Goodness-of-fit tests for discrete data: a review and an application to a health impairment scale

We review the advantages and disadvantages of several goodness-of-fit tests which may be used with discrete data: the multinomial test, the likelihood ratio test, the X2 test, the two-stage X2 test and the discrete Kolmogorov-Smirnov test. Although the X2 test is the best known and most widely used of these tests, its use with small sample sizes is controversial. If one has data which fall into ordered categories, then the discrete Kolmogorov-Smirnov test is an exact test which uses the information from the ordering and can be used for small sample sizes. We illustrate these points with an example of several analyses of health impairment data.