qqplot result and normality for ANOVA The independent variable is promotion, and it is assigned to 3 groups. The dependent variable is sales revenue. I have 172 observations of sales revenue for promotion group 1, 188 for group 2 and 188 for group 3.
With this dataset, I want to know which promotion is more effective at increasing sales revenue.
I first thought of doing ANOVA with Tukey test to answer the question.
To see whether to do ANOVA I checked the normality assumption with qqplot and the plot seems to tell me that residuals are a bit bimodally distributed and Shapiro-Wilk's null hypothesis is rejected as well (Is the sample size is too big for shapiro-wilk test?).
I want to know if I can continue to do ANOVA with the data. If not, what statistical analysis is best for this case? Also, when the sample size is only 22 with the same qqplot result (say, null hypothesis of Sapiro-Wilk is not rejected) , can I do ANOVA?

Thanks!
 A: You don't need ANOVA (and its distributional assumptions) to compare three groups. You can use the Kruskal-Wallis test to check whether the three promotions have the same revenue distribution. Validity of assumptions aside, both the parametric (ANOVA) and the non-parametric (Kruskal-Wallis) tests only tell you whether to accept or reject the null hypothesis that the three promotion strategies have the same effect on revenue. If the null hypothesis is rejected, you don't actually learn which promotion is most effective.
Then you dive deeper and test the pairwise differences among the groups. However, it is possible that all pairwise comparisons are insignificant even though the overall test is significant, or the other way around. You also have to consider how to adjust for doing all these tests. See here and here.
Instead of hypothesis testing you can do estimation. Here is how this analysis might proceed in three steps.
Start by plotting your data. Three (aligned) histograms, one for each promotion strategy, will be wonderfully informative and show if the revenue distributions are qualitative different and how. The how cold be important since a promotion might increase sales on average or perhaps it might induce a few customers to spend a lot more.
Say the exploratory data analysis suggests to compare the center of the revenue distribution. Then you can decide to use the mean (sensitive to big spenders) or the median (more robust). You choose the median.
Finally you use the bootstrap to estimate the median revenue for each promotion strategy. This will give you so much more information than an accept/reject statement of the null hypothesis that the promotion strategies are equally effective.
