Power and sample size calculation for 2×2 tables under multinomial sampling with random loss

Abstract

Multinomial sampling, in which the total number of sampled subjects is fixed, is probably one of the most commonly used samplig schemes in categorical data analysis. When we apply multinomial sampling to collect subjects who are subject to a random exclusion from our data analysis, the number of subjects falling into each comparison group is random and can be small with a positive probability. Thus, the application of the traditional statistics derived from large sample theory for testing equality between two independent proportions can sometimes be theoretically invalid. On the other hand, using fisher's exact test can always assure that the true type I error is less than or equal to a nominal α-level. Thus, we discuss here power and sample size calculation based on this exact test. For a desired power at a given α-level, we develop an exact sample size calculation procedure, that accounts for a random loss of sampled subjects, for testing equality between two independent proportions under multinomial sampling. Because the exact sample size calculation procedure requires intensive computations when the underlying required sample size is large, we also present an approximate sample size formula using large sample theory. On the basis of Monte Carlo simulation, we note that the power of using this approximate sample size formula generally agrees well with the desired power on the basis of the exact test. Finally, we propose a trial-and-error procedure using the approximate sample size as an initial estimate and Monte Carlo simulation to expedite the procedure for searching the minimum required sample size.