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Buchstabendarstellung für alle paarweisen multiplen Vergleiche

All pairwise-comparisons are a recurring task in many statistical applications, perhaps the most

prominent being the analysis of variance.

Users of all pairwise comparison procedures are accustomed to displays, in which treatment means that are not significantly different, are followed by a common letter. Such displays are readily available in most software packages for linear models, e.g. with the MEANS tatement with PROC GLM of the SAS System.

Unfortunately, availability of such displays so far has been mainly restricted to balanced data and to linear models with a single homoscedastic normally distributed error term (Westfall et al., 1999, p.69). Users are often faced with data not meeting these assumptions. A few

examples are:

- adjusted means in an unbalanced incomplete block design (lattices, -designs)

- adjusted means in an analysis of covariance

- adjusted means from a spatial analysis of a variety trial

- treatment effects in a generalized linear (mixed) model or a nonlinear (mixed) model

- weighted means in a mixed model analysis of longitudinal data or of a series of experiments

- multiple comparison of parameter estimates, which are not means (e.g., variances,  or

- multiple comparison by non-parametric methods

In any of these cases, one is faced with a set of t(t1)/2 significance statements (p-values) corresponding to all pairwise comparisons, where t is the number of treatments, possibly adjusted for multiplicity. For example, the LSMEANS statement of the GLM procedure of the SAS System produces such a set statements when invoked with the PDIFF option. Such sets of significance statements may be distractingly hard to interpret, particlarly  when t is large. In my experience, this often prompts users of statistical procedures not to do any multiple comparisons at all. What they ask for in consultation is the familiar letters display.


In this paper, I will present two simple algorithms suited for displaying

any set of t(t-1)/2 all-pairwise p-values. The methods are exemplified

using artificial examples as well as real data sets.