Rank Procedures for the Two-Factor Mixed Model

Abstract

The test problem of fixed treatment effects is considered in the two-factor mixed model with interaction and unequal cell frequencies when the classical assumptions of normality do not hold. An explicit form of a test statistic is derived using a partial rank transform (ranking all observations within each block), and the asymptotic distribution of the statistic is determined under the assumption that the number of blocks tends to infinity and the cell frequencies are bounded. The statistic reduces to Friedman's statistic if no interactions are involved in the model and all cell frequencies are equal; hence the proposed test can be regarded as a generalization of Friedman's test for repeated observations when the cell frequencies are not equal. The test is compared to a corresponding test that can be used under the assumption of normality by the criterion of asymptotic relative efficiency. In the case of two treatments, the exact conditional distribution is determined and estimators and confidence intervals for the shift effect are proposed.

Key Words:

• Analysis of variance
• Generalized Friedman's test
• Linear rank statistic
• Mixed model
• Rank transformation