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Two-Sample Tests
Abstract
Chapter 4 describes connections, equivalencies, and relationships relating to two-sample tests of null hypotheses. First, Student’s conventional two-sample t-test is described. Second, a permutation two-sample test is presented and the connection linking the two tests is established. An example analysis illustrates the differences in the two approaches and the connection linking the tests. Third, measures of effect size for two-sample tests are presented for both Student’s two-sample t-test and a permutation two-sample test and the connections linking the various measures are set out. Fourth, the Wilcoxon–Mann–Whitney two-sample rank-test is presented with a permutation alternative for rank-score data and the connection linking the two tests is described. Finally, the connection between a conventional two-sample z-test for proportions and Pearson’s chi-square test of independence is delineated.
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This research monograph provides a synthesis of a number of statistical tests and measures, which, at first consideration, appear disjoint and unrelated. Numerous comparisons of permutation and classical statistical methods are presented, and the two methods are compared via probability values and, where appropriate, measures of effect size.
Permutation statistical methods, compared to classical statistical methods, do not rely on theoretical distributions, avoid the usual assumptions of normality and homogeneity of variance, and depend only on the data at hand. This text takes a unique approach to explaining statistics by integrating a large variety of statistical methods, and establishing the rigor of a topic that to many may seem to be a nascent field in statistics. This topic is new in that it took modern computing power to make permutation methods available to people working in the mainstream of research.
This research monograph addresses a statistically-informed audience, and can also easily serve as a textbook in a graduate course in departments such as statistics, psychology, or biology. In particular, the audience for the book is teachers of statistics, practicing statisticians, applied statisticians, and quantitative graduate students in fields such as psychology, medical research, epidemiology, public health, and biology.
The prime object of this book is to put into the hands of research workers, and especially of biologists, the means of applying statistical tests accurately to numerical data accumulated in their own laboratories or available in the literature.
The comparison of two treatments generally falls into one of the following two categories: (a) we may have a number of replications for each of the two treatments, which are unpaired, or (b) we may have a number of paired comparisons leading to a series of differences, some of which may be positive and some negative. The appropriate methods for testing the significance of the differences of the means in these two cases are described in most of the textbooks on statistical methods.
Consider a population in which sexual selection and natural selection may or may not be taking place. Assume only that the deviations from the mean in the case of any organ of any generation follow exactly or closely the normal law of frequency, then the following expressions may be shown to give the law of inheritance of the population.
The accuracy of an estimate of a normally distributed quantity is judged by reference to its variance, or rather, to an estimate of the variance based on the available sample. In 1908 “Student” examined the ratio of the mean to the standard deviation of a sample.1 The distribution at which he arrived was obtained in a more rigorous manner in 1925 by R.A. Fisher,2 who at the same time showed how to extend the application of the distribution beyond the problem of the significance of means, which had been its original object, and applied it to examine regression coefficients and other quantities obtained by least squares, testing not only the deviation of a statistic from a hypothetical value but also the difference between two statistics.
Let x and y be two random variables with continuous cumulative distribution functions f and g. A statistic U depending on the relative ranks of the x's and y's is proposed for testing the hypothesis . Wilcoxon proposed an equivalent test in the Biometrics Bulletin, December, 1945, but gave only a few points of the distribution of his statistic. Under the hypothesis the probability of obtaining a given U in a sample of and is the solution of a certain recurrence relation involving n and m. Using this recurrence relation tables have been computed giving the probability of U for samples up to . At this point the distribution is almost normal. From the recurrence relation explicit expressions for the mean, variance, and fourth moment are obtained. The 2rth moment is shown to have a certain form which enabled us to prove that the limit distribution is normal if go to infinity in any arbitrary manner. The test is shown to be consistent with respect to the class of alternatives for every x.
A formula is developed for the correlation between a ranking (possibly including ties) and a dichotomy, with limits which are always ±1. This formula is shown to be equivalent both to Kendall's and Spearman's.
A class of multiple sample tests based on empirical coverages is proposed which is a generalization of Greenwood's and Sherman's one-sample goodness-of-fit test statistics. The asymptotic normality of the tests is established by embedding the empirical coverages into a stationary process satisfying the strong mixing condition.
The articles published by the Annals of Eugenics (1925–1954) have been made available online as an historical archive intended for scholarly use. The work of eugenicists was often pervaded by prejudice against racial, ethnic and disabled groups. The online publication of this material for scholarly research purposes is not an endorsement of those views nor a promotion of eugenics in any way.
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Permutation tests: Precision in estimating probability values
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A simple statistical significance test
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