# Can Wilcoxon be used in large sample with non-normal distribution?

I am doing my undergrad research, aiming to know the difference of before and after an intervention. our sample size is 37 which is already considered as a large sample right? However, when we test the normality of our data it is not normal which violates the assumption. Is it right to use Wilcoxon signed test or go for parametric test like t-test?

• If $n=37$, the t-test usually has good power to test for a mean difference in highly non-normal samples relative to the Wilcoxon, unless the data are bizarrely skewed - and if they are the effect is super difficult to interpret. Alternately, many kinds of count data are often log transformed before doing testing. Apr 24 at 4:47
• I agree with the basic sentiment (that the one-sample t often performs fairly well), though it may be more strongly expressed than I'd agree with. Both tests are typically biased for skewed parent distributions though (albeit with paired data under shift alternatives this is usually not an issue) and if that's the issue I'd tend to avoid both unless the sample size is large (where the t is okay). To my eye, it doesn't take very heavy skewness to clearly see the bias in the one sample t-test at n=37. Apr 24 at 7:01
• With symmetric distributions that are somewhat heavier-tailed than normal (but not necessarily heavy-tailed per se; exponential tails will do, for example), the Wilcoxon signed rank test will often outperform the t under shift alternatives. Apr 24 at 7:01
• @Chilenesa What sort of quantity are you measuring? What potential values could this variable take? (i.e. what's possible in the population, not what's in the sample). Are these differences of paired data or is it inherently one-sample? Does your hypothesis relate to a population mean? Apr 24 at 7:03
• It is a paired data and one sample only Apr 25 at 6:42

I would not call n=37 large (!).

Yes, you can use a Wilcoxon signed rank test, but there are some substantial caveats. It's not suitable for all situations. Then again, the t-test might not be either.

1. If you're interested in a hypothesis about population means (and if not, why the t-test?), don't test something else. It's possible for the signed rank test and the t-test to see opposite directions of difference from a hypothesized value because they measure changes differently, so idly swapping from one to the other is typically not what I'd look to do unless I had a pretty good handle on the properties of the parent distribution. Under symmetry of the parent distribution (not necessarily the sample) you should typically be okay, though.

2. The properties of the Wilcoxon signed rank test can be pretty sensitive to the symmetry assumption. It may be noticeably biased (that is, have a lower rejection rate than $$\alpha$$ at some non-zero effect size).

3. In small samples like this, the one-sample t test can be somewhat sensitive to the symmetry assumption as well, but often less so than the signed rank test.

4. You can improve the accuracy of the significance level of the t for non-normal parent distributions, if that symmetry assumption holds.

5. The two sample t is more robust against non-normal distributions, but for one sample tests the t-test can have some issues, especially in smallish samples (and relative power may still be poor even in large ones). I'd typically I'd urge a more careful selection of a suitable choice of distributional assumption (without reference to the particular data values you want to use in the test), after very carefully making the population parameter in the hypotheses explicit, and explicitly identifying the form of the alternative (people often default to a simple location shift but that's often untenable with many kinds of data; it generally won't make sense for a bounded variable, for example).

However, if the parent distribution would be no more than mildly skew and not very heavy tailed, the t-test should be fine. With larger samples greater skewness could be tolerable (in that the impact on significance level and test bias will be reduced, it won't help asymptotic relative efficiency but perhaps with larger samples you may be in a circumstance where you're less bothered about efficiency).

• Also check Glen_b's answer here. I am sure, if you dig in, there would be other related threads too. Apr 26 at 3:56
• Yes, though I tend to focus more on the two sample case there. I hope I didn't say anything very inconsistent between them. Apr 26 at 4:11