nonparametric test for sample and control

Question

Hi I have to use an appropriate test (and justify my choice) to explore whether there is an underlying difference in location between the concentration of an Enzyme in rats exposed to a specific toxin and rats that are not exposed to the toxin.

---------------------
|   toxin | control |
|   543   | 535     |
|   523   | 385     |
|   431   | 502     |
|   635   | 412     |
|   564   | 387     |
|   549   |         |
---------------------   

So my idea was to use the Mann-Whitney test but I'm not sure if i'm get the right result. Using R with the function wilcox.test(toxin,control,paired=FALSE) i'm getting a W = 27. Manually I'm getting a W = 48. Basically manually what I did was:

• pool both samples as sorted_r_vector = sort(c(toxin,control))
• then calculate the ranks for all toxin and sum them.

I'm getting 48. What I'm doing wrong?

Answer 1

I also get a rank sum of of $48$ and $18$ respectively, when I calculate them by hand. The thing is that R does not report the Wilcoxon test $W$, but instead the equivalent Mann-Whitney $U$. The relationship between the two is $$ U=km+\frac{k(k+1)}{2}-W $$ where $k$ is the number of observations in the one group and $m$ is the number of observations in the other group. Using $W=48$ with $k=5$ and $m=6$ we get that $$ U=3 $$ and with $W=18$ and $k=5$ and $m=6$ we get $$ U=33 $$ which is what R reports. Running wilcox.test(y,x) in R we also get $U=3$.