H. A. David

The Paired t Test Under Artificial Pairing

Abstract Suppose that in the situation of a paired t test natural pairing, such as the use of twins, is not possible. Reduction in variability is then often achieved artificially, for example by pairing animals of similar birth weight. This article points out that, unless such pairing is ineffective, the usual assumptions underlying the paired t test are violated. Nevertheless, simulation indicates that, with randomization in the allocation of treatments, the standard procedure gives good results. Our bivariate normal model provides the factor by which the length of the confidence interval for the mean treatment difference is reduced as a result of the pairing. Another form of pairing sometimes used is shown to be incorrect. Nonparametric analogs are also briefly considered.

First (?) Occurrence of Common Terms in Mathematical Statistics

Abstract A list of over 300 terms commonly used in mathematical statistics is presented with their apparent first occurrence in print. Some of the more interesting problems encountered in preparing the list are described.

Selection Through an Associated Characteristic, With Applications to the Random Effects Model

Abstract The problem of choosing the best k objects out of n is considered when, instead of measurements yi (i = 1, …, n) of primary interest, only associated measurements xi are readily obtainable. Large measurements are regarded as desirable. It is assumed that the n pairs (xi , yi ) are a random sample from a continuous population. A general expression is developed for the probability π that the s objects with the largest X-values include the k objects (k ≤ s) with the largest Y-values. When X and Y are bivariate normal with correlation ρ, a table of π is presented for n ≤ 15; this table gives immediately the smallest s for which π ≥ P*, where P* is preassigned. Several applications are treated, especially one to the random effects model for n varieties with r replications. It is shown how reasonable values of r may be found to provide a reduction in the number of varieties from n to s, such that the chosen s contain the k best varieties with at least probability P*. A method conditional on the observe...

Advances in Biometry.

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Order Statistics for Discrete Populations and for Grouped Samples

Abstract The aim of this paper is two-fold: (1) To give a unified treatment of the theory of order statistics when the parent distribution is not necessarily continuous. (2) To assess the effects of grouping on the distribution of order statistics and to indicate the convenience, under suitable conditions, of using order statistics for the estimation of parameters from grouped data with or without censoring.

The Beginnings of Randomization Tests

Randomization (or permutation) tests became known through R.A. Fishers 1935 demonstration that no assumption of normality was needed for the analysis of Darwins paired t test data: Assuming normality was simply more convenient than tedious enumeration of cases more removed from the null hypothesis of no difference than were the experimental results. We describe an interesting precursor, often overlooked or merely cited, to Fishers treatment. Next, we outline the analytical development of randomization theory by Welch and Pitman, including the latters realization of broad possibilities for tests without distributional assumptions. Finally, we briefly indicate the current status of randomization tests which are no longer tedious to carry out with modern computers.

A Note on Order Statistics for Dependent Variates

Abstract Sathe and Dixit showed that a basic recurrence relation for cdfs of order statistics can be modified to hold under any dependence structure of the original variates. A greatly shortened proof is given in this note. Balakrishnan, Bendre, and Malik have used the relation to obtain a general expression for the cdf of an order statistic. This result is here linked to a classical formula in probability theory.

First (?) Occurrence of Common Terms in Probability and Statistics—A Second List, with Corrections

Abstract An annotated list is presented containing presumed first occurrences in print of terms commonly used in probability and statistics. The list supplements and provides some corrections to a longer list published in volume 49 of The American Statistician.

Nonparametric Analysis of Unbalanced Paired-Comparison or Ranked Data

Abstract Suppose that we have t objects C 1, …, Ct and that Ci and Cj , are judged pairwise in n ij independent comparisons, for i, j = 1, …, t; i ≠ j. In the simplest of such “paired-comparison” designs, all pairs of objects are compared an equal number of times (i.e., all nij = n). Yet it is often impractical to carry out such a “completely balanced” experiment: most observational data and even many planned experiments are highly unbalanced, and some are “incomplete” in that nij = 0 for some pairs (i, j). Most of the available methods for analysis of unbalanced paired-comparison data are parametric, in the sense that a (paired-comparison) linear model generates, for each pair of objects, the “preference probability” π ij with which Ci is preferred to C j . David (1987) proposed a simple nonparametric method of scoring objects from unbalanced paired-comparison data that takes into account differences in the strength of the opposition encountered by each object as well as possible differences in the {nij ...

A Multi-Stage Procedure for the Selection of the Best of Several Populations

Abstract This paper proposes a multi-stage procedure for the selection of the best of several populations, allowing more observations to be made on the more successful treatments. To this end, a weighting function operates on the results accumulated up to any one stage to determine the proportion of observations to be taken from each population for that stage. The paper is restricted to consider only binomial populations. Simulation experiments utilizing five somewhat arbitrary weighting functions indicate that this method has a higher probability of selecting the best population than conventional equal sample size experiments.