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I have a sample of size 30, being studied for effect of a surgical procedure on certain substance concentration in blood,

In other words, for comparing a measurement change before and after a procedure in a sample of size 30, which test should I use? Wilcoxon signed-rank test or paired t-test?

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  • $\begingroup$ Sample size is not a primary consideration when looking at what test to use; the properties of the power curve and significance level under various possible situations might be more relevant. Sample size won't save the power curve. What's the actual hypothesis of interest? Do you know anything about how the concentration is distributed? (Many such measurements may be mildly right skew, but it's not always the case) $\endgroup$
    – Glen_b
    Feb 10, 2017 at 23:48

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The Wilcoxon signed rank test is appropriate because there is no reason to believe that the data are even approximately normally distributed and the sample size is only 30.

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  • $\begingroup$ Thanks Sir.. another question, please.. How can I determine whether my data are normally distributed? I need the most simplified way for that. $\endgroup$
    – Fatma
    Feb 9, 2017 at 22:55
  • $\begingroup$ You would look to see whether or not the density is The Wilcoxon signed rank test is relatively efficient even when the distribution is approximately normal.. $\endgroup$
    – Michael R. Chernick
    Feb 9, 2017 at 23:12
  • $\begingroup$ @Fatma The concentrations wouldn't need to be approximately normal; under the null hypothesis the differences would need to be. This is a considerably weaker assumption. That said, I'd be inclined to use the signed rank test; even when it has less power than the t-test it's only a little less. The tail of the distribution of differences only needs to be a little heavier than normal for the signed rank test to start having better power. $\endgroup$
    – Glen_b
    Jun 5, 2017 at 0:21
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With this sample size it will be usually not possible to reliably detetmine whether the regression residuals are normally distributed. However, there may be information in the literature on this. Without knowing more, I would suspect you may have a non-normal case, because many such outcomes are often best analyzed e.g. using ANCOVA after a log-transformation (of course I do not know whether that is the case here). Standard out-of-the-box rank tests might be "safe", but involve some (often small) loss of power and flexibility (in terms of e.g. accounting for covariates).

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