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Conceived and designed the experiments: WCL. Performed the experiments: LHC. Analyzed the data: LHC WCL. Contributed reagents/materials/analysis tools: LHC WCL. Wrote the paper: LHC.

The authors have declared that no competing interests exist.

Randomization is a hallmark of clinical trials. If a trial entails very few subjects and has many prognostic factors (or many factor levels) to be balanced, minimization is a more efficient method to achieve balance than a simple randomization. We propose a novel minimization method, the ‘two-way minimization’. The method separately calculates the ‘imbalance in the total numbers of subjects’ and the ‘imbalance in the distributions of prognostic factors’. And then to allocate a subject, it chooses—by probability—to minimize either one of these two aspects of imbalances. As such, it is a method that is both treatment-adaptive and covariate-adaptive. We perform Monte-Carlo simulations to examine its statistical properties. The two-way minimization (with proper regression adjustment of the force-balanced prognostic factors) has the correct type I error rates. It also produces point estimates that are unbiased and variance estimates that are accurate. When there are important prognostic factors to be balanced in the study, the method achieves the highest power and the smallest variance among randomization methods that are resistant to selection bias. The allocation can be done in real time and the subsequent data analysis is straightforward. The two-way minimization is recommended to balance prognostic factors in small trials.

Random allocation of subjects is a hallmark of clinical trials. The simplest allocation method is the ‘simple randomization’ (complete randomization with equal allocation) where the recruited subjects are assigned to treatment group or control group based entirely on probabilities (say, using random numbers, or computer-generated random variates)

If a trial entails very few subjects and has many prognostic factors (or many factor levels) to be balanced, one may need to resort to a more sophisticated method of ‘minimization’

In this paper, we propose a novel minimization method, the ‘two-way minimization’. The method separately calculates the ‘imbalance in the total numbers of subjects’ and the ‘imbalance in the distributions of prognostic factors’. And then to allocate a subject, it chooses—by probability—to minimize either one of these two aspects of imbalances. We perform Monte-Carlo simulations to compare the performances of the two-way minimization with five existing randomization methods.

Consider an arbitrary point during the trial. Let

Suppose that a total of

At the beginning, we let the trial adopt a simple randomization scheme for allocating subjects. After

The proposed two-way minimization is an adaptive randomization procedure

(A1) minimizing

If

(A2) minimizing

Let

We let chance dictate which rule (A1 or A2) to use for allocating a new subject. To be precise, we define a parameter

Furthermore, the parameter

We assume that there are a total of

In the simulation, the treatment effects are set at

We consider two different sample sizes:

In each round of the simulation, we perform a multiple linear regression with the dependent variable being the trial response, and the independent variables, the

In addition to the power and the variance described above, predictability of treatment allocation is also an important criterion for evaluating a trial (especially when perfection in masking/concealment is difficult to achieve). If the allocation in a trial can somehow be predicted, the study will be prone to selection bias. In our simulation study, we derive two indices of predictability: Predictability-I: defined as the probability that the next subject is allocated to the group different from the one the previous subject allocated to; and Predictability-II: defined as the probability that the next subject is allocated to the group with fewer subjects already allocated to.

Sample Size | Number and Type of Prognostic Factors | Treatment Effect | ||

0.0 | 0.5 | 1.0 | ||

Bias | ||||

20 | Three binary prognostic factors | 0.0031 | 0.0013 | 0.0073 |

Six binary prognostic factors | −0.0042 | 0.0007 | −0.0012 | |

Three polytomous prognostic factors | −0.0061 | 0.0007 | 0.0016 | |

40 | Three binary prognostic factors | −0.0095 | −0.0021 | 0.0023 |

Six binary prognostic factors | 0.0052 | 0.0032 | −0.0012 | |

Three polytomous prognostic factors | −0.0039 | 0.0032 | −0.0028 | |

Variance of estimates/Average of estimated variances | ||||

20 | Three binary prognostic factors | 0.2117/0.2123 | 0.2136/0.2124 | 0.2112/0.2128 |

Six binary prognostic factors | 0.2306/0.2287 | 0.2253/0.2298 | 0.2289/0.2270 | |

Three polytomous prognostic factors | 0.2699/0.2741 | 0.2758/0.2711 | 0.2773/0.2720 | |

40 | Three binary prognostic factors | 0.1009/0.1016 | 0.1029/0.1017 | 0.1042/0.1018 |

Six binary prognostic factors | 0.1024/0.1034 | 0.1011/0.1031 | 0.1036/0.1035 | |

Three polytomous prognostic factors | 0.1109/0.1080 | 0.1081/0.1073 | 0.1059/0.1074 |

Sample Size | Number and Type of Prognostic Factors | Type I Error Rate | Power | |

Treatment Effect = 0.5 | Treatment Effect = 1.0 | |||

20 | Three binary prognostic factors | 0.0491 | 0.1738 | 0.5338 |

Six binary prognostic factors | 0.0511 | 0.1569 | 0.4946 | |

Three polytomous prognostic factors | 0.0511 | 0.1457 | 0.4154 | |

40 | Three binary prognostic factors | 0.0495 | 0.3339 | 0.8639 |

Six binary prognostic factors | 0.0483 | 0.3268 | 0.8537 | |

Three polytomous prognostic factors | 0.0507 | 0.3192 | 0.8421 |

The treatment effect is set at 1.0.

The treatment effect is set at 1.0.

As for the allocation predictability (panels, C, D, G, H, K and L, in

In this study, we focused on trials with small sample sizes. We showed that the proposed two-way minimization has the correct type I error rates. It also produces point estimates that are unbiased and variance estimates that are accurate. We compared the performances of the new method with several existing methods. Four methods can maintain stable performances as the effects of prognostic factors increase, namely: 1) the stratified randomization; 2) the biased coin minimization; 3) the deterministic minimization; and 4) the proposed two-way minimization. However, the first three methods have drawbacks: the stratified randomization and the biased coin minimization perform less than ideally when they are charged with balancing more prognostic factors/levels; the deterministic minimization is rather easy to predict and is therefore prone to selection bias. By comparison, the proposed two-way minimization is a better method for balancing prognostic factors in small trials.

For a large trial, it is generally held that even a simple randomization suffices. But there is no reason why one cannot force balance a large trial using the two-way minimization. In fact in doing so, he/she will be rewarded with even higher statistical performances as compared to leaving everything to chance. For example in a trial with

The two-way minimization may appear to be a fancy allocation procedure that is unduly complex. Yet, the entire algorithm of it can actually be incorporated into a simple spreadsheet program (available from the authors). Then, all that a trial researcher has to do is to simply feed in the prognostic-factor information for the subjects consecutively recruited in the trial. The allocation for them shall be produced one by one from the program fully automatically. The two-way minimization also calls for simple analysis despite its complex allocation scheme—a regression adjustment for the force-balanced prognostic factors is all that is needed. Further studies are warranted to extend the two-way minimization to deal with unbalanced designs where the treatment and the control groups are not to be of equal sample size due to ethical or logistical considerations. More work is also needed to study the performances of two-way minimization for other types of trial response, such as non-normal, binary, Poisson, and time-to-event data, etc, and whether the optimal value for the tuning parameter of 0.05 that was identified remains optimal for these other response types.

Recently, Perry et al.

In conclusion, the proposed two-way minimization has desirable statistical properties and is resistant to selection bias. The allocation can be done in real time and the subsequent data analysis is straightforward. The two-way minimization is recommended to balance prognostic factors in small trials.