Hypothesis Testing – Appropriate Normality Tests for Small Samples

goodness of fithypothesis testingnormality-assumptionsmall-sample

So far, I've been using the Shapiro-Wilk statistic in order to test normality assumptions in small samples.

Could you please recommend another technique?

Best Answer

The fBasics package in R (part of Rmetrics) includes several normality tests, covering many of the popular frequentist tests -- Kolmogorov-Smirnov, Shapiro-Wilk, Jarque–Bera, and D'Agostino -- along with a wrapper for the normality tests in the nortest package -- Anderson–Darling, Cramer–von Mises, Lilliefors (Kolmogorov-Smirnov), Pearson chi–square, and Shapiro–Francia. The package documentation also provides all the important references. Here is a demo that shows how to use the tests from nortest.

One approach, if you have the time, is to use more than one test and check for agreement. The tests vary in a number of ways, so it isn't entirely straightforward to choose "the best". What do other researchers in your field use? This can vary and it may be best to stick with the accepted methods so that others will accept your work. I frequently use the Jarque-Bera test, partly for that reason, and Anderson–Darling for comparison.

You can look at "Comparison of Tests for Univariate Normality" (Seier 2002) and "A comparison of various tests of normality" (Yazici; Yolacan 2007) for a comparison and discussion of the issues.

It's also trivial to test these methods for comparison in R, thanks to all the distribution functions. Here's a simple example with simulated data (I won't print out the results to save space), although a more full exposition would be required:

library(fBasics); library(ggplot2)
set.seed(1)

# normal distribution
x1 <- rnorm(1e+06)   
x1.samp <- sample(x1, 200)
qplot(x1.samp, geom="histogram")
jbTest(x1.samp)
adTest(x1.samp)

# cauchy distribution
x2 <- rcauchy(1e+06)
x2.samp <- sample(x2, 200)
qplot(x2.samp, geom="histogram")
jbTest(x2.samp)
adTest(x2.samp)

Once you have the results from the various tests over different distributions, you can compare which were the most effective. For instance, the p-value for the Jarque-Bera test above returned 0.276 for the normal distribution (accepting) and < 2.2e-16 for the cauchy (rejecting the null hypothesis).

Related Question