We are working on some analyses of the data we have. We have some inquiries regarding these analyses. The data that we have are continuous data collected from 34 patients at two distinct time points. We want to assess whether there are any statistically significant differences between these two periods for several variables, with each variable being analyzed individually.
Given that the data is not normally distributed, we applied the Wilcoxon test on the mean. However, after discussing with the team, somebody suggested a further assessment of statistical significance using the median, Q1, and Q3 values due to data skewness.
Here are my questions:
1. Given that the Wilcoxon test was already employed (which should address data skewness), is there a need to also test the statistical significance of the median, Q1, and Q3?
2. If the answer to the first question is yes, should we use bootstrapping to determine confidence intervals for differences in the median, Q1, and Q3 of the paired data and checked if these intervals include zero?
The Wilcoxon signed rank test is not usually a test of mean.
It's useful to think about how paired tests operate. For a Wilcoxon signed rank test, the first step is to subtract the paired values from one another. It's equivalent to a one sample test, testing the signed ranks of the differences again zero.
I mention this, because even on a test of the means, or the median, say, in a paired test, you are not comparing the mean of one group to the mean of the other group. Instead, you are comparing the mean of the differences to zero. (Like a paired t-test).
A sign test will work for paired data to test the median. It's easy to modify this test to assess Q3, and so on.
You could also use quantile regression (presumably with a random factor to account for the pairing in the data) to look at various quantiles.
Or you could use a permutation test for various quantiles.
But again, you have to consider what is actually being tested in the paired situation.
I think looking at quantiles is a good idea. Maybe you don't really care about the median value of a group, but it's really the 90th percentile that is of interest.
If your response variable is some kind of concentration of some substance, then the logarithms will presumably be approximately normal distributed.
The "Wilcoxon test on the mean" does not exist, as Sal already mentioned. Your data seems to be paired (measurements at two time points from the same patient, and it's supposedly the difference between these two time points you are interested in), so the sign test would be a possibility (or the paired t-test using the logarithms).
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