccard指数和Spearman秩相关系数研究了山西阳城崦山自然保护区侧柏群落优势种群的种间关系。结果表明：（1）群落优势种总体呈正关联,但关联性较弱,种的分布具有相对独立性。（2）Fisher精确检验与Jaccard指数结合使用可以取得较好的研究结果。（3）Jaccard指数结果表明在乔、灌、草各层中占据明显优势地位的种群其种间联结程度较大。（4）根据研究结果把侧柏群落的优势种群划分为3个生态种组：白皮松生态种组、侧柏生态种组和白莲蒿生态种组。     

Multiple species effects and spatial autocorrelation in detecting species associations

Abstract. The traditional approach to the analysis of species association within a community, based upon co-occurrence in sampling units such as quadrats, has been to test all pairs of species, using a 2 × 2 contingency table for each pair. It has long been recognised that all these tests are not independent of each other, but there is an additional problem in that the association between any particular pair may depend on the combination of the other species that are present or on the environmental factors that determine that combination. We use a 2^k contingency table to examine this problem and find that pairwise associations are not independent of the other species. The second problem that we consider is the effect of spatial autocorrelation in the data which makes the statistical tests too liberal. In the absence of a derived solution for a deflation factor to correct the test statistic calculated from a 2^k table, we describe a Monte Carlo approach that provides an approximate solution to this problem. In our data the amount of deflation that is necessary for a 2^k table is small compared to the amount required for the 2 × 2 tables used to test pairwise association.     

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.     

Exact Power of Conditional Tests for Several Binomial Populations

An efficient recursive polynomial multiplication method is proposed for exact unconditional power calculation for unordered 2 × K contingency table with up to moderate sample size. Our method can be applied to the family of cell-additive statistics which includes the Freeman-Halton statistic, the Pearson χ² statistic and the likelihood ratio statistic. We illustrate our proposed method by several numerical examples.     

On Conditions for Validity of the Approximations to Fisher's Exact Test

A 2 × 2 table is analyzed from the conditional viewpoint, using Fisher's exact test, for which there is abundant software nowadays (StatXact, SPSS with the module “Exact Test”, …). Nevertheless, because it is well-nigh impossible to work the test out “by hand”, it is customary in many books to analyze the table, in an approximate fashion, by using the classic chi-square test with the appropriate continuity correction, and this is what many researchers do in practice. Unfortunately little research has been carried out on the validity conditions of the test (remember the classic advice that the minimum expected quantity E should be larger than or equal to 5), so that it is applied indiscriminately with the obvious danger of obtaining erroneous results. In this paper the exact validity conditions (which depend not only on E, but also on the real P-value, on the size of the sample, on the number of tails of the test and on the continuity correction used) are determined, showing that the classic condition E > 5 can be either very strict or quite liberal (depending on the circumstances) and it can even be necessary for the test to have E > 20. A similar study is made of the case of using the binomial approximation, while the Poisson approximation is also discussed. The article ends with a list of rules and precautions to be borne in mind by researchers using the approximate methods. The strict rules are complex enough to make the researcher want to use the exact test, but simplified (and conservative) rules are also given for those unable to do so.     

A Test for Homogeneity of Odds Ratios in Ordered 2 × 2 Tables

Consider K ordered 2 × 2 contingency tables. A new test of the null hypothesis that the odds ratios of these tables are equal vs the alternative hypothesis that the odds ratios are nondecreasing, is recommended. The test is exact (non‐asymptotic), is easily carried out (software is available), and has other favorable properties. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some New Aspects of 2 × 2 Table Measures and Their Applications

The paper deals with the analysis of 2 × 2 table measures and describes several new possibilities for applying them. An estimate is given for HELLMICH's asymmetric K-measure and a possible extension of the contingency concept in a special 2×2 case is presented. The concept introduced and the recommended computer application of the 2×2 table measures is supported by new results obtained for the relationship between jaw joint pains and noises examined earlier by F. SCHMID and C. ZSCHEGE. With a larger number of observations the opportunity arises for a deeper analysis of this dental problem using path analysis. The plan of evaluation and the conncetion between the variables examined is illustrated by a path-diagram. The association tables determined on the basis of the (C) values introduced during the examination of jaw joint disturbances using association analysis draw attention to the special role of the left side of mastication. The paper raises ideas for further generalisations and indicates where further research is needed.     

九连山常绿阔叶林乔木优势种群的种间关联性分析

九连山国家级自然保护区分布着原生的典型常绿阔叶林, 且已形成不同的优势群落, 目前尚不清楚群落乔木层中优势物种之间的作用关系。采用2×2联列表, 通过方差分析, χ²检验、Pearson相关系数和Spearman秩相关系数检验, 对九连山常绿阔叶林乔木层中重要值较高的28个优势种群、378个种对间的关联性进行定量研究。方差分析表明: 28个优势种群的总体种间关联性呈显著的正关联, 反映该群落处于较稳定的顶极阶段。不同检验结果表明: χ²检验结果有140个种对呈正相关, 238个种对呈负相关, 正负关联比为0.588; Pearson相关系数检验有104个种对呈正相关, 274个种对呈负相关, 正负关联比为0.380; Spearman秩相关系数检验有144个种对呈正相关, 234个种对呈负相关, 正负关联比为0.615; 与Pearson相关系数检验方法相比, Spearman秩相关系数检验具有较高的灵敏度。378个种对中, 绝大多数种对的联结关系未达到显著水平, 种对间的独立性相对较强, 这种种间联结的松散性可能与群落目前的发展阶段及物种本身的生态学特性有关, 各群落正处于稳定的顶极阶段。根据28个优势种群对环境的适应方式和主导生态因素, 可将它们划分为阳生植物和阴生植物两大生态种组。     

An empirical assessment of saddlepoint approximations for testing a logistic regression parameter.

Saddlepoint methods provide quick and easy approximations to significance levels for conditional tests of logistic regression parameters. We evaluate the accuracies of saddlepoint approximations for three well-known conditional tests: Bartlett's test for no three-factor interaction in a 2 x 2 x 2 table, the test for trend in a series of probabilities, and the exact test of no association in stratified 2 x 2 tables with a common odds ratio. General recommendations are suggested regarding the use of saddlepoint approximations for exact conditional significance levels.     

Testing the Equality of the Intra‐Subject Treatment Covariance Matrices in a Crossover Study

In a 2 × 2 crossover bioavailability study, the sets of estimates of the pharmacokinetic parameters quite often have a symmetric covariance structure between the two treatments. For testing the equality of the intra‐subject covariance matrices for the two treatments in such studies, we suggest in this paper some statistical tests. When the response vectors are bivariate, we propose an exact test. Since the statistical procedures depend on the assumption of a symmetric covariance structure between the two treatments, we put forth some statistical tests for this assumption. We then apply the discussed tests to real data from a crossover bioavailability trial.     

缙云山风灾迹地恢复群落主要乔木树种种间联结性

应用方差比率法,并通过2×2列联表χ2检验法及利用联结系数AC指数测定法研究了缙云山自然保护区风灾迹地恢复群落自然恢复样地和人工恢复样地中主要乔木树种的种间联结性。结果表明:(1)自然恢复样地主要乔木种群间在总体上呈一定的正关联;而人工恢复样地种群间在总体上呈负关联;(2)2×2列联表的χ2检验共有5个(3)种对为显著正关联(括号内为人工恢复群落的种对数,下同),2个(1)种对为显著负关联;(3)联结系数(AC)指数标量出了自然恢复和人工恢复样地中种对间的联结程度和相伴随出现的机率,两种恢复方式中的多数乔木种群间的联结性不显著且联结程度较低,绝大多数乔木树种之间无显著关联,而且人工恢复比自然恢复无关联性更强;(4)种对间的正关联,主要是由于它们具有相近的生物学特性以及对生境具有相似的生态适应性所致,种对间的负关联主要是由于物种间的空间排斥,适应不同的微环境的结果;(5)人工恢复和自然恢复所在的群落处于演替的不稳定阶段。     

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.     

Conditional Maximum Likelihood Estimate and Exact Test of the Common Relative Difference in Combination of 2 × 2 Tables under Inverse Sampling

On the basis of the conditional distribution, given the marginal totals of non-cases fixed for each of independent 2 × 2 tables under inverse sampling, this paper develops the conditional maximum likelihood (CMLE) estimator of the underlying common relative difference (RD) and its asymptotic conditional variance. This paper further provides for the RD an exact interval calculation procedure, of which the coverage probability is always larger than or equal to the desired confidence level and for investigating whether the underlying common RD equals any specified value an exact test procedure, of which Type I error is always less than or equal to the nominal α-level. These exact interval estimation and exact hypothesis testing procedures are especially useful for the situation in which the number of index subjects in a study is small and the asymptotically approximate methods may not be appropriate for use. This paper also notes the condition under which the CMLE of RD uniquely exists and includes a simple example to illustrate use of these techniques.