Exact Tests for Comparing Two Paired Proportions with Incomplete Data

Various asymptotic test procedures have been developed previously for testing the equality of two binomial proportions with partially incomplete paired data. Test procedures that discard incomplete observations have been shown to be less powerful than those procedures that utilize all available observations. On the other hand, asymptotic test procedures that utilize all available observations may not be reliable in small‐sample problems or sparse data structures. In this article, unconditional exact test procedures are proposed for testing the equality of two paired binomial proportions with partially incomplete paired data under a random mechanism. The proposed unconditional exact test methods are illustrated with real data from a neurological study. Empirical studies are conducted to investigate the performance of these and other test procedures with respect to size and power. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A New Rank Test Against Ordered Alternatives in K-Sample Problems

The distribution-free test against ordered alternatives proposed by Jonckheere (1954) is based on the Kendall's rank correlation coefficient τ. A new rank test is proposed and illustrated with a numerical example. The proposed test is based on Spearman's σ and has similar functional structure as the Kruskal-Wallis test. A useful by-product is a test for departure from a trend.     

A Unifying,Analytic Model for Certain Linear Rank Tests. Part II: Exact Distribution Algorithms

On the basic of the analytic properties of the unifying model developed in the earlier Part I of this presentation, exact distribution algorithms are developed for the Wilcoxon 2-sample sum-of-rank test, the Freund-Ansari scale test, and the Wilcoxon signed-rank test. For the signed-rank test, provisions are made for zero-differences.     

Signed-Rank Tests in the Presence of Ties (with extended tables)

We begin with a review of the areas of application of the signed-rank tests (SRTs) and we conclude that the results are exact only if no ties of non-null differences exist. In order to apply the SRTs according to WILCOXON and according to PRATT also in the presence of ties, by assigning midranks, we derive their null distributions. As special cases the null distributions for the problem without ties are obtained. In order to save the practising statistician the time-consuming calculations of the distribution functions, we compute tables of critical values (for reasons of volume they will be published as part of the reprints only). For N₀ = 0 (1) 5 null differences and M = = 1(1) 10 non-null differences the critical values of all distributions with all possible tie vectors are calculated. Instructions are provided and an example serves to illustrate the use of the table. The extension of the tables are obtained by means of counting formulas given in the text. Approximations are provided in order to make the application of tests possible for larger samples as well. It is shown that the approximation of the null distribution in the presence of ties by the null distributions under the assumption of no ties in some cases overstates and sometimes understates the exact rejection probability. For N₀ = 0 (1) 10 and M = 1 (1) 10 all distributions with all possible tie vectors for the SRTs with WILCOXON and PRATT ranking are examined with respect to the lattice type of the test statistic. The result is given in table 6. It is evident that the portion of PRATT -distributions with lattice character decreases as the number of null differences increases. Continuity corrections are obtained for the asymptotic normal distribution which take into account the lattice character of the distribution of the test statistic.     

A Coefficient of Agreement Adjusted for Bias in Paired Ordered Categorical Data

The quality of ordered categorical recordings is determined from repeated measurements on the same subject in order to assess the level of agreement between raters, scales or occasions. The presented rating-invariant method for ordered categorical data provides means of analysing the quality of single-item rating scales, irrespective of the number of possible response values and the marginal distributions. Marginal heterogeneity implies systematic disagreement, so-called bias. An augmented ranking approach is the basis for the separation of inter-rater disagreement into systematic and random components. Correlation between pairs of augmented rank values provides a measure of agreement to the best common ordering of paired classifications, given inter-rater bias. The essential differences in interpretation and applicability of the proposed coefficient of agreement and the Spearman rank-order correlation for ordered categorical data are discussed.     

Exact,Distribution Free Confidence Intervals for Late Effects in Censored Matched Pairs

When comparing censored survival times for matched treated and control subjects, a late effect on survival is one that does not begin to appear until some time has passed. In a study of provider specialty in the treatment of ovarian cancer, a late divergence in the Kaplan–Meier survival curves hinted at superior survival among patients of gynecological oncologists, who employ chemotherapy less intensively, when compared to patients of medical oncologists, who employ chemotherapy more intensively; we ask whether this late divergence should be taken seriously. Specifically, we develop exact, permutation tests, and exact confidence intervals formed by inverting the tests, for late effects in matched pairs subject to random but heterogeneous censoring. Unlike other exact confidence intervals with censored data, the proposed intervals do not require knowledge of censoring times for patients who die. Exact distributions are consequences of two results about signs, signed ranks, and their conditional independence properties. One test, the late effects sign test, has the binomial distribution; the other, the late effects signed rank test, uses nonstandard ranks but nonetheless has the same exact distribution as Wilcoxon's signed rank test. A simulation shows that the late effects signed rank test has substantially more power to detect late effects than do conventional tests. The confidence statement provides information about both the timing and magnitude of late effects (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Formulas for the Computation of the WILCOXON Test and other Rank Statistics

A simple program for the computation of the WILCOXON test is introduced, which is suitable for implementation on a programmable desk computer. The allocation of ranks R_i is not necessary (see BERCHTOLD, 1979). In an analogous manner any ranks respectively rank sums also can be calculated, so that any test criterion or statistical estimate called rank statistic, may be completely constituted by computer programs without explicitely ranging of values.     

EDISON-WMW: Exact Dynamic Programing Solution of the Wilcoxon–Mann–Whitney Test

In many research disciplines, hypothesis tests are applied to evaluate whether findings are statistically significant or could be explained by chance. The Wilcoxon–Mann–Whitney(WMW) test is among the most popular hypothesis tests in medicine and life science to analyze if two groups of samples are equally distributed. This nonparametric statistical homogeneity test is commonly applied in molecular diagnosis. Generally, the solution of the WMW test takes a high combinatorial effort for large sample cohorts containing a significant number of ties. Hence, P value is frequently approximated by a normal distribution. We developed EDISON-WMW, a new approach to calculate the exact permutation of the two-tailed unpaired WMW test without any corrections required and allowing for ties. The method relies on dynamic programing to solve the combinatorial problem of the WMW test efficiently. Beyond a straightforward implementation of the algorithm, we presented different optimization strategies and developed a parallel solution. Using our program,the exact P value for large cohorts containing more than 1000 samples with ties can be calculated within minutes. We demonstrate the performance of this novel approach on randomly-generated data, benchmark it against 13 other commonly-applied approaches and moreover evaluate molecular biomarkers for lung carcinoma and chronic obstructive pulmonary disease(COPD). We foundthat approximated P values were generally higher than the exact solution provided by EDISONWMW. Importantly, the algorithm can also be applied to high-throughput omics datasets, where hundreds or thousands of features are included. To provide easy access to the multi-threaded version of EDISON-WMW, a web-based solution of our algorithm is freely available at http://www.ccb.uni-saarland.de/software/wtest/.     

A Generalized Intrinsic Rank Test for k Independent Samples

The original intrinsic rank test is generalized in that the sizes of the k samples may now be arbitrary, and the number of intrinsic rank intervals need not equal the number of samples. Furthermore, the size of these intervals can be made variable, subject only to relatively mild constraints. These generalizations permit the formulation and testing of more specific hypotheses concerning the commonality of the sample distributions. A generalized intrinsic rank function is used to transform the usual ordinal ranks, obtained from the combined samples, into intrinsic ranks. Original sample identity and intrinsic ranks are then cross-tabulated and evaluated as 2-way contingency table.     

Exact Test Procedures for Generalized Kendall Correlation Coefficients

An exact test of KENDALL'S tau is presented, useful in small samples when the asymptotic test may not be adequate. The procedure is proposed especially for situations when ties and/or censored observations are present in the variables to be analyzed by KENDALL'S tau. The paper describes generalizations of the procedure, presents calculated examples and offers a computer program, which facilitates the usage of the proposed procedure.     

Generalized Wilcoxon Rank Tests and Sensitivity Curves

In this paper we use the sensitivity curves of TUKEY (1977) and the change of decision point (cdp) (a modified version of the breakdown point of YLVISAKER, 1977), supplemented by simulation studies to acquire knowledge about sensitivity in generalized Wilcoxon rank test statistics. Sensitivity depends on balanced or unbalanced sample size cases, censoring, combinations of failure distributions and sources of errors in the data. It is important to consider the quality of the data, and the results show that cdp and some properties of the sensitivity curves may serve as a hint when selecting a test statistic and when making a decision for a given test statistic.     

An Extended Threshold Model for Analyzing Ordered Categorical Data

This paper presents an extended threshold model for analyzing ordered categorical data. The model admits interactions between the position of the thresholds and the levels of the effective factors. These interactions are described according to the approach of Milliken and Graybill (1970). Especially important for practical application is the special assumption that there is a linear relation between interactions and thresholds, and that the slopes of the concerning regression lines may be different for samples. This means that the latent variables are distributed according to the same type of distributions, but may have different expectations and variances. Underlying this submodel, the estimation of parameters and the testing of hypotheses according to the maximum likelihood method is described. The procedure is illustrated by a numerical example, and an outline is given about a cluster analysis using model parameters.     

Analysis of ordered categorical data: two score-independent approaches

SUMMARY: A trend test is often employed to analyze ordered categorical data, in which a set of increasing scores is assigned a priori. There is a drawback in this approach, because how to choose a set of scores is not clear. There have been debates on which scores should be used (e.g., Graubard and Korn, 1987, Biometrics 43, 471-476; Ivanova and Berger, 2001, Biometrics 57, 567-570; Senn, 2007, Biometrics 63, 296-298). Conflicting conclusions are often obtained with different sets of scores. Two approaches, which have been applied to genetic case-control studies, are appealing for ordered categorical data, because they take into account the natural order in the data, are score independent, and not contingent on asymptotic theory. These two approaches are applied to a prospective study for detecting association between maternal drinking and congenital malformations.     

A sampling study of the size and power of tests for paired survival data

The paired-t, sign, and signed rank tests were compared for samples from a bivariate exponential distribution. Each is a valid α-level test. One test was not uniformly more powerful than the others for all sample sizes, α levels, correlations, and alternative hypotheses considered, but the signed rank test did well consistently. It was always preferable to the sign test and never was appreciably worse than the paired-t test. The relative performance of the tests depends on α as well as the sample size.