Comparing sample size formulae for trials with unbalanced allocation using the logrank test

This paper compares the sample size formulae given by Schoenfeld, Freedman, Hsieh and Shuster for unbalanced designs. Freedman's formula predicts the highest power for the logrank test when the sample size ratio of the two groups equals the reciprocal of the hazard ratio. The other three formulae predict highest powers when sample sizes in the two groups are equal. Results of Monte Carlo simulations performed for the power of the logrank test with various sample size ratios show that the power curve of the logrank test is almost flat between a sample size ratio of one and a sample size ratio close to the reciprocal of the hazard ratio. An equal sample-size allocation may not maximize the power of the logrank test. Monte Carlo simulations also show that, under an exponential model, when the sample size ratio is toward the reciprocal of the hazard ratio, Freedman's formula predicts more accurate powers. Schoenfeld's formula, however, seems best for predicting powers with equal sample size.