How to capitalize on a priori contrasts in linear (mixed) models: A tutorial

August 16, 2018

(This article was first published on Shravan Vasishth's Slog (Statistics blog), and kindly contributed to R-bloggers)

We wrote a short tutorial on contast coding, covering the common contrast coding scenarios, among them: treatment, helmert, anova, sum, and sliding (successive differences) contrasts.  The target audience is psychologists and linguists, but really it is for anyone doing planned experiments.
The paper has not been submitted anywhere yet. We are keen to get user feedback before we do that. Comments and criticism very welcome. Please post comments on this blog, or email me.

Factorial experiments in research on memory, language, and in other areas are often analyzed using analysis of variance (ANOVA). However, for experimental factors with more than two levels, the ANOVA omnibus F-test is not informative about the source of a main effect or interaction. This is unfortunate as researchers typically have specific hypotheses about which condition means differ from each other. A priori contrasts (i.e., comparisons planned before the sample means are known) between specific conditions or combinations of conditions are the appropriate way to represent such hypotheses in the statistical model. Many researchers have pointed out that contrasts should be “tested instead of, rather than as a supplement to, the ordinary `omnibus’ F test” (Hayes, 1973, p. 601). In this tutorial, we explain the mathematics underlying different kinds of contrasts (i.e., treatment, sum, repeated, Helmert, and polynomial contrasts), discuss their properties, and demonstrate how they are applied in the R System for Statistical Computing (R Core Team, 2018). In this context, we explain the generalized inverse which is needed to compute the weight coefficients for contrasts that test hypotheses that are not covered by the default set of contrasts. A detailed understanding of contrast coding is crucial for successful and correct specification in linear models (including linear mixed models). Contrasts defined a priori yield far more precise confirmatory tests of experimental hypotheses than standard omnibus F-test.

 Full paper:

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