Non-parametric, non-homogeneous variance ANOVA?

[Will P]

Hello, 

I recently ran a repeated measures ANOVA for some behavioural data. The within subject factor was one factor with two levels (conditions). The between subject factor was group, and there were 3 groups. 

There was a main effect of condition, a main effect of group, and also a condition by group interaction. 

A subsequent one-way anova showed that the groups differed significantly with respect to one condition more than another. 

However, a Levene’s test showed that for both levels, the variance was not homogenous between groups (p= 0.01, and p =0.008). Further, using separate one-way ANOVAs for each condition (level), a Shapiro Wilks test for Normal distribution showed that the data for one group in each condition, was not normally distributed (for one group/condition, the skewness and kurtosis was -2 < +2, for the other group/condition, the skewness and kurtosis were 2.03, and 5.6).

The group sizes were 19, 19 and 23 subjects for the 3 groups. 

I therefore have 3 questions i’d be grateful if you could help me with:

1) I understand that if you violate the ANOVA assumption of…

i) a Normal distribution, you should use a Kruskal-Wallis test, with a post-hoc Mann-Whitney with Bonferroni correction test,

ii) the Homogeneity of Variance, you should use a Welch F test, with a post-hoc Games-Howell test.

What test do you use therefore if the data is neither parametric, nor has homogenous variance?

2) I have actually run both alternatives (i and ii) listed above, and I get the same outcome results as the original repeated measures ANOVA.

Hence, should I just stick with the original repeated measures anova results since the departure from a normal distribution or variance homogeneity does not seem to have such an impact?

3) Since ANOVA’s are typically quite robust to departures from a normal distribution, can I report the original results of the repeated measures ANOVA (i.e. main effect of group, condition, condition-by-group interaction), but specify in the subsequent one-way anova’s exploring the pairwise comparisons within each condition, that the Mauchy’s test was significant, and therefore my post-hoc analyses are Games-Howell corrected?

Thanks very much for your time,

Will

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