Evaluation of empirical type I error rates of F and normality tests under different variance and mean conditions in multi-treatment CRDs

Hypothesis tests, such as normality tests, are extensively employed in Agricultural Sciences to evaluate the normality assumption of the F test in the Analysis of Variance (ANOVA) when large sample sizes are unavailable. Nonetheless, researchers conducting these tests are exposed to the risk of committing type I or type II errors, with probabilities that are influenced by different experimental conditions. This study assesses the empirical type I error rate of hypothesis tests by considering the equality (inequality) of treatment means, the homogeneity (heterogeneity) of variances, and different numbers of repetitions per treatment. Applying Completely Randomized Designs (CRD), sub-scenarios were simulated for each experimental scenario, with 10,000 iterations performed for each sub-scenario. Response variable values and experimental residuals were generated and subjected to appropriate tests. The results demonstrate that when the assumption of homogeneity of variances is violated, both the F and normality tests (excluding the Kolmogorov-Smirnov test) exhibit higher empirical type I error rates. Additionally, for normality tests, these error rates increase with the number of repetitions. Conversely, without such violations, the error rates remain stable and closely approximate the theoretical significance level for all analyzed hypothesis tests.