Choosing appropriate covariance matrices in a nonparametric analysis of factorials in block designs

The standard nonparametric, rank-based approach to the analysis of dependent data from factorial designs is based on an estimated unstructured (UN) variance–covariance matrix, but the large number of variance– covariance terms in many designs can seriously affect test performance. In a simulation study for a factorial arranged in blocks, we compared estimates of type-I error probability and power based on the UN structure with the estimates obtained with a more parsimonious heterogeneous-compound-symmetry structure (CSH). Although tests based on the UN structure were anti-conservative with small number of factor levels, especially with four or six blocks, they became conservative at higher number of factor levels. Tests based on the CSH structure were anti-conservative, and results did not depend on the number of factor levels. When both tests were anti-conservative, tests based on the CSH structure were less so. Although use of the CSH structure is concluded to be more suitable than use of the UN structure for the small number of blocks typical in agricultural experiments, results suggest that further improvement of test statistics is needed for such situations.