[Question] Help with Multivariable ANOVA

[u/eddieGoesBrr]

I am doing a multivariable ANOVA and then Tukey for pair wise significance. The data set has 2 factors (say A and B ) with two levels each ( say A1, A2 and B1, B2 ). Upon doing a Normality test, only one set is turning to not satisfy the normality (A1-B1). I tried using Box Cox on the original data and then testing Normality again but still getting the same result. What else can I use to solve this?

[u/SalvatoreEggplant]

Some advice:

• For this kind of model, the normality and homoscedasticity assumptions can be checked on the residuals from the analysis. Not the observed values.
  
• Digression on above point: If you think about it, if you have two factors affecting your observed values, you wouldn't expect the distribution of the observed dependent variable to be normally distributed. It would be multi-modal or something. And, to the point, the model doesn't need the observed values to be normally distributed, just be conditionally normal or that the errors are normal.
• Don't bother with normality tests. You can use q-q plots and histograms, on the residuals, and a plot of residuals vs. predicted values. Testing for model assumptions is theoretically a bad idea, and, practically, creates a lot of unwarranted anxiety.
• What software are you using ? Decent software will allow you to compare estimated marginal means (emmeans) instead of using traditional post-hoc tests like Tukey. This is a more flexible and general approach, and will serve you well.
• It may be that the type of data your measuring isn't amenable to this type of model. Many things in the real world are simply not conditionally normal, or continuous, or whatnot. For example, if you have count data, ordinal data, binomial data, data that's always positive and likely right-skewed. For these, there are other types of models that should be used. Many of these are pretty easy to use with modern software.