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Let's say I run an experiment with one dependent variable and three experimental groups. In the end I would like to report both the results of a statistical test to know if there are any group effects, but also some descriptive statistics (center and spread) for the individual populations. The data won't be normally distributed (the dependent variable is continuous but bounded between 0 and 1), so I plan to use a Kruskal-Wallis test and report the median and interquartile range. However, I've been reading (here, and elsewhere) that the Kruskal-Wallis test isn't necessarily appropriate to use in cases of normality violations.

So if I use the more standard ANOVA instead, would it still be appropriate to report the median and interquartile range, or would that not 'fit' with the test?

[Edit]: The data look similar to (truncated) Gamma distributions.

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    $\begingroup$ The Kruskal-Wallis test is a fine choice if your data aren't normal; the ANOVA would be more suspect. Can you say more about what your data actually are? How is it they are bound by 0 & 1? $\endgroup$ – gung - Reinstate Monica Aug 14 '15 at 0:22
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    $\begingroup$ In addition to what @gung requested also include the n for each group and state more specifically what you'd like to say about your data. You don't just change the descriptive statistics based on the data. It depends on what you want to say about them because they mean different things. $\endgroup$ – John Aug 14 '15 at 0:37
  • $\begingroup$ @John - I'm surprised that you say that you wouldn't change descriptive statistics based on the data. If I want to describe the center of a distribution, my intuition is that mean would be appropriate if the data are approximately normal (or at least continuous and not skewed), whereas the median is more appropriate otherwise. Similarly for spread and standard deviation vs. iqr/mad. $\endgroup$ – Evan Aug 14 '15 at 13:46
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    $\begingroup$ Evan, you and @John both have good points. Descriptive statistics ought to be motivated by what you intend to say about the data--and this can be determined before seeing the data. After obtaining the data, though, you might want to add to that description to describe important or interesting departures from your expectations or assumptions. $\endgroup$ – whuber Aug 14 '15 at 13:51
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    $\begingroup$ The two things have different purposes. You use resistant statistics, such as median and IQR, in order to insulate yourself from being misled by outlying data, etc. There's nothing contradictory in also planning to test additional assumptions you might want to evaluate, such as symmetry of the distribution, its normality, etc. In fact, you can do anything you like--mix and match frequentist and Bayesian approaches, make different (incompatible) assumptions at different times, etc. You have to be careful about how you compute p-values if you're going to do any formal testing, that's all. $\endgroup$ – whuber Aug 14 '15 at 14:21
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Consistent with @whuber, the data are not usually capable of telling you how Gaussian a variable's distribution is. So it makes sense to employ statistics that are always descriptive, such as the empirical cumulative distribution function, extended box plots (showing more quantiles than the 3 quartiles) and quantiles. The median is always descriptive and interpretable for a continuous variable; the mean may not be. For an asymmetrically distributed variable, the SD is not interpretable or helpful in most cases whereas outer quantiles (including quartiles) and Gini's mean difference are interpretable and meaningful. Couple those with the number of missing and number of non-missing observations.

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  • $\begingroup$ I take the phrase "employ statistics that are always descriptive" to imply that it would not be inappropriate to report (for example) median and iqr while using an ANOVA (or other parametric test). Is that correct? $\endgroup$ – Evan Aug 14 '15 at 15:05
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    $\begingroup$ No I wouldn't say that. These are still descriptive. If I felt strongly enough that the raw data are normally distribution so that I would use parametric ANOVA, I would report mean and SD in addiiton. But I would report quantiles or ECDF in every case. $\endgroup$ – Frank Harrell Aug 14 '15 at 15:28

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