StATS: Parametric versus nonparametric tests (July 30, 2001)

Dear Professor Mean, When should I use a parametric test versus a non-parametric test?

A parametric test, of course, is a test that requires a parametric assumption, such as normality. A nonparametric test does not rely on parametric assumptions like normality.

Whether to choose a parametric versus nonparametric test is a matter of judgement. But keep in mind several things.

First a nonparametric test protects against some violations of assumptions and not others. The two sample t-test requires three assumptions, normality, equal variances, and independence. The non-parametric alternative, the Mann-Whitney-Wilcoxon test, does not rely on the normality assumption, but you better make sure that you still meet the equal variances and independence assumptions.

Second, you can often use a transformation to better match some of the assumptions like normality and equal variances. The log transformation can sometimes give you data that is much better behaved.

Third, you might select your statistic on the basis of what others in your field have used. Is there a lot of precedent for the type of data you are collecting to be non-normal?

Fourth, are you comfortable with the fact that the non-parametric test may be evaluating a different measure than the parametric test? The Mann-Whitney-Wilcoxon test, for example, provides you with an estimate of P[X>Y], probability that a randomly selected patient from your first population has a larger value than a randomly selected patient from the second population. This may be more interesting (or it may be less interesting) than a two-sample t-test which provides you with an estimate of the difference in the average between the first and the second populations.

Fifth, do you have a very large sample size? Large sample sizes are more likely to show significant deviations from normality, but these are the situations where non-normality is the least of your worries (thanks to the Central Limit Theorem). Some statisticians reserve non-parametric tests for situations where the sample size is small.

What would I recommend?

I'm comfortable with a variety of statistical approaches. If you have already specified the statistical analysis in your protocol, then stick with your protocol unless your data suggests very strongly that a different approach is called for. Don't deviate from your pre-specified analysis without a good reason.

If you are in the protocol writing stage, I would suggest that you propose using a paired t-test, unless you have evidence (such as through a normal probability plot) that the data are non-normal. In this case, you will use a transformation of the data or a non-parametric test instead. Don't use a formal test of normality like the Shapiro-Wilk test. A formal test of normality has the very little power for small sample sizes, but this is when non-normality is most troublesome. Even worse, a test has lots of power for large sample sizes, but this is when non-normality is least troublesome.

Finally, if you are in middle of the peer review process, listen to your peer reviewers. They have a good feel for the standards and practices in the area you are working in. Also, there's safety in numbers. If you adopt an approach that the peer reviewers like, you are probably using an approach that others in your discipline would like.

Further reading

Parametric Tests. Chong-Ho Yu (accessed 7/26/01)

Measurement theory: Frequently asked questions. Warren S. Sarle (accessed 7/26/01)

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