On the use of Friedman-type statistics in a randomly incomplete two-way design

The Friedman test is a widely used distribution-free test for the equality of treatment effects versus a general alternative when a set of judges independently rank each of the treatments. A generalized form of the Friedman statistic, employing a real-valued rank function of the ranks, has been presented by Sen (1968). This rank function may assign nonzero values to only a subset of the ranks, reflecting an incomplete design setting. The asymptotic relative efficiency (A.R.E.) of the Sen test with respect to the classical F test will depend on the rank function and on the distribution of the observations which underlie the ranks. If the underlying distribution is fixed, one can find an "optimal" rank function which maximizes the A.R.E. When the distribution is uniform, exponential, or logistic, the optimal rank function assigns nonzero values to ranks in a manner which closely reflects the information provided by the sufficient statistic for the location parameter.;When the judges randomly omit ranks, the unranked treatments can be assigned ranks for a given judge through some defined imputation scheme. The Friedman statistic is then calculated on the completed design. Different imputation schemes are evaluated through exact and simulated power studies under alternatives employing the Feigin-Cohen rank model. These power studies require the use of null distributions for the completed Friedman statistic which depend of the imputation scheme used. For three considered schemes, the power studies reveal little or no difference in powers. Meanwhile, when the existing null Friedman tables such as those in Hollander & Wolfe (1973) are used as approximations to the exact null tables, the resulting power estimates were substantially incorrect.;When different judges specify different subsets of ranks, one can obtain a test based on Sen statistics by partitioning the judges based on the structure of assigned ranks. The Sen statistic can be calculated on each group of judges, and the statistics for all of the groupings can be combined through a weighted summation having an asymptotic weighted chi-square distribution.