Algorithm for balancing both continuous and categorical covariates in randomized controlled trials

Minimization as proposed by Pocock and Simon for balancing categorical covariates in clinical trials with individual-level, consecutive randomization has been increasingly used. An extension of the method exists that uses the symmetric Kullback-Leibler divergence index to balance both categorical and continuous covariates, albeit for two-arm randomized controlled trials only. To date, the algorithm has not been made widely available to researchers via publicly accessible software. We adapted Endo et al.'s algorithm to randomized trials with two or more arms. In addition, our algorithm incorporates Efron's biased coin method to decrease the predictability of assignment even when a predefined threshold of difference in the numbers of subjects between treatment arms is reached, whereas Endo et al.'s algorithm assigns the next subject to the treatment of smaller size with certainty. We developed code in the free statistical software R to make the algorithm readily available to trialists. While there are no definitive answers regarding the optimal choices for certain statistical parameters that must be defined prior to algorithm application (P(k), D(n), and p_D(n)), we provide guidance on these. We conducted simulations with actual data from a three-arm randomized trial to compare the modified algorithm and R code to another published biased coin minimization method that can accommodate continuous and categorical covariates in multi-arm trials. The three-arm trial used three categorical covariates (sex, race/ethnicity, and online personal health record access) and four continuous covariates (age, fasting blood glucose, body mass index, and waist circumference). All of the continuous and categorical covariates were well balanced, and the results were comparable to the comparison method.