A Default Bayes Factor for Testing Null Hypotheses About the Fixed Effects of Linear Two-Level Models

Testing null hypotheses of the form “β = 0,” by the use of various Null Hypothesis Significance Tests (rendering a dichotomous reject/not reject decision), is considered standard practice when evaluating the individual parameters of statistical models. Bayes factors for testing these (and other) hypotheses allow users to quantify the evidence in the data that is in favor of a hypothesis. Unfortunately, when testing equality- contained hypotheses, the Bayes factors are sensitive to the specification of prior distributions, which may be hard to specify by applied researchers. The paper proposes a default Bayes factor with clear operating characteristics when used for testing whether the fixed parameters of linear two-level models are equal to zero. This is achieved by generalizing an already existing approach for linear regression. The generalization requires: (a) the sample size for which a new estimator for the effective sample size in two-level models containing random slopes is proposed; (b) the effect size for the fixed effects for which the so-called marginal R2 for the fixed effects is used. Implementing the aforementioned requirements in a small simulation study shows that the Bayes factor yields clear operating characteristics regardless of the value for sample size and the estimation method. The paper gives practical examples and access to an easy-to-use wrapper function to calculate Bayes factors for hypotheses with respect to the fixed coefficients of linear two-level models by using the R package bain. Translational Abstract When researchers fit two-level models to nested data, they are usually interested in whether the overall effect of a predictor, expressed through the values of the so-called fixed coefficient, is different from zero. This is, even more, the case, when having a small variance of the random effect of that particular coefficient. Null Hypotheses Significance Tests only allowresearchers to retain or reject the null hypotheses that state no effect of the predictor. Additionally, in the context of two-level models, obtaining a p-value requires some form of approximation of the degrees of freedom or comparing the fit of two models that differ only in the fixed effects of interest. Bayes factors allow users to quantify the evidence in the data that is in favor of the null hypothesis, while also avoiding many of the limitations and problems associated with p-values. However, when testing null hypotheses, the Bayes factor is sensitive to the specification of so-called prior distributions, which are not of interest to researchers. This paper proposes a Bayes factor where the prior sensitivity has been dealt with. The paper shows that this Bayes factor does not depend on the sample size value and the estimation method. Additionally, the paper offers an alternative to calculating the effective sample size in two-level models containing varying slopes, as well as offering practical examples and access to an easyto- use wrapper function to calculate this Bayes factor for continuous predictors.