Let $X_1$ be a random value from distribution 1 and let $X_2$ be a random value from distribution 2. I thought that the null hypothesis for the Mann-Whitney test was $P(X_1 < X_2) = P(X_2 < X_1)$.

If I run simulations of the Mann-Whitney test on data from normal distributions with equal means and equal variances, with $\alpha=0.05$, I get Type I error rates which are very close to 0.05. However, if I make the variances unequal (but leave the means equal), the proportion of simulations in which the null hypothesis is rejected becomes larger than 0.05, which I didn't expect, since $P(X_1 < X_2) = P(X_2 < X_1)$ still holds. This happens when I use wilcox.test in R, regardless of whether I have exact=TRUE, exact=FALSE, correct=TRUE, or exact=FALSE, correct=FALSE.

Is the null hypothesis something different from what I've written above, or is it just that the test is inaccurate in terms of Type I error if the variances are unequal?


1 Answer 1


From Hollander & Wolfe pp 106-7,

Let $F$ be the distribution function corresponding to population 1 and $G$ be the distribution function corresponding to population 2. The null hypothesis is: $H_O: F(t)=G(t)$ for every $t$. The null hypothesis asserts that the $X$ variable and the $Y$ variable have the same probability distribution, but the common distribution is not specified.

Strictly speaking this describes the Wilcoxon test, but $U=W-\frac{n(n+1)}{2}$, so they're equivalent.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.