# Equivalence of Mann-Whitney U-test and t-test on ranks

What does it mean when people say that a t-test performed on ranked data is equivalent to a Mann-Whitney U-test? Does that mean they just test the same hypothesis/are useful in the same situations or are they are supposed to give the exact same p-values? The reason I ask is I tried both in R and compared two groups with very small sample sizes (3 and 4). I got completely different answers: one significant and one not.

The two groups are A=(1,2,3) and B=(4,5,6,7).

t-test: p = 0.01

Mann-Whitney U-test: p = 0.06

• It's impossible to get a significant MW result with Ns of 3 and 4. – Jeremy Miles Apr 5 '13 at 23:02
• @JeremyMiles At least if you work at the 5% level, since the smallest attainable level is 5.7% -- but if your n's are perforce 3 and 4 respectively (rather than something you can change), one might reasonably criticize insisting on $\leq$5% in such circumstances; indeed, one might well mount an argument for a substantially higher $\alpha$, such as, oh, something more like, say about 11.4%. There's little point in keeping $\alpha$ very low if $\beta$ is really high. – Glen_b Jan 2 '15 at 1:29
• @Glen_b good point, I should have qualified that. (And nice solution to use a higher alpha). – Jeremy Miles Jan 2 '15 at 16:53
• @JeremyMiles the tradeoff between the two error rates depends heavily on context of course. – Glen_b Jan 2 '15 at 22:37

Does that mean they just test the same hypothesis/are useful in the same situations or are they are supposed to give the exact same p-values?

It means:

(i) the test statistics will be monotonic transformations of each other.

(ii) they give the same p-values, if you work out the p-values correctly.

Your problem is that the t-test on ranks doesn't have the distribution you use when you use t-tables (though in large samples they'll be close). You need to calculate its true distribution to correctly calculate the p-value. It matters most for small samples ... but they're also the ones where you can calculate the actual distribution most easily.

• So correct me if I am wrong but, what you are saying is the M_W test is giving the correct p-value while the t-test on ranks is giving a value which is slightly inaccurate. OK that helps. Thanks. – Jimj Apr 6 '13 at 0:53
• Yes, that's it exactly. Indeed, if you can see that the two statistics are monotonic versions of each other, it's easy to see that this cannot be otherwise. Showing it is not difficult algebraically. – Glen_b Apr 6 '13 at 1:37
• Well, it's possible that the M-W may be giving an approximation as well, depending on the circumstances, but in small samples most packages give the exact p-values. In larger samples they may switch to normal approximation (which tends to be pretty accurate unless there's a lot of ties). Where both are approximations, the M-W will generally be very close to the true p-value. – Glen_b Apr 6 '13 at 1:37
• And the reason a ttest on ranks is not giving the exact right pvalue is because ranked data is not normal? – Jimj Apr 7 '13 at 18:24
• Correct. Not only is ranked data not normal, which would be sufficient to make the result not have a t-distribution, but given the sample sizes, you also know the variance under the null. – Glen_b Apr 7 '13 at 22:51

N.B. Not really an answer per se...

Jonas Kristoffer Lindeløv wrote a recent blog post here that addresses relationships between linear models and group tests (such as t-test, Mann-Whitney, Wilcox, Kruskal-Wallis, ANOVA, etc.). He also created a somewhat limited simulation to assess the differences between the t-test and MW directly.

It would be nice to either have 1) an analytic calculation or 2) a strong set of simulations to create some good rules-of-thumb for when we can use a linear model on ranks as opposed to a nonparametric test, as these are sometimes more convenient.