Quantifying the exact allocation ratios in randomized clinical trials with biased coin randomization

The biased coin randomization approach is frequently adopted in randomized clinical trials to control the balance in overall treatment assignments. In this article, an algorithm is developed to theoretically determine the exact allocation ratios for all the patients that will be enrolled in a randomized clinical trial with biased coin randomization based on the order in which they are randomized. Our results show that the exact allocation ratios can significantly deviate from the ratio as specified for the trial, which poses challenges to the enrollment of the trial and to the interpretation of the results. In order to maintain a constant allocation ratio throughout the trial, a modification of the widely adopted permuted block randomization is proposed, which is shown to achieve better balance for not only the overall treatment assignments but also the baseline stratification variables that are desired to be balanced at the end of the trial.