The Wilcoxon and sign tests - nonparametric tests for a bigger picture

Consider the following example data of differences in 17 clients between baseline and a follow-up after an intervention therapy:

Clients 1 to 9 score 1 (at T1) and 4 (at T2); Clients 10 to 17 have scores of 3 (at T1) and 1 (at T2).

Analysing this data using the two related samples in SPSS nonparametric tests gives a (Wilcoxon) signed-rank z=-1.97, p=0.048 (with a medium-high effect size of z/sqrt(17)=-0.48 as an effect size) but a sign test has a p-value of 1.00 (9 of 17 increase whereas 8 of the 17 decrease suggesting an almost equal split in numbers increasing and decreasing). The Wilcoxon has a high z-value because the largest changes are associated with increases from time 1 to time 2 even though almost half the changes are decreases between times 1 and 2.

Using the sign test, in addition to the Wilcoxon signed rank test, looks at the prevalence of change over time by assessing the proportion of people showing change in the same direction e.g. asking what proportion of the 17 clients above had scores which increased between two time points such as baseline and follow-up.

The Wilcoxon, looks at patterns over time in the size of differences, whereas the sign test looks at the prevalence of any direction of difference (ignoring magnitude) across the clients.