Goodness-of-Fit Statistics for Discrete Multivariate Data
Larntz, Kinley, Journal of the American Statistical Association
Timothy R. C. Read and Noel A. C. Cressie. New York: Springer-Verlag, 1988. xi + 211 pp. $44.
Read and Cressie have written an easy-to-read, informative book that is required reading for anyone analyzing categorical data. They build on their excellent series of papers (Cressie and Read 1984; Read 1984a, b), incorporating their results into a complete picture of recent and past literature on what might be called the "chi-squared problem." They offer unequivocal advice to the practitioner and unsolved problems for the researcher.
The chi-squared problem concerns the applicability of the chi-squared approximation for the statistic
[X.sup.2] = [SIGMA] [(observed - expected).sup.2]/expected. (1)
This arises in counted data problems including contingency tables, popular log-linear models for multidimensional contingency tables, logistic regression, and Poisson regression (Bishop, Fienberg, and Holland 1975; Fienberg 1980). [X.sup.2] is often called the Pearson chi-squared statistic. Alternative statistics include the log-likelihood ratio chi-squared statistic
[G.sup.2] = 2[SIGMA] observed log (observed/expected), (2)
and many variants thereof. Fisher (1970), Cramer (1946), and Cochran (1952) are just a few who have offered advice on this subject.
A major breakthrough in the study of the chi-squared problem was the work of Cressie and Read (1984) in unifying the alternative statistics into a power family [a la Box and Cox (1964)]. The power-divergence family of goodness-of-fit statistics, indexed by a power parameter [lambda], is given by 2/[lambda]([lambda] + 1) [SIGMA] observed [(observed/expected).sup.[lambda]] - 1]. The Pearson chi-squared statistic (1) corresponds to [lambda] = 1. The log-likelihood ratio chi-squared statistic (2) corresponds to [lambda] = 0. Read and Cressie have found that [lambda] = 2/3 yields a goodness-of-fit statistic with good small-sample properties for null hypothesis situations (i.e., it is well approximated by the nominal chi-squared distribution) and adequate power against interesting alternatives.
One beauty of the power-divergence unification is that results for one statistic in the family can be easily adapted to others in the family. In addition, comparisons among members of the family are more easily understood as functions of [lambda]. The power comparison of "bump" and "dip" alternatives to a multinomial null hypothesis illustrates this perfectly. Koehler and Larntz (1980), comparing [X.sup.2] and [G.sup.2], note that the Pearson statistic is more powerful for bump alternatives and the log-likelihood ratio statistic is more powerful for dip alternatives. Read and Cressie take this further, showing that the power-divergence statistic with [lambda] large gives high power against bump alternatives, whereas for dip alternatives taking [lambda] as small as possible results in highest power. …