Frequency Matched Study with Equal Numbers of Cases and Controls and Strata Specific Exposures
Ref: Gail M. The determination of sample sizes for trials involving several independent 2 x 2 tables. Journal of Chronic Diseases 1973; 26: 669-673.
Medical experiments commonly consist of k independent 2 x 2 trials. For example suppose subjects are naturally stratified into k risk groups (stages) and then within each risk group are randomly assigned to treatment A or B and the proportions responding to the respective treatments are observed. Such data are used to draw inferences on the relative odds of success:
where p1j and p0j are the proportion of successes for the two treatments within stratum j.
This program computes sample sizes required to attain a specified power and size when planning k independent 2 x 2 trials. The calculations are done assuming that the subjects are divided evenly between the two "treatments" within each stratum and that one of three functions of the p's is constant across all strata.
Computations are done assuming that the alternative hypothesis delta is either a constant difference, odds ratio or relative risk across all strata. The variance of delta is computed using a Taylor series approximation and the asymptotic normality properties of maximum likelihood estimates. The approximations are reasonably accurate for p's between 0.1 and 0.9 and sample sizes larger than 30.
When most of the p0j are less than .1 or the weighted average of the p0j is less than .1, the alternate formula suggested by Gail can be used. The user can optionally select to use either of the two formulae. Whenever some of the p1j are greater than .9 the regular formula will be used, but the result may be overly conservative.
By redefining 'success' to be the development of a disease this program can be used to design studies on the relative risk of developing the disease in each of two populations. Although j has been considered a single stratifying variable, j can in general index strata based on several cross-classifying variables and the computations are appropriate whenever "delta" is constant over all the strata.
The program requires the user to specify the number of strata and the frequency of subjects in each strata, fj; the fj must add to unity. The probability of "success" in each strata must also be provided; this must be the smaller of the two p's.
Then, given any three of the following:
the program can compute the fourth.