# Should I use a contingency table or Wilcoxon Rank-Sum Test? (is chi squared nonparametric)

I have a debate over whether a contingency table or Wilcoxon Rank Sum Test (WRST) should be used in a particular situation. There are several problems with the contingency table, but I will mention one at the end having to do with whether the chi squared test is parametric or nonparametric that I am especially interested in resolving. I favor the WRST, but I am open to arguments in favor of the contingency table.

In this situation, N numbers were generated by process 1 and M numbers were generated by process 2, where N > M. The numbers are between 0 and 1. (Or they are percents.) I want to know which process tends to produce bigger numbers. In fact, process 1 basically always produces bigger numbers than process 2, but I want to know if N and M are large enough that this difference is statistically significant. (Actually, N = 8 and M = 6.) I know of no way to find out the probability distribution of process 1 or process 2, so it seems to me that a nonparametric test is called for. I want to use the Wilcoxon Rank-Sum Test.

However, another person wants me to use a contingency table. They say that WRST is only useful when process 1 and process 2 are known to have the same distribution, differing only by a shift. This seems untrue, according to Wikipedia's entry on the Mann-Whitney U test, another name for the WRST (see #4 under "assumptions and formal statement of hypotheses" which clearly does not assume the distributions differ only by a shift.)

At any rate, the contingency table seems potentially wrong just because if N > M, it is unclear what the size of the table would be. (2 x M or 2 x N?) However, say for the sake of argument that I could coerce the data into a table, maybe by throwing out data from process 1 until they are equal in size or by grouping the data into bins and averaging them. (There are problems with either approach, but bear with me.) There is still a problem in that the data are most naturally expressed as numbers between 0 and 1, but the contingency table I think expects integers as input. Still, for the sake of argument let's say I somehow deal with that problem too. There remains a huge problem (in my opinion) in that I think the chi squared test is parametric. (I think contingency tables are often based on chi squared?) I believe it assumes that the integers in the table are generated by a Poisson or multinomial process; but I don't know if process 1 and process 2 are Poisson or multinomial; they almost certainly are not - there is no reason for them to be.

Here is the crux of the problem: the thing that is causing arguments is that some sources say chi square is nonparametric:

http://benbaab.com/salkind/ChiSquare.html

While other sources say a contingency table assumes its cells are Poisson:

http://data.princeton.edu/wws509/notes/c5.pdf (see page 5)

Can anyone tell me whether I should use a contingency table or Wilcoxon Rank-Sum Test - and how I can resolve this contradiction between the two websites quoted above in such a way as to prove who is right (both to myself and to the other person)? Are the websites somehow both right?

Thanks for any help you can give.

• 1. Are these values between 0 and 1 discrete (e.g. a count divided by a total count) or continuous (each number could be any value in [0,1] at all, rather than an individual number only being able to take a limited number of distinct values)? 2. "Nonparametric" has a couple of (related) meanings (about the number of parameters needed to specify one or another thing) and in addition, some books use it to mean "distribution-free", which is not quite the same thing, but is somewhat related to one of the meanings. (Lastly, and quite wrongly, some people use it to mean anything non-normal.) ... ctd – Glen_b Sep 16 '15 at 19:32
• ctd... What are you actually after when you ask about a nonparametric procedure, and why do you want one? – Glen_b Sep 16 '15 at 19:33
• 3. When you say " In fact, process 1 basically always produces bigger numbers than process 2" do you simply mean "all 6 values from process 1 were bigger than all 8 from process 2" or something else? 4. The signed rank test is for paired data. If you have 6 and 8 observations, what's paired with what? 5. Note that there's never really any way to find out the functional form of a distribution from data; people make simple assumptions which are often vaguely plausible from external considerations and which don't seem so badly wrong they'll give misleading conclusions. – Glen_b Sep 16 '15 at 19:38
• 6. How is your colleague going to be calculating a chi-square on numbers between 0 and 1? – Glen_b Sep 16 '15 at 19:55
• @Jvonkof You may try test of difference between proportions for process 1 and Process 2. – Subhash C. Davar Sep 17 '15 at 14:33