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In the case described above, our Professor instructs as to interpret the simple effects by using MSwithin of the two-way ANOVA. In case the Levene's test is significant though, he instructs us to perform one-way ANOVA for each level of the independent variables (i.e. use the MSwithin of a one-way ANOVA). Does anyone know why in case of equal population variances we can use MSwithin of the two-way ANOVA?

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  • $\begingroup$ First off, the ANOVA is robust to violations of equal variances when the sample sizes are equal. If they are not equal, then unequal variances can lead to a biased result. Second, formal testing for equal variances should only be a guide as it will fail to detect a problem with small sample sizes but will be overly sensitive with large sample sizes. Third, I'm not sure how doing one-way ANOVAs for the two main effects solves the problem. It may for one factor but it can't for both unless you are using a modified ANOVA that includes an adjustment for heteroscedasticity, such as a Welch t-test. $\endgroup$ – dbwilson Jun 29 '18 at 23:05
  • $\begingroup$ Thank you for the reply! Probably my question was not clear; if we have an interaction and a significant Levene's, we look for the simple effects by conducting one-way ANOVA for each of the factors. But if Levene's is not significant, we use MSwithin of the two-way ANOVA for the simple effects. This is what I do not understand. Why MSwithing of the two-way ANOVA if the Levene's is not significant? $\endgroup$ – D. Kalli Jul 1 '18 at 10:40
  • $\begingroup$ I am not familiar with this approach, so am not entirely sure what the justification is or what it is based on. However, the MSwithin from the two-way ANOVA with the interaction (it is possible to have a two-way ANOVA without the interaction) is the smallest MSwithin possible and thus provides a statistically more powerful tests. I'm assuming these are unbalanced designs, otherwise there would be no need to do this. I recommend reading up on Type I and II sums-of-squares. It seems like your professor's approach accomplishes a similar goal as a Type II SS two-way ANOVA model. $\endgroup$ – dbwilson Jul 1 '18 at 12:13
  • $\begingroup$ Your answer helps me a lot! I see your point! Thank you very much! $\endgroup$ – D. Kalli Jul 1 '18 at 13:49

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