Which nonparametric test should I use

I am currently doing an experiment to test whether there is a significant difference between test results of a control group of steel bars of good shape and another group of corroded steel bars. I do have 38 measurements in total and need to know which nonparametric test should I use since the test results of both groups are not normally distributed.

The control group of steel bars has been soaked in water for few weeks and then been retested for few weeks and therefore I do have now 20 measurements before and after soaking, which nonparametric test should I use for this case since the data are not normally distributed too?

If there are several tests, which is the most accurate one for the case studies presented above?

Thanks

• instead of non-parametric test, is it possible for you to transform the data to normal distribution? – ceoec Jun 28 '16 at 5:05
• I haven't tried transformation, but why? – Ali Sultan Jun 28 '16 at 7:01
• non-parametric is less powerful, plus you data is continuous so transformation probably would work. And if your data can be transformed to normal distribution so you can run parametric test, it would be a better option. – ceoec Jun 28 '16 at 7:12
• Thanks a lot. Is there any free tool I can use to transform the data? I searched for box-cox but couldn't find anything. – Ali Sultan Jun 28 '16 at 7:17
• By the way, do you think that according to the central limit theorem that I can use parametric tests if the sample size is 38 for the first case study even if the data are not normally distributed? – Ali Sultan Jun 28 '16 at 7:21

Remember that the assumption of many parametric tests are that the errors are normally distributed, not the observations.

If you have reached the conclusion that your data is not normally distributed on the basis of the observations alone then I would suggest verifying this by, for example, modelling the effect of group on your measurement using linear regression and then inspecting the distribution of the model residuals.

If the model residuals are normally distributed, then a parametric test will be appropriate.

If the model residuals are not normally distributed, then inspecting the shape of the distribution may help you to determine which, if any, transformation would be appropriate.

• No, I verified this using many normality tests such as Shapiro-Wilk, Anderson-Darling, Chen-Shapiro, and some others. – Ali Sultan Jun 29 '16 at 23:28
• Ah, well as I said, visually inspecting the errors of the model should give an indication of whether it would be possible to transform the data to be something more normal. If a transformation isn't possible, then I might look into a non-parametric method. – Ian_Fin Jun 30 '16 at 8:02
• Well, this was my question, which ones are the most efficient for the case studies I explained above. Thanks for your interest. – Ali Sultan Jul 1 '16 at 8:40
• Without knowing what the distribution actually looks like, e.g. its skewness and kurtosis, it's impossible to say. – Ian_Fin Jul 1 '16 at 9:44
• But as far as I know that the nonparametric tests are distribution free, am I wrong? – Ali Sultan Jul 1 '16 at 17:07