Nonparametric methods are used when the assumptions of normality of the residuals and equal variation in each group, required for parametric methods such as the ttest and ANOVA, are not met and can not be met by a suitable transformation of scale.
The more widely used methods replace each observation by its rank in the total set of observations. This comes with a cost. In general, nonparametric tests are not as powerful as parametric tests because, in transforming the data to ranks, they throw out some useful information. This means that a nonparametric test may fail to detect a true treatment effect which could be detected by a parametric method. Therefore, where possible, parametric methods should be used. Brief notes on three commonly used tests are:
The MannWhitney test
This the the nonparametric equivalent of Student's ttest, but it is a test of whether the medians (rather than the means) are the same in each group. Like the ttest it is only appropriate when comparing two groups
The KruskalWallis test
This is the nonparametric equivalent of the oneway ANOVA. It can be used to compare the medians of two or more groups, the null hypothesis being that all are samples from the same population. If the overall differences are statistically significant, posthoc comparisons are done by comparing each pair of medians in turn using the same test.
The Friedman's test
This is the nonparametric equivalent of the twoway ANOVA without interaction, appropriate for a randomised block experimental design. Again, it tests the null hypothesis that all treatment groups came from the same population, and is a test of group medians, removing any block effect.
Numerical examples are not given here. The methods are described in most statistical textbooks and are available in most computer packages.
