# ANOVA:How to detect non-normality with a QQPlot in the presence of non-homogeneous variance

This is a pretty general question but, I often find statistical textbooks claiming that, in order to justify the within groups normality assumption of a one way ANOVA, you can look at a QQ plots of the residuals. However, the qq plots can only detect non-normality when the variance (or standard deviation) across all groups is homogeneous. I was wondering how I could use this type of plot when I use Welch's ANOVA. For example, in the case of Welch's ANOVA, you don't need to adhere to any assumption of homogeneous variance, but if your variance is not homogeneous, how can you use a QQ plot to test for normality? I was thinking of standardizing each of the residuals by their within group standard deviations first so that all variances would be equal. I feel like this approach is reasonable but I have not seen it discussed in any textbook or anywhere online. Lets just assume the sample sizes of each group are unequal and too small to invoke the central limit theorem.

For example, in the case of Welch's ANOVA, you don't need to adhere to any assumption of homogeneous variance, but if your variance is not homogeneous, how can you use a QQ plot to test for normality?

If the individual groups are not too small you could plot each one.

I was thinking of standardizing each of the residuals by their within group standard deviations first so that all variances would be equal.

They will no longer be normal. For one-way unbalanced ANOVA they'd have different t-distributions. You could work out the t-distribution each should follow and plot against the relevant t-scores I suppose.

Edit: If the sample size in each subgroup is large, you can treat it as normal. If the sample size in each subgroup is constant, they should share the same t-distribution. In either case, you should be able to get a reasonable combined plot.

• Actually I should add that I was also curious what would happen IF the sample sizes were too small to plot individually. Anyway thanks for your answer regarding the t-distributions. If that is the case, I think that pretty much takes care of my question and throws my suggestion of standardizing the variances out the window. I'm not 100% sure why the distributions would no longer be normal and the t-distributions would be different but that's probably due to of my lack of knowledge in the area. If you could elaborate that would be great! – Jimj Mar 15 '13 at 6:33
• Basically, because of this fact. – Glen_b -Reinstate Monica Mar 15 '13 at 7:55
• Hey Glen_b, Ok can I get an explanation or example of how to carry out your suggestion and get a reasonable plot? Thanks so much for your help. Really appreciate the time you put into this. – Jimj Mar 18 '13 at 23:35