ABSTRACT
When the inferences to be drawn are related only to the population of differences of the paired observations, the first step in the analysis usually is to take the differences of the paired observations, leaving only a single set of observations. Therefore, this type of data can be legitimately classified as a one-sample problem. It is easy to show that the distribution of a standardized chi-square variable with large degrees of freedom can be approximated by the standard normal distribution. Ideally, no ties should occur because of the assumption of continuous populations. Ties do not present a problem in counting the number of runs unless the tie is across samples; that is, two or more observations from different samples have exactly the same magnitude. The very generality of the test weakens its performance against specific alternatives. Asymptotic power can be evaluated using the normal distribution with appropriate moments under the alternative, which are given in Wolfowitz.
