Alan Stuart

Practical Nonparametric Statistics

A self-contained introduction to the theory and methods of non-parametric statistics. Presents a review of probability theory and statistical inference and covers tests based on binomial and multinomial distributions and methods based on ranks and empirical distributions. Includes a thorough collection of statistics tables, hundreds of problems and references, detailed numerical examples for each procedure, and an instant consultant chart to guide the student to the appropriate procedure.

Data-Dredging Procedures in Survey Analysis

Introduction and Summary 1. It is a commonplace of the statistical design of experiments that the hypotheses to be tested should be formulated before examining the data that are to be used to test them. Even in experimental situations, this is sometimes not possible, and in the last decade or so some progress has been made toward the development of more flexible testing procedures which allow the data to be dredged for hypotheses in certain ways. In survey analysis, which is commonly exploratory, it is rare for precise hypotheses to be formulable independently of the data. It follows that normally no precise probabilistic interpretations can validly be given to relationships found among the survey variables. In practice, this has not prevented survey practitioners from reporting probability levels as if they were precisely meaningful. Most investigators are so accustomed to making probability statements that a survey report looks naked without them, but we fear that many survey reports are wearing the Emperors clothes. This paper offers a classification of data-dredging procedures and some comments on their use.

Asymptotic Relative Efficiencies of Distribution-Free Tests of Randomness Against Normal Alternatives

S EVERAL writers, notably Hotelling and Pabst [5], have explicitly assumed that the relative efficiency of two test statistics is to be measured by their estimating efficiencies. While this seems reasonable, it is by no means obvious, since if the two tests are consistent, the ratio of their powers against any fixed alternative hypothesis must tend to unity with increasing sample size n, and it may easily be shown that for any n, the less efficient estimator may provide a more powerful test (Sundrum [14]). Pitman [11] has proposed a measure of the asymptotic relative efficiency of consistent tests. Given that the two statistics, t1 and t2, have normal limit distributions with variances of order n-1, and that certain general regularity conditions are satisfied, he considered a limiting process in which the alternative hypothesis Hi differs from the null hypothesis Ho by a quantity of order n-12, so that as n increases, Hi tends to Ho. Under these conditions, he showed that the reciprocal of the ratio of sample sizes required to attain equal power against the same alternative was, in the limit,

The Power of Two Difference-Sign Tests

SITUATIONS arise in which we are unable to make the assumptions necessary for the application of standard theory based on the normal distribution. Most of the distribution-free tests which have been proposed for such situations are based on statistics which are very easy to compute, and this ease of computation goes some way to compensate for any information, available in the sample, which may be ignored. Quite apart from such situations, it is interesting to investigate the performances of distribution-free tests when the standard normal situation in fact holds good, for if a test is consistent (i.e. if the probability of rejecting a false alternative hypothesis tends to unity with increasing sample size) there must come a point, for any set of alternatives, where the loss of power involved in its use is negligible. In this paper two very simple tests are examined in this light. A distribution-free test of serial independence of N (unequal) observations ordered in time, proposed by Moore and Wallis [6], consists in counting the number of positive first differences in the series. On the null hypothesis that the observations came from the same (continuous) population, every ordering of the observations is equally probable, so that the mean value and variance of the statistic are very simply obtained, and its distribution can easily be shown to be asymptotically normal. A lower bound for the power of the test against a general class of alternatives, implying a trend in the observations, was obtained by Mann [5]. This paper considers its power in the particular case where the alternative is a normal regression model with coefficient ( and residual variance 2. The loss of power entailed by the use of this test at the 95% level of significance is unimportant when either N ? 25, P/caN/2 _ .5, or N> 75, P/crV/2 _ .3. The difference-sign test is easily generalized to the bivariate case for use in testing the correlation between two series. The approximate power of this second test is tabulate(d below against the alternative hypothesis that the pairs of observations were drawn from a bivariate normal population with non-zero correlation p. Much larger sample sizes are required for the power of this test to approach that of the test based on Fishers transformation of the correlation coefficient. For

The asymptotic relative efficiencies of tests and the derivatives of their power functions

Abstract For comparing two consistent tests of a simple null hypothesis H 0 : θ = θ0 against a given alternative hypothesis H 0 : θ = θ1, the measure most frequently used is the asymptotic relative efficiency (ARE), due to Pitman (1948). The ARE is defined as the limit of the reciprocal of the ratio of sample sizes required to attain the same power. The limit is taken as sample sizes tend to infinity and simultaneously θ1 → θ0, this being necessary to keep the powers of the tests bounded away from 1.

Calculation of Spearman's Rho for Ordered Two-Way Classifications

1. In a letter published in the December 1962 issue of this journal, Gilbert M. Oster gave a formula which he had derived empirically for calculating Spearmans rank correlation coefficient from observations classified into an ordered two-way table with the same number of rows and columns. This note shows that a more general version of his formula (which was first derived for an application about three years ago) holds whether or not the number of rows and of columns is the same. 2. We follow the notation indicated in the schematic table below. There are r rows and c columns in the classification. niC is the frequency observed in the ith row and jth

Basic Ideas of Scientific Sampling (2nd Edition)

Basic Ideas of Scientific Sampling (2nd edition). By Alan Stuart. London, Griffin, 1976. 106 p. 21·5 cm. £2·20. (Griffins Statistical Monographs and Courses No. 4.)

Too Good to be True

In using the goodness‐of‐fit test should a very small value of χ2 occasion a rejection of the null hypothesis ? Mr Stuart presents the conflicting views on this question and answers it: No.

The Advanced Theory of Statistics. Volume 1, Distribution Theory.

A method for continuously effecting reactions in a liquid phase in the presence of a gas and of a finely divided solid catalyst in a bubble column-cascade reactor with little or no liquid back-mixing, dwell time in the reactor being dependent on the liquid and gas throughputs, said reactor comprising a vertical column and a plurality of equidistantly-spaced, horizontally mounted, uniformly-perforated plates therein, the aperture area of each plate being dependent on the cross-sectional area of the column and the plate spacing within the column being such that adjacent plates are separated vertically by a distance at least three times the diameter of said columnar reaction zone.