Comparing Multiple Proportions

Abstract

Chi-squared is a useful and convenient statistic for testing hypothe­ses about multinomial distributions (see Sections 2.1 and 4.1). This is important because a wide range of applied problems can be formu­lated as hypotheses about “cell frequencies” and their underlying ex­pectations. For example, in Section 4.12, Silver Butterfly Straddles, the question arises whether the price change data are distributed normally; this question can be reduced to a multinomial problem by dividing the range of price changes into subintervals and counting how many data points fall into each interval. The cell probabilities are given by the normal probabilities attaching to each of the in­tervals under the null hypothesis, and these form the basis of our expected cell frequencies. Chi-squared here, like chi-squared for the four-fold table, is the sum of the squares of the differences between observed and expected cell frequencies, each divided by its expected cell frequency. While slightly less powerful than the Kolmogorov- Smirnov test—in part because some information is lost by grouping the data—chi-squared is easier to apply, can be used in cases where the data form natural discrete groups, and is more widely tabled.