**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) seamonkey.ed.asu.edu/~alex/teaching/WBI/parametric_test.html

Measurement theory: Frequently asked questions. Warren S. Sarle (accessed 7/26/01) ftp://ftp.sas.com/pub/neural/measurement.html

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