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

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.

Oscillatory and aperiodic neural activity jointly predict language learning

“…Raincloud plots were produced to visualise behavioural data using the code provided by (Allen et al, 2019). For linear models, contrasts for categorical variables were sum-to-zero contrast coded, with coefficients reflecting differences to the grand mean (Schad et al, 2020).…”

Slow oscillation-spindle coupling predicts sequence-based language learning

“…A beta regression was used to assess the relationship between Statistical Learning Ability and behavioural performance on the sentence judgement tasks, while effects were plotted using the package effects (Fox et al, 2019) and ggplot2 (Wickham, 2016). Categorical factors were sum-to-zero contrast coded, meaning that factor level estimates were compared to the grandmean (Schad et al, 2020). Further, an 83% confidence interval (CI) threshold was used given that this approach is more conservative than the traditional 95% CI threshold and corresponds to the 5% significance level with non-overlapping estimates (Austin & Hux, 2002;MacGregor-Fors & Payton, 2013).…”

Morphological decomposition in Bantu: a masked priming study on Setswana prefixation

“…Parameters were estimated with restricted maximum likelihood. In all models, contrasts for the fixed effects were computed from the generalised inverse function (Schad, Hohenstein, Vasishth, & Kliegl, 2020).…”