A generalization of Friedman's rank statistic

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Abstract
Abstract  In this paper a very natural generalization of the two‐way analysis of variance rank statistic of Friedman is given. The general distribution‐free test procedure based on this statistic for the effect of J treatments in a random block design can be applied in general two‐way layouts without interactions and with different numbers of the continuous observations per cell provided the design scheme is connected. The asymptotic distribution under the null hypothesis of the test statistic is derived. A comparison with the method of m rankings of Benard and van Elteren is made. The disadvantage of Benard and van Elteren's test procedure is that the number of observations per block does influence the statistic twice, namely firstly by the number itself, as it should, and see ondly by the level of the ranks which will be different in different blocks if the numbers of observations per block are different. The proposed test statistic is not sensitive to differences in the levels of the ranks caused by the different numbers of observations per block. The test is derived from considerhg the Kruskal‐Wallis statistics per block.Finally, the results of simulation experiments are given. The simulation is carried out for three designs and a number of normal location alternatives and gives some information about the power of the suggested test procedure. A comparison is made with Benard and van Elteren's test and with the classical analysis of variance technique. For some simple orthogonal designs the exact null distributions of Benard and van Elteren's test and the proposed test are compared.