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

TL;DR

This tutorial explains how to properly specify and interpret a priori contrasts in linear and mixed models, emphasizing their importance over omnibus F-tests for testing specific hypotheses in factorial experiments.

Contribution

It provides a comprehensive explanation of various contrast types, their properties, and implementation in R, including the use of the generalized inverse for custom hypotheses.

Findings

Contrasts offer more informative tests than omnibus F-tests.

Proper contrast coding is essential for accurate hypothesis testing.

Reproducible R code demonstrates practical application.

Abstract

Factorial experiments in research on memory, language, and in other areas are often analyzed using analysis of variance (ANOVA). However, for effects with more than one numerator degrees of freedom, e.g., 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. Because 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" (Hays, 1973, p. 601). In this tutorial, we explain the mathematics underlying different kinds of contrasts (i.e., treatment, sum, repeated, polynomial, custom, nested, interaction 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 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 useful confirmatory tests of experimental hypotheses than standard omnibus F-test. Reproducible code is available from https://osf.io/7ukf6/.