Wilcoxon Rank Sum Test – Advanced

Property 1: Suppose sample 1 has size n1 and rank sum R1 and sample 2 has size n2 and rank sum R2, then R1 R2 = n(n+1)/2 where n = n1 n2.

Proof: This is simply a consequence of the fact that the sum of the first n positive integers is \frac{n(n+1)}{2}. This can be proven by induction. For n = 1, we see that \frac{n(n+1)}{2} = \frac{1(1+1)}{2} = 1 = n. Assume the result is true for n, then for n + 1 we have,  1 + 2 + … + n + (n+1) = \frac{n(n+1)}{2} + (n + 1) = \frac{n(n+1)+2(n+1)}{2}\frac{(n+1)(n+2)}{2}

Property 2: When the two samples are sufficiently large (say of size > 10, although some say 20), then the W statistic is approximately normal N(μ, σ) where

image945

Proof: We prove that the mean and variance of W = R1 are as described above. The normal approximation was proven in Mann & Whitney (1947) of Bibliography and we won’t repeat the proof here.

Let xi = the rank of the ith data element in the smaller sample. Thus, under the assumption of the null hypothesis, by Property 1

image3534

By Property 4a of Expectation

image3535

As we did in the proof of Property 1, we can show by induction on n that

image3536image3537

From these it follows that

image3538

We can now calculate the following expectations:

image3539

Also where i ≠ j

image3540image3541

By Property 2 of Expectation

image3542

By Property 3 of Basic Concepts of Correlation

image3543

By an extended version of Property 5 of Basic Concepts of Correlation

image3544

image3545

image3546

image3547

4 Responses to Wilcoxon Rank Sum Test – Advanced

  1. Orukpe Lewis says:

    I’m grateful for dis better understanding

  2. Nnamdi says:

    Thanks. Good one

  3. shark says:

    thanks 😀

  4. fra says:

    thanx! really helpful!

Leave a Reply

Your email address will not be published. Required fields are marked *