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I have a paired sample data with n=21. The two values V1 and V2 are from two different time points for the same group. I want to see if there is any improvement/reduction in values at time 2(V2). Of course, the first thing that comes to my mind is the paired t-test for which differences (D=V2-V1) have to be normally distributed. These are boxplot, histogram and qqplot of the differences data. The data looks slightly skewed but I'm not sure if I should go with t-test as the sample size is small (n=21). From Shaipro Wilk test, I'm getting p value as 0.025 and W=0.89 which rejects the null hypothesis that the data is normal. I'm thinking of an alternative to t-test such as Wilcoxon- signed rank sum test. Not sure which method is more appropriate. Also, how important is it for the data to be symmetrical for Wilcoxon test? From the box plot, it doesn't look so symmetrical. I'm kinda confused as to how to proceed and would really appreciate any help.

This is how the data looks like: V1 V2 D(V2-V1) 1 2.5 2.0 -0.5 2 3.5 1.5 -2.0 3 2.0 2.0 0.0 4 1.5 4.0 2.5 5 4.0 3.5 -0.5 6 3.5 4.0 0.5 7 3.0 3.0 0.0 8 2.5 2.0 -0.5 9 4.0 3.5 -0.5 10 3.5 2.5 -1.0 11 3.5 3.5 0.0 12 2.5 1.5 -1.0 13 2.0 2.0 0.0 14 3.0 3.0 0.0 15 1.5 2.5 1.0 16 1.5 1.5 0.0 17 1.5 1.5 0.0 18 2.0 2.5 0.5 19 3.5 2.5 -1.0 20 1.5 1.5 0.0 21 3.0 2.0 -1.0

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  • $\begingroup$ Wilcoxon should work for this; however paired t might be a stronger test. Do you have any reason to believe your data should be normally distributed? If so, a larger sample might help, but it looks like the critical value for that test was off by a large amount. $\endgroup$
    – MikeP
    Commented Jul 5, 2018 at 17:49
  • $\begingroup$ Thanks! No, I have no reason to believe that data should be normally distributed. I think Wilcoxon is the best choice to go with. Yes, the critical value for the test is way off. $\endgroup$
    – Mohamad Sahil
    Commented Jul 5, 2018 at 18:03

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First off all: boxplot, histogram etc. are useless because of a small sample size. Secondly the normal tests are of limited useful because not rejecting null hypothesis doesn't mean that you accept it. So if you don't have reasons believe otherwise use t-test - it's pretty robust and what it's more important is more powerful. You can also use resampling methods so as to find the distribution of a statistics. In my opinion I would not use either tests just build confidence intervals for the differences. You will give you more information about uncertainty.

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  • $\begingroup$ While the downvote isn't mine, I think I can provide some explanation for it. Your answer doesn't seem to offer good reasons for your recommendations. For example, while the t-test is reasonably level robust as long as you don't go a long way from normality, it is only more powerful in a fairly limited set of circumstances and only by a little. It's the most powerful test at the normal where its asymptotic relative efficiency (relative to the signed rank test) is $\pi/3$. That is, in a large sample from a normal distribution it's like having your sample size multiplied by 1.047 ... ctd $\endgroup$
    – Glen_b
    Commented Jul 6, 2018 at 1:10
  • $\begingroup$ ctd... which isn't that great an advantage; like taking n=21 to n=22). If the tails are even a little heavier its power quickly starts to fall behind, and with heavy tails it can be arbitrarily bad (like throwing out almost all the information in the data). I'd suggest beginning this sort of choice between tests by carefully elucidating the precise null and alternatives of interest and any assumptions that you'd want to make (e.g. whether restricting to a location-shift alternative or something more general) and then choosing a test that should perform well at that task. $\endgroup$
    – Glen_b
    Commented Jul 6, 2018 at 1:10
  • $\begingroup$ Thanks! You mean...I should do both the tests and see which gives the best result? I'm using R to do both the tests. wilcox.test from the base stat package gives results with a warning message that says that exact p-value cannot be computed with ties and zeroes. Since my data is small, I cannot afford to discard the ties. How do you think should I go about this issue? $\endgroup$
    – Mohamad Sahil
    Commented Jul 6, 2018 at 3:57

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