H.P. Piepho (University Kassel, Germany)
Multiple treatment comparisons in linear models when the standard error of a difference is not constant
Users of analysis of variance (ANOVA) procedures are accustomed to an ANOVA table, followed by a table of means. When the underlying fixed effects linear model is variance-balanced, i.e. the standard error of a difference is constant for all pairwise comparisons, non-significant differences can be indicated by underlining. Unfortunately, when the design is unbalanced, i.e. the standard error of a difference is not constant over pairs of treatments, it may turn out to be impossible to consistently represent significant differences by underlining. The same problem occurs, e.g., in linear mixed models and in generalized linear models. This paper proposes a simple, conservative approach, which allows a lines-representation of treatment comparisons. The price for the improved display of results is a potential loss of significances, though a loss of more than one significance is rarely observed in practice. Very frequently, there is no such loss at all. Lost significances may be reported separately.