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I am currently writing my Master's Thesis and I have a question in regards to the Kolmogorov-Smirnov test (ks.test in R), Anderson-Darling test (ad.test in R), and Wilcox/Mann-Whitney U test (wilcox.test in R).

With KS and AD I am testing whether two samples are drawn from the same distribution and with the MWU test I am testing the means of two samples.

However, I was wondering whether I can actually draw any conclusions/interpret the respective test statistics themselves (not p-values)? Could anyone provide me with an explanation and maybe with an economic intuition when I analyze return data?

Many thanks for your help, highly appreciated!

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  • $\begingroup$ It would be wise first to understand what these tests are testing. The KS and AD tests are comparing one sample to a reference distribution. See, for instance, the R manual page for ks.test {stats} for a clear warning against using it to compare two samples. The MW U test can be construed as testing medians but not the means (unless you make an unusually restrictive assumption of symmetry of both distributions). Because comparing distributions is complex, people usually use graphical comparisons such as QQ plots rather than single-number statistics to interpret the differences. $\endgroup$
    – whuber ♦
    Commented Mar 24, 2017 at 20:03
  • $\begingroup$ @whuber there are two-sample versions of both the KS and AD test (due to Smirnov and Pettit, respectively); and the MW doesn't really test medians unless you make almost-as-restrictive assumptions as you would need to make it a test of means. (Indeed it's possible to create samples that the MW rejects in one direction but the medians are the other way around.) $\endgroup$
    – Glen_b
    Commented Mar 25, 2017 at 3:55
  • $\begingroup$ Hey guys, thanks a lot for your answers. Big sorry for not mentioning it earlier, as @Glen_b states, I am applying the two-sample versions of the KS and AD test. Is there any interpretation of the test statistic itself? $\endgroup$
    – tiba14ab
    Commented Mar 25, 2017 at 8:52
  • $\begingroup$ While it's definitely the case that some level of interpretation can be offered for what these statistics actually measure/respond to, it's not clear what an "economic intuition" might consist of; indeed I expect that wouldn't be especially clear to most statisticians. Specifically economic intuitions -- whatever those will consist of -- might be better sought elsewhere. $\endgroup$
    – Glen_b
    Commented Mar 26, 2017 at 0:21
  • $\begingroup$ Thanks for your answer @Glen_b. That is a valid point. Could you, however, help me with a statistical interpretation of the respective test statistics? I am struggling a bit to understand what exactly the statistics tell me by looking at the way they are computed. Many thanks! $\endgroup$
    – tiba14ab
    Commented Mar 27, 2017 at 10:07

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I think that the only intuition behind the test statistics of those tests would be related to the p value (significant or not) - the conclusions to be drawn depend on the research question. However, if you see how such test statistics are calculated you can try to get some additional intuition: e.g., if you compare two cumulative density functions you can determine the intervals, namely values, for which the CDFs are similar and intervals, for which the CDFs diverge - this gives you additional inference, e.g. you can see for which intervals the test statistic reaches its maximum (and therefore the CDFs diverge). VK

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