Type I error in multiple comparison tests in analysis of variance

In a hypothesis test, a researcher initially fixes a type I error rate, that is, the probability of rejecting the null hypothesis given that it is true. In the case of means tests, it is important to present a type I error that is equal to the nominal pre-fixed level, such that this error remains unchanged across various scenarios, including the number of treatments, number of repetitions, and coefficient of variation. The purpose of this study is to analyse and compare the following multiple comparison tests for the control of both conditional and unconditional type I error rates, depending on a significant F-test in the analysis of variance: Tukey, Duncan, Fisher's least significant difference, Student-Newman-Keuls (SNK), and Scheffe. As an application, we present a motivation study and develop a simulation study using the Monte Carlo method for a total of 64 scenarios. In each simulated scenario, we estimate the comparison-wise and experiment-wise error rates, conditional and unconditional on a significant result of the overall F-test of analysis of variance for each of the five multiple comparison tests evaluated. The results indicate that the application of the means tests based only on the significance of the F-test should be considered when determining the error rates, as this can change them. In addition, we find that Fisher's test controls for the comparison-wise error rate, the Tukey and SNK tests control for the experiment-wise error rate, and the Duncan and Fisher tests control for the conditional experiment-wise error rate. Scheffe's test does not control for any of the error rates considered.