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I have read a lot about when to use a (paired) t-test or a Wilcoxon Signed Rank Test as the non-parametric alternative but I need your help:

I have gathered some paired data and want to perform some hypothesis tests on them. Then I've calculated the differences between both groups and made a density-plot which looks like this:

Density plot of Differences between both (paired) samples

As this looks kind of normally distributed, I have also performed a Shapiro-Wilk and Anderson-Darling normality test. Both deliver p-values greater than .05 (p = .1101 and p = .0955).

Although the p-values are just a bit above the significance level and the graph is rather skewed and has no "nice" bell-shape, can I assume that the data is normally distributed based on this output? So should I perform a paired t-test or should I use the Wilcoxon Signed Rank test instead?

Thank you very much!

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    $\begingroup$ You can always do both! On this evidence, the P-values should be very similar. The question is never are my data normally distributed (short answer: they aren't) but always are my data close enough to normal that the difference doesn't matter. $\endgroup$ – Nick Cox Jan 13 '16 at 19:52
  • $\begingroup$ The fact that two normality tests failed to reject tells you more about your sample size than whether the distribution is close enough for your purposes to use some procedure which assumes normality. Different procedures are not all equally sensitive to this kind of deviation from normality, and individuals tolerance for impact on say p-values or coverage also differ. In this case, the one-sample t is probably not so problematic, but your mileage may vary. $\endgroup$ – Glen_b -Reinstate Monica Jan 13 '16 at 21:43
  • $\begingroup$ Thank you very much! After performing both a t-test and wilcoxon signed rank test the p-values are indeed quite similar (.029 vs .039). I am just a bit confused how to argue why i used which test. Theory says that for underlying "normally distributed data" usually the t-test has quite good statistical power whereas for "not that normally distributed data" researchers suggest to use the wilcoxon signed rank test. The easiest thing would be looking at the density-plot and using normality test(s). If the null of the normality test(s) get(s) failed to reject I'd prefer one test over the other. $\endgroup$ – martl Jan 13 '16 at 22:35
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Although the p-values are just a bit above the significance level, can I assume that the data is normally distributed based on this output?

First, are you sure the Anderson-Darling test should be compared against 0.05? Wikipedia lists a variety of cutoffs depending on your understanding of the distribution.

I'm not sure if your data is normally-distributed (it looks skewed, but we all know that density plot is an estimate), but both Shapiro-Wilk and Anderson-Darling are null hypothesis tests. A high p-value simply means you fail to reject the hypotheses set forth by each test. In both cases, you cannot necessarily accept the distribution's normality.

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  • $\begingroup$ Thank you very much for your comment! Well, I just read that it would be appropriate to use a normality test in order to check which hypothesis test to use. A few papers then suggests to use Shapiro-Wilk, but additionally the Anderson-Darling can be used - therefore I simply tried both and compared their p-values without thinking too much :-) But as far as I understand it, I fail to reject the null hypotheses in this case and can assume (based on the normality test) that the underlying data is normally distributed and therefore use the t-test? At least I can argue like this, right? $\endgroup$ – martl Jan 13 '16 at 22:58

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