The Power of Two Difference-Sign Tests

Abstract

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 Fisher's transformation of the correlation coefficient. For