Sample size determination for comparing two therapies taking both the α (type I or false positive) error and the β (type II or false negative) error into consideration

The sample size required for determination of a given degree of difference between two therapies or modalities A and B being compared with each other, supposing that the sample sizes are equal, is given by Here C_α/2 is the upper α/2 percentile point of the standard normal distribution, C_1?β is the abscissa cutting off the proportion 1?β in its upper tail and β in its lower tail; p_A and p_B are the probabilities of favorable response to A and B; if p?_A and p?_B are the corresponding measured proportions, then $\text{p}$ = $\text{1}\text{2}$(p?_A + p?_B); and the Q's are 1 minus the p's. When it is desirable to use unequal size groups, the sample sizes n₁