A rank test Rn has been proposed by Cronholm and Revusky which in effect increases the number of observations per subject by the device of randomly assigning a subject to a treated group in each of a series of sub-experiments. In this note further results on the asymptotic normality of the Rn test are discussed both under the null hypothesis H0 of randomness and under a class of alternatives to H0. The limiting power of the test is also discussed.