NAG Fortran Library Routine Document G08AGF

The Wilcoxon one-sample signed rank test may be used to test whether a particular sample came from a population with a specified median. It is assumed that the population distribution is symmetric. The data consists of a single sample of n observations denoted by x1; x2; . . . ; xn. This sample may arise from the difference between pairs of observations from two matched samples of equal size taken from two populations, in which case the test may be used to test whether the median of the first population is the same as that of the second population. The hypothesis under test, H0, often called the null hypothesis, is that the median is equal to some given value ðXmedÞ, and this is to be tested against an alternative hypothesis H1 which is H1 : population median 61⁄4 Xmed; or H1 : population median > Xmed; or H1 : population median < Xmed, using a two-tailed, upper-tailed or lower-tailed probability respectively. The user selects the alternative hypothesis by choosing the appropriate tail probability to be computed (see the description of argument TAIL in Section 5). The Wilcoxon test differs from the Sign test (see G08AAF) in that the magnitude of the scores is taken into account, rather than simply the direction of such scores. The test procedure is as follows (a) For each xi, for i 1⁄4 1; 2; . . . ; n, the signed difference di 1⁄4 xi Xmed is found, where Xmed is a given test value for the median of the sample. (b) The absolute differences jdij are ranked with rank ri and any tied values of jdij are assigned the average of the tied ranks. The user may choose whether or not to ignore any cases where di 1⁄4 0 by removing them before or after ranking (see the description of the argument ZEROS in Section 5). (c) The number of non-zero di is found. (d) To each rank is affixed the sign of the di to which it corresponds. Let si 1⁄4 signðdiÞri. (e) The sum of the positive-signed ranks, W 1⁄4 X