TY  - JOUR
A1  - Schad, Daniel
A1  - Vasishth, Shravan
A1  - Hohenstein, Sven
A1  - Kliegl, Reinhold
T1  - How to capitalize on a priori contrasts in linear (mixed) models
BT  - a tutorial
JF  - Journal of memory and language
N2  - 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-tests. Reproducible code is available from https://osf.io/7ukf6/.
KW  - contrasts
KW  - null hypothesis significance testing
KW  - linear models
KW  - a priori
KW  - hypotheses
Y1  - 2019
U6  - https://doi.org/10.1016/j.jml.2019.104038
SN  - 0749-596X
SN  - 1096-0821
VL  - 110
PB  - Elsevier
CY  - San Diego
ER  -