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I'd like to compare the (continuous) responses of two unpaired groups. The end objective is to test whether one of the groups will be 'superior' to the other (either by a mean or a median comparison).

However, I've come across 4 issues in the data:

  1. Non-normality: Visual approaches such as QQ-plot/histograms, and statistical ones such as Shapiro-Wilk test. Standard transformations (e.g. box-cox, log,etc) on the purchase amounts are hindered by the fact that the response can take negative values as well. (Cube root transformation did not help).

  2. Different variances between the two groups (based on Levene's test)

  3. Very imbalanced group sizes (~1:300), but there are over 1000 responses in the smaller group.

  4. Extreme outliers - there are a few values far below the 1%ile and 99%ile values of the data. These are not erroneous.

Based on these issues, my understanding is that parametric approaches (e.g. Welch's t-test) are inappropriate.

I've looked through Cross Validated and I haven't come across a conclusive answer in this context, particularly when it comes to points #3 and #4.

Based on other questions asked about Wilcoxon Ranked Sum test over here, it looks like it is resistant to at least some of the issues stated above, and its median-comparison approach would be particularly well suited to the extreme values observed in the data.

Am I on the right track here? Are there any alternative approaches I should be looking at?

I would really appreciate some help from this amazing community!

Some of the CV references used by me so far:

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    $\begingroup$ With data, that has large outliers as true values, you should make the decision of whether a mean or a median or both are relevant statistics depending on the subject, not on the easiest to perform test. Throwing a third hat in the ring, you could set up a permutation test for median, for mean, for 5% trimmed means or any other statistic that suites your substantial question best. I am not arguing in favour of a permutation test, just saying, you have the freedom to decide, which statistic should be tested based on the substance of your research. $\endgroup$
    – Bernhard
    Commented Feb 1, 2018 at 6:50

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