Sample Size Determinations for r x c Comparative Trials

Sample size determination for r x c comparative trials is described based on existing tables of the non-central $\chi^2$ distribution and the expression for the non-centrality parameter of the appropriate $\chi^2$ statistic. In addition, the optimal allocation of sample elements among the treatment groups is explored through the evaluation of the expression for noncentrality under the comparative trial model. It is shown that a general rule for sample allocation, such as the "square root rule," does not apply uniformly. The non-central procedure is then compared to existing procedures for sample size determinations for 2 x 2 trials and it is shown that this procedure yields results in close agreement to those for the widely employed Halperin (twp-tailed) procedure. Finally, sample size determinations for other contingency table models are discussed.