Comparing Multiple Proportions
Chi-squared is a useful and convenient statistic for testing hypotheses about multinomial distributions (see Section 4.2 at p. 103). 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 4.6.2, 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 fourfold 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.