## **SUPPLEMENTARY MATERIAL** ##
### **Figures S1** ###
**Figure S1 |** All simulated combinations of the dependent variable *Y* and the predictor *X* that were fitted in linear regression models for sample sizes *N* = 10, 25, 50, 100, 250, 500, 1000. Subfolders specify sample sizes.
Figure names are constructed as “X” <Distribution name of X> “_Y” <How many distributions of Y> “Distributions_N” <Sample size> “_Sim” <Number of simulation runs>, such that the file “XD0_Y10Distributions_N10_Sim50000” shows results from all 50,000 simulation runs where the predictor *X* was normally distributed, the independent variable *Y* had ten different distributions (D0–D9) and the sample size was *N* = 10.
In each of the 7 × 10 = 70 figures, the leftmost column depicts the distribution of *Y*, the second column depicts the distribution of *X*, the third column a QQ-plot of the residuals and the forth column a QQ-plot of the -log10(*P*-values). Residuals were distributed as the dependent variable *Y* because the regression coefficient *b* was on average zero.
### **Figures S3** ###
**Figure S3 |** All simulated combinations of the dependent variable *Y* and the last of four predictors *X* that were fitted in linear regression models. The first three predictors were normally distributed and the distribution of the last one was varied. The sample size was *N* = 100. For a description of file names and content see **Figure S1**.
### **Figures S4** ###
**Figure S4 |** All simulated combinations of the dependent variable *Y* and the predictor *X* fitted in linear random-intercept models. We simulated *N* = 100 independent samples each of which was sampled twice, such that the single random effect explained roughly 30% of the variation in *Y*. For a description of file names and content see **Figure S1**.
### **Figures S7** ###
**Figure S7 |** All simulated combinations of the dependent variable *Y* and the predictor *X* fitted in generalized linear models with a Poisson error structure. The sample size was *N* = 100. For a description of file names and content see **Figure S1**.
### **Simulation code** ###
Folder **Simulation code** contains the R scripts used to start the simulation runs. For these scripts to work, TrustGauss (see below) needs to be installed first.
### **Simulation raw data** ###
Folder **Simulation raw data** contains raw summaries across 50000 simulation runs for assessing type I and type II error rates.
### **TrustGauss package** ###
Folder **TrustGauss package** contains the R package which bundles all functions written for this study. It is available either as a tar.gz file or .zip file and requires an R version > 3.4.3. Version 0.2 contains functions for simulating type II errors. The documentation of the functions is provided in the Supplementary Text.
