single sample Wilcoxon Signed Rank Test as an alternative to single sample t-test

Question

I know that the t-test tests the hypothesis $H_0: \mu = 0 $ vs. $H_a: \mu \neq 0$ where $\mu$ is the population mean

The single sample wilcoxon signed rank test tests the hypothesis that the median is equal to a certain value, in our case here lets suppose that median value is 0. The single sample wilcoxon signed rank test is seen as a nonparametric alternative to the t-test.

My question is: why is the wilcoxon signed rank test considered an alternative to the t -test when one tests for the mean parameter, and the other for the median?

Answer 1

1) When you say "the t-test...tests for the mean parameter" it is, in this context, the paired t-test uses the t-statistic from the mean value of the difference between the pairs divided by the standard error of the mean, for t-table lookup significance of difference from a zero mean. This test requires the difference function to be normally distributed.

2) The Wilcoxon signed-rank test does no such thing, and, despite confusion to the contrary, does not test different medians either. This test requires the differences to be from a symmetrical distribution. When I looked at the test for difference of median values, I found it confusing that the median values do not agree with the Wilcoxon parameters for the same data, but, they do not agree. Instead it finds the Wilcoxon W-statistic of the signed ranks. In some implementations, e.g., R-language but not S-PLUS, the Wilcoxon W-statistic is the same as the calculation for the Mann-Whitney U-statistic but for signed rank differences. The U-statistic in turn does not use medians as a measure of location, and in general, which measure of location is optimal for what situation depends on the relevant characteristics of the distribution modeled.

So how exactly is the W-statistic used to calculate a probability? This reference says:

Confidence interval to the median difference. The differences between pairs of observations are first arranged in rank order. A triangular matrix of the Walsh averages (the means of all possible pairs of values) is then constructed. The Hodges-Lehmann estimate of the median difference is given by the median of these values. (Note, this may be wherein the confusion lies. Median values, unlike mean values, do not maintain their properties when threading through a single list compiled from two lists. That is, a median of differences of signed ranks is certainly not a difference of medians of ranks.)

The upper and lower 95% confidence limits to this median are obtained by counting in a specified number of Walsh averages from each end of the array. The required number of averages is given by the quantile of the Wilcoxon matched-pairs signed-ranks statistic for n observations at P = 0.025.