Showing papers on "Wilcoxon signed-rank test published in 1970"

A generalized Kruskal-Wallis test for comparing K samples subject to unequal patterns of censorship

Abstract: SUMMARY A generalization of the Kruskal-Wallis test, which extends Gehan's generalization of Wilcoxon's test, is proposed for testing the equality of K continuous distribution functions when observations are subject to arbitrary right censorship. The distribution of the censoring variables is allowed to differ for different populations. An alternative statistic is proposed for use when the censoring distributions may be assumed equal. These statistics have asymptotic chi-squared distributions under their respective null hypotheses, whether the censoring variables are regarded as random or as fixed numbers. Asymptotic power and efficiency calculations are made and numerical examples provided. A generalization of Wilcoxon's statistic for comparing two populations has been proposed by Gehan (1965a) for use when the observations are subject to arbitrary right censorship. Mantel (1967), as well as Gehan (1965b), has considered a further generalization to the case of arbitrarily restricted observation, or left and right censorship. Both of these authors base their calculations on the permutation distribution of the statistic, conditional on the observed censoring pattern for the combined sample. However, this model is inapplicable when there are differences in the distribution of the censoring variables for the two populations. For instance, in medical follow-up studies, where Gehan's procedure has so far found its widest application, this would happen if the two populations had been under study for different lengths of time. This paper extends Gehan's procedure for right censored observations to the comparison of K populations. The probability distributions of the relevant statistics are here considered in a large sample framework under two models: Model I, corresponding to random or unconditional censorship; and Model II, which considers the observed censoring times as fixed numbers. Since the distributions of the censoring variables are allowed to vary with the population, Gehan's procedure is also extended to the case of unequal censorship. For Model I these distributions are theoretical distributions; for Model II they are empirical. Besides providing chi-squared statistics for use in testing the hypothesis of equality of the K populations against general alternatives, the paper shows how single degrees of freedom may be partitioned for use in discriminating specific alternative hypotheses. Several investigators (Efron, 1967) have pointed out that Gehan's test is not the most efficient against certain parametric alternatives and have proposed modifications to increase its power. Asymptotic power and efficiency calculations made below demonstrate that their criticisms would apply equally well to the test proposed here. Hopefully some of the modifications they suggest can likewise eventually be generalized to the case of K

Sequential Signed-Rank Test

Abstract: A one-sample nonparametric test is proposed which sequentially examines the values of the Wilcoxon signed-rank statistic. The test rejects the null hypothesis of symmetry about zero if for any n between 1 and N the Wilcoxon signed-rank statistic for X 1, …, X n differs from its null mean by |z| aN standard deviations; otherwise the null hypothesis is accepted. The critical constant |z| a N is determined by Monte Carlo sampling.

A distribution-free test for parallelism

Abstract: This article suggests a distribution-free signed rank test for the parallelism of two regression lines. Pitman efficiency comparisons of the signed rank test and a normal theory t-test are presented. Monte Carlo sampling is used to compare the signed rank test with a nonparametric test proposed by Potthoff [7].

The effects of dependence on nonparametric detection

Abstract: This paper investigates the effects of dependence on rank tests, in particular on a class of recently defined nonparametric tests called "mixed" statistical tests. It is shown that the mixed test statistic is asymptotically normal for Gaussian processes with mild regularity properties justifying the use of asymptotic relative efficiency (ARE) as a figure of merit. Results are presented in terms of variations on three well-known statistics--the one-sample Wilcoxon, the two-sample Mann-Whitney, and the Kendall \tau . It is found that the effects of dependence on ARE with respect to a parametric test can be offset to some extent by appropriately grouping sample values. If, however, a constant false-alarm rate is to be attained, either the form of the dependence must be known or some learning scheme must be applied.

A nonparametric test for relative spread ink unpaired samples

Abstract: TheSiegel-Tukey test is generalized so that it may be applied to more than two samples. Consequently, the originalT statistic [Wilcoxon] of theSiegel-Tukey test is to be replaced by theH statistic [Kruskal andWallis]. Differences in location are to be controlled. A numerical example is given.

On Nonparametric Estimation and Testing for Interactions in Factorial Experiments

Abstract: Some optimum nonparametric tests and estimates for contrasts of interactions in two-factor models with replications are proposed and discussed. These procedures are based on nonparametric estimates of the type considered by Hodges and Lehmann [4]. Unbiasedness and joint asymptotic normality of the estimates are proved and the asymptotic efficiencies relative to the least squares procedure is found. It is shown that, in the special cases of Wilcoxon and normal scores, these methods are more robust than least squares against changes in the underlying distribution. Also, confidence intervals based on the above procedures are investigated (c.f. [12]). An example is given to explain the computations.

Linear Interpolation within McCornack's Table of the Wilcoxon Matched Pair Signed Rank Statistic

Abstract: For 20 ≤ N ≤ 100 cumulative probabilities of the Wilcoxon signed rank statistic T were approximated by linear interpolation with interpolation based on the table of critical values given by McCornack [3]. It was found that the exact probability was never under-approximated and that relative errors were less than ten percent for all values of T corresponding to probability levels of at least .0025 and N >30. This approximation was superior to the normal approximation at the .0005 and .005 levels, but inferior at the .025 level. Neither approximation appeared to exhibit any sustained superiority at the .05 level for 30 < N ≤ 100.

Tables for Wilcoxon's Rank Sum Test in Randomized Blocks

Abstract: The Wilcoxon two-sample test carried out in randomized blocks is discussed and exemplified. Tables of critical values are given for 1-sided (2-sided) significance levels of 0.001 (0.002), 0.005 (0.01), 0.01 (0.02), 0.025 (0.05), 0.05 (0.10), and 0.10 (0..

Properties of a rank test of cronholm and revusky.

Abstract: Cronholm and Revusky [4] have proposed a rank test based on independent subexperiments for the comparison of a treatment effect with a control when the treatment is destructive but the control produces at most a transient effect. This article provides a thorough study of its performance under two sampling situations. With the aid of a computer, the exact small sample power of the test is evaluated for some important alternatives. A modification of the test is proposed for the situation where numerical measurements are available rather than merely the ranks within each stage. The modification is found to substantially improve the power. Both tests are shown to be unbiased. Finally, the Pitman asymptotic efficiencies are obtained and comparisons are made with the appropriate Wilcoxon tests.