An interactive FORTRAN IV program for calculating aspects of power with dichotomous data

Abstract

Dunlap (1981) described an interactive program that calculates power, necessary sample size, or detectable differences among means with data appropriate for analysis of variance. The present paper describes a similar interactive program (see Appendix) for helping plan experiments with dichotomous data when the usual method of analysis is chi square. Rather than solvingfor power with contingency tables of any arbitrary dimensionality, this program focuses on contingency tableswith two rows and two columns, where the null hypothesis can be expressed in terms of the difference between two population proportions. Although power calculations with higher order contingency tables are not particularly difficult (see Cohen, 1977), the present program was limited to 2 by 2 tables for several reasons. First, in higher order tables, specifying the anticipated pattern of frequencies from among the almost limitless number of possible patterns seemed a confusing task for the user. Why not target on two crucial groups with the simplest form of discrete data, dichotomous? Second, with the 2 by 2 table, exact power can be computed for the chisquare test, providing an index of accuracy for the various power-approximating algorithms. The question of accuracy is of no small concern for chi-square power calculations. Finally, three distinct power-approximating algorithms exist for the 2 by 2 problem; only one was found for higher order tables. The chi-square test applied to frequency data is an approximate rather than exact test and involves three distinct steps of approximation in its derivation (see Fry, 1938). Because of this fact, an investigator can have a perfectly accurate power computation for the chisquare distribution, and still have poor accuracy in using the distribution in approximating probabilities associated with binomial data. Also, the commonly recommended Yate's correction for continuity isactually not appropriate for 2 by 2 contingency tables of the type ordinarily encountered by researchers, in that its use results in serious underestimates of Type I error rates and, thus, unnecessary loss of power (see Camilli & Hopkins, 1978). For this reason, power calculations with the program presented here would not be appropriate for the continuity-corrected 2 by 2 chi-square procedure. The first step in writing this program was to compare the various algorithms in terms of accuracy. To accomplish this, it was necessary to compute exact power values. Exact Power Computations. To obtain exact power values for the 2 by 2 chi-square test, a program was written that generates all possible outcomes for an experiment with fixed column frequencies; row frequencies are free to vary as a function of the data sampled. This is the most commonly encountered 2 by 2 experimental design, labeled Case 2 by Pearson (1947). As an example, compare a group of 10 females and 10 males on a binomial variable such as cigarette smoking (smokers vs. nonsmokers). For the males and for the females, there are 11 possible frequency patterns, ranging from all smokers to no smokers. Thus, a total of 121 possible patterns of outcome exist for the study, all of which are easily enumerated by the computer. Given theoretical population proportions for the groups, one can compute the probability of each group frequency pattern, and also, the joint probability for the particular table as the product of the group probabilities via the binomial distribution, as follows: