I would like to test the normality of data and homogeneity of variances for a factorial experiment. For example, given three fixed crossed factors i.e. A, B, C, with 3 replicates (1,2,3).
Should data normality be tested per each group (A, B, C) or on the whole dataset?
For the homoscedasticity test, should Levene's test be preferred for non-normal data, and Bartlett's test for normal ones?
Many papers have found that testing the homoscedasticity or normality is ineffective because such tests generally do not have enough power to detect situations where these assumptions are unsatisfactory (e.g., Wilcox, 2012, Modern Statistics for the Social and Behavioral Sciences, CRC Press). Modern heteroscedastic methods perform nearly as well as homoscedasitc methods when indeed there is homoscedasticity. Classic rank-based methods have been improved substantially. And there are many robust methods for comparing measures of location, which are also summarized in this book. As you will see, even small departures from normality can cause power problems, but modern methods deal with this in a very effective manner.
The normality of data and homogeneity of variances shouldn't be performed per each treatment, but per each variable. For those variables that show normality and homogeneity, you can do ANOVA (i.e. PROC GLM of SAS). For those variables that don't show normality or homogeneity you can perform a non-parametric analyses, i.e. Kruskal-Wallis test. The disadvantage of non-parametric analyses is the impossibility to do interaction analyses, which are important in a factorial assay design; so, you can transform data (log transformation, arc-sin transformation, square root transformation) and perform ANOVA. The type of transformation that you should use will depend of variable characteristics.
I hope I helped you.
Normality is the initial condition of error term; so to check it, you should apply your model, obtain the residuals and then test whether the residuals accept or reject normality. Clearly, the residuals assumed to be iid for all groups.
To check homogeneity of variances, there are 3 famous tests: Levene's test, Brown-Forsythe test and Bartlett's test. Bartlett's test is not robust with respect to the normality, in the sense that when the population distribution is not normal, then this test may go wrong. Since Brown-Forsythe test works with median instead of mean, this test is more robust than Levene's test or Bartlett's test.
Note that similar to normality, the assumption of homogeneity of variances can be checked from residuals; for instance, comparing box-plot of residuals for groups may help to check the equality of variances.
Can you make a practical example in SAS using the attached dataset?
Many papers have found that testing the homoscedasticity or normality is ineffective because such tests generally do not have enough power to detect situations where these assumptions are unsatisfactory (e.g., Wilcox, 2012, Modern Statistics for the Social and Behavioral Sciences, CRC Press). Modern heteroscedastic methods perform nearly as well as homoscedasitc methods when indeed there is homoscedasticity. Classic rank-based methods have been improved substantially. And there are many robust methods for comparing measures of location, which are also summarized in this book. As you will see, even small departures from normality can cause power problems, but modern methods deal with this in a very effective manner.
I have published recently a paper about the problem of homogeneity and normality in ANOVA. There are several bibliographic reference also, that give contrastisting opinion about your problem. But I think that if there are suspicious of not normality or heterogeneity an NP test (accordig with Randy Wilcox) are very powerfull. (my paper is: 19. Acutis M., Scaglia B., Confalonieri R., 2012. Perfunctory analysis of variance in agronomy, and its consequences in experimental results interpretation. European Journal of Agronomy, 43, 129-135. Doi: 10.1016/j.eja.2012.06.006)
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