Subsets not significantly different but superset is I have various clinical data on participants in a study. I'm looking at a continuous variable ("A") and a (binary) categorical variable (group) ("O"). I used a Wilcoxon test in R (the data are not normally distributed) to see if "A" is significantly different between the two groups. I got a borderline p-value of 0.054.  
If I run the Wilcoxon again but include only the males (30 of 72), the p-value is ~0.3; for females only it's ~0.25.  
How is it possible that there is no difference in "A" between the groups for males and females separately, but when combined there is a difference?
 A: It seems to be a question of test power. If you only look at a subset you have a lot less participants and therefore a lot less power to find an effect of similar size.
With a reduced sample size you can only find a much bigger effect. So it is NOT recommended to only look at the subsets in this case. Unless there is an interaction (i.e., do the results point into the same direction for both men and women?).
Furthermore, there is no need to use a Wilcoxon test only because your data is not normally distributed (unless it heavily deviates). Probably you can still use the t.test (for example one of the user here, whuber, recently advocated the t.test in a similar case, because the normally assumption does not necessarily hold for the data but for the sampling distribution. quoting him: "The reason is that the sampling distributions of the means are approximately normal, even though the distributions of the data are not").
However, if you still don't want to use the t.test there are more powerful 'assumption free' parametric alternatives, especially permutation tests. See the answer to my question here (whubers quote is also from there): Which permutation test implementation in R to use instead of t-tests (paired and non-paired)?
In my case the results were even a little bit better (i.e., smaller p) than when using the t.test. So I would recommend this permutation test based on the coin package. I could provide you with the necessary r-commands if you provide some sample data in your question.
Update: The effect of outliers on the t-test
If you look at the help of t.test in R ?t.test, you will find the following example:
t.test(1:10,y=c(7:20))      # P = .00001855
t.test(1:10,y=c(7:20, 200)) # P = .1245    -- NOT significant anymore

Although in the second case you have a much extremer difference in the means, the outlier leads to the counterintuitive finding that the data is not significant anymore. Hence, a method to deal with outliers (e.g. Winsorizing, here) is recommended for parametric tests as the t if the data permits.
A: This is not necessarily an issue of statistical power; it could also be an example of confounding.  
Example:  


*

*One category of $O$ is more common in males but the other is more common in females  

*The distribution of $A$ differs between males and females   

*Within each sex separately, the distribution of $A$ is exactly the same for both $O$ categories


Then there will still be an overall difference in the distribution of $A$ between the $O$ categories.
