Some Methods for Strengthening the Common χ 2 Tests

Since the x2 tests of goodness of fit and of association in contingency tables are presented in many courses on statistical methods for beginners in the subject, it is not surprising that x2 has become one of the most commonly-used techniques, even by scientists who profess only a smattering of knowledge of statistics. It is also not surprising that the technique is sometimes misused, e.g. by calculating x2 from data that are not frequencies or by errors in counting the number of degrees of freedom. A good catalogue of mistakes of this kind has been given by Lewis and Burke (1). In this paper I want to discuss two kinds of failure to make the best use of x2 tests which I have observed from time to time in reading reports of biological research. The first arises because x2 tests, as has often been pointed out, are not directed against any specific alternative to the null hypothesis. In the computation of x2, the deviations (fi mi) between observed and expected frequencies are squared, divided by mi in order to equalize the variances (approximately), and added. No attempt is made to detect any particular pattern of deviations (fi mi) that may hold if the null hypothesis is false. One consequence is that the usual x2 tests are often insensitive, and do not indicate significant results when the null hypothesis is actually false. Some forethought about the kind of alternative hypothesis that is likely to hold may lead to alternative tests that are more powerful and appropriate. Further, when the ordinary x2 test does give a significant result, it does not direct attention to the way in which the null hypothesis disagrees with the data, although the pattern of deviations may be informative and suggestive for future research. The remedy here is to supplement the ordinary test by additional tests that help to reveal the significant type of deviation. In this paper a number of methods for strengthening or supplementing the most common uses of the ordinary x2 test will be presented and illustrated by numerical examples. The principal devices are as follows: