Scott Hamilton, PhD

In large multi-center clinical trials, stratifying the randomization by center leads to overall treatment group balance with high efficiency. However, rare diseases can have a prevalence lower than one in 2,000 making large trials impossible to enroll. There are 6,000 to 8,000 rare diseases that affect 30 million people in the EU.

Studies in rare diseases require a smaller sample size and many investigational sites to recruit enough patients to meet recruitment goals. Thus, studies will be necessarily small, perhaps in the range of 50 to 100 patients. Stratifying the randomization by site in small to medium studies significantly increases the probability of treatment group imbalances. In this article, a novel adaptive-block method of randomization to maximize the efficiency of overall treatment group balance while maintaining balance at investigational centers in small- to medium-sized studies is proposed.

The most common method for proactively balancing the treatment groups in a clinical trial among the known and unknown prognostic factors of the primary outcome is to stratify the randomization.

 In Hamilton’s experience with the randomization of approximately 2,500 clinical trials, stratification was used almost 100% of the time. Furthermore, the method most often used to control the balance among the unknown prognostic factors was to stratify by site (aka investigator or center). This practice is supported by the regulatory guidance document ICH E9 which reads “It is advisable to have a separate random scheme for each center, i.e. to stratify by center or to allocate several whole blocks to each center.”

 European regulatory guidance documents mention that “most multicenter trials are stratified by center (or investigator) for practical reasons or because center (or investigator) is expected to be confounded with other known or unknown prognostic factors.”

 It is the author’s experience involved in many studies where the FDA has explicitly requested stratification by investigator to balance the known and unknown prognostic factors among the treatment groups. For these reasons, the lion’s share of clinical studies are stratified by site/investigator. Indeed, a survey of medical studies in leading journals found that over 70% were stratified by site/investigator.

Since most randomized studies utilize list-based randomization, permuted block randomization is by far the most common method to implement site stratification. With permuted blocked randomization, stratifying by investigator works well in large multicenter trials, i.e. balance among treatment groups is achieved within site and in the overall total. However, it can lead to over-stratification in small to medium sized studies. That is, the chance of overall total study imbalance can be greater with the stratification than it would otherwise be with no stratification. Therneau showed that when the number of distinct factor level combinations exceeds n/k the probability of overall imbalance is greater than that with no stratification, where k indicates the number of treatment groups. A real-life example is a study where two treatments are compared across 80 subjects and they are enrolled across 20 centers. If permuted blocked randomization with a block size of four and stratified by site is used, our simulations show that there is a 0.46 probability of a significant imbalance exceeding 10%; essentially flipping a coin that the overall treatment group imbalance will be worse than 36 in one group and 44 in the other. Thus, stratifying by site in small- to medium-sized studies can cause the overall treatment group totals to end up out of balance, especially in rare diseases when many sites are needed for recruitment goals.

Given the need to meet recruitment timelines and the large number of studies in rare diseases, multi-center trials will be utilized for small to medium sized studies, and statisticians will design randomization plans with site stratification.

 Therefore, we need a stratification strategy that will accommodate a large number of sites without overly elevating the risk of overall treatment group imbalance. Certainly, dynamic randomization is a recommended method to effectively balance treatment groups among many prognostic factors and levels.

 However, experience indicates most sponsors are reluctant to utilize dynamic randomization for fear of complexity and perceived resistance from regulatory authorities. Other IRT providers have encountered similar reluctance.

 Given its wide acceptance, it would be optimal to adapt permuted block randomization with a constraint that would accommodate a large number of prognostic factors/levels and maintain the overall balance.

The ideas presented grow out of a collaboration between Hamilton and Alster. We worked on the randomization plan for a study where 90 subjects were to enroll at 18 sites and randomized to one of two treatments. The risk of overall treatment group imbalance was fairly high, especially at the interim analysis scheduled for enrollment of 60 subjects. The method described herein is a proposed solution. The research in this paper is intended to help statisticians and programmers designing randomizations when faced with a highly stratified randomization. To that end, we performed an experiment to provide results that show the properties of this adaptive block method compared to standard permuted block randomization. Hamilton designed the experiment that is presented in the next section and Alster programmed the simulations in Python version 3.7.

The essence of the proposed method is that the blocks are not constructed in advance. They are built as the randomization progresses with the assignment to treatment of each subject. Let’s look again at our example above where there are two treatments (A, B) randomized in a 2:2 ratio. There is no pre-defined randomization list. Therefore, a centralized IRT, such as a web-based randomization system, would be employed. When study site personnel access the IRT to randomize a subject, the system recognizes the site and registers the overall treatment group balance (e.g. Treatment group A total vs. Treatment group B total). If there are more subjects in one group then the system looks at the current block at that site that is randomizing to see whether the other group can be assigned within the constraints of the block position. For example, if there are more subjects in group A than B in the overall total and the subject will be randomized in the first position of the current block it would randomize the subject to group B. However, in our example the constraint of the block position is that standard blocks of size four with a 2:2 ratio have to be created. Thus, if the block position was 3 or 4 and 2 previous slots in the block had been filled by group B, the system would randomize the subject to A to maintain the correct blocked structure of the developing randomization list.

Algorithmically, the blocks are constructed in real-time as the subjects randomize in the following manner. Note that each block referenced in the algorithm is within a stratum (site).

To assess the effectiveness of this adaptive-block randomization method Lance and I executed a simulation study where we used the example of 80 subjects with a randomization plan that stratifies by 20 sites. Based on our experience with randomization plans from approximately 2,500 studies, I chose to investigate the treatment group/randomization ratios encountered in over 85% of the studies where Signant Health has managed the randomization. They are listed in Table 1.

Two important properties of randomization are predictability and efficiency. The term “predictability”, sometimes called selection bias, refers to the probability study personnel could predict the next assigned treatment. For a more complete description of predictability please see Shum, Hamilton, and Lo

, or for a more detailed explanation see Blackwell and Hodges.

 Since the proposed method is an enhancement to regular permuted blocked randomization, the predictability remains the same. Thus, we focused our experiment on efficiency of treatment group balance. I have already described the adaptive-block specifications for scenario one in the shaded box above. The specifications for scenarios two to five are detailed in the Appendix. To examine the efficiency of our proposed method, Alster executed 100,000 Monte Carlo samples to randomize through the two methods: 1) standard site-stratified permuted block randomization and 2) the proposed adaptive-block randomization. Each sample consisted of samples of 78 or 80 subjects with random allocation to 20 sites. The measure of imbalance consisted of the largest treatment group ratio among all the treatment groups normalized by the randomization ratio and converted to a percentage. With two treatment groups, the imbalance measure is simply the normalized ratio. With three treatment groups we choose the largest of the three ratios. We present results for the overall balance and average balance within sites. The software used to program the simulations was Python version 3.7.

Recall that standard permuted block randomization stratified on site will directly balance treatment groups within sites, and that overall balance is expected as a result of the site balancing. That is, when treatment groups are balanced at all the sites there should be overall balance. However, as shown above the overall balance can break down in small to medium sized studies. Thus, we expected the adaptive-block method to perform better at maintaining overall balance. The important question was whether the adaptive-block method would maintain balance within sites as well. The results in Table 2 report the percent of trials that had perfect overall balance, imbalance by one to three subjects, and imbalance of 10% or greater. Most statisticians would agree that a 10% or greater imbalance would be a failure of the randomization. We also report average imbalance within the sites. Table 2 displays these results for each of the five scenarios studied. The results show that the adaptive-block method is far more efficient and was always much better than standard permuted block randomization at maintaining overall balance. For instance, the adaptive-block method maintained perfect balance in 55% to 92% of the trials vs. 3% to 23% with standard permuted block randomization, respectively. Perhaps more importantly, the adaptive-block method essentially guaranteed no imbalances of 10% or greater (only 1:1:1 showed greater than 10% imbalance 4% of the time), whereas standard permuted block randomization produced significant imbalances more than 50% of the time in four of the five scenarios studied (range of 8% to 64%). Reassuringly, the adaptive-block method-maintained balance within sites as well and sometimes more efficiently than standard permuted block randomization.

Additionally, we examined the impact of the adaptive-block method on the permutation distribution. As discussed in Hamilton and Shum, permutations that have strings of repeated treatments produce more treatment group imbalances. Thus, we would expect that the adaptive-block method would create fewer of these permutations. Indeed, the results showed that in the adaptive-block randomizations the blocks with strings of repeated treatments were created approximately a third of time less than the uniform distribution created by standard permuted block randomization. We explored the impact on overall balance using permuted block randomization and reducing the blocks with strings of repeated treatments by a third, however the balancing efficiency did not increase. As expected, the position of the permutations in relation to the enrollment distribution was the key to the efficiency of the adaptive-block method.

In this paper, we propose a novel adaptive-block randomization method that greatly increases the efficiency of overall treatment group balance without sacrificing efficiency within site. Our method is most useful in small to medium sized studies and in rare diseases where many sites are needed to meet recruitment goals. Blazynski

 reported that approximately 1,350 industry sponsored phase II trials were completed in 2018. Since most Phase II studies enroll 100 or fewer subjects from several investigational sites, this method has the potential for high impact in phase II studies as well. This paper shows that for some very common designs standard permuted block randomization can result in 10% or greater imbalances more than 50% of the time. This is clearly unacceptable and must be remedied. Fortunately, we have shown that the fairly simple adaptive-block method will maintain excellent overall and within site balance in these small to medium size studies.

Stratifying by site is the most common cause for a large number of stratification factor levels. However, even with a reasonable number of sites when additional prognostic factors are added to the stratification, the number of levels gets large. Fortunately, the proposed method is agnostic to what the stratification factors represent. It will be highly efficient at balancing treatment groups with a large number of stratification levels regardless of what they are.

, is the Principal Biostatistician for Signant Health. 

is the Manager of SAS programming for Fibrogen.

Assman SF, Pocock SJ, Enos LE, Kasten LE. (2000) Sub-group analysis and other (mis)uses of baseline data in clinical trials. Lancet. 355:1064–1069.

Blackwell D, Hodges JL. (1957). Permutation for the Control of Selection Bias. Annals of Mathematical Statistics, 28: 449-460.

Blazynski, C, Runkel, L. (2019) 2018 Completed Trials: State of Industry Sponsored Clinical Development. https://pharmaintelligence.informa.com/~/media/informa-shop-window/pharma/2019/files/whitepapers/jn2403-clinical-trials-whitepaper-update_lr.pdf

Buyse, M., McEntegart, D.J, “Achieving Balance in Clinical Trials: An Unbalanced View from EU Regulators, “Applied Clinical Trials, May 2004.

CPMP, C. f. (2003). Points to Consider on Adjustment for Baseline Covariates. http://www.emea.eu.int/pdfs/human/wep/286399en.pdf.

ICH-E9. (1998). International Conference on Harmonization, E9 document, Guidance on Statistical Principles for Clinical Trials. Federal Register, 63: 49583-49598.

Hamilton S., Shum C. (2016) Building Blocks for Better Balance, Applied Clinical Trials https://www.appliedclinicaltrialsonline.com/view/building-blocks-better-balance

McEntegart, D. (2008). Blocked Randomization. In R. S. D'Agostino, Wiley Encyclopedia of Clinical Trials (p. DOI:10.1002/9780471462422.eoct301). Hoboken: John Wiley & Sons, Inc.

McEntegart, D.J. The Pursuit of Balance Using Stratified and Dynamic Randomization Techniques: An Overview. Ther Innov Regul Sci 37, 293–308 (2003). https://doi.org/10.1177/009286150303700305

N.W. Scott, G. M. (2002). The method of minimization for allocation to clinical trials: a review. Controlled Clinical Trials, 23, 662-674.

Rosemberger, W. F. (2015). Randomization in Small Clinical Trials. https://www.newton.ac.uk/files/seminar/20150710110011451-409338.pdf.

Shum C., Hamilton, SA., and Lo, KH., (2016). A Comparison of Techniques for Creating Permuted Blocked Randomization Lists. Applied Clinical Trials, 23, 662-674.

Adaptive Block Treatment Assignment Specifications

2 treatments (A, B) ratio 1:1 and a blocksize of 4.

1st and 2nd Position: The algorithm looks at the overall balance and registers which treatment has the lowest total and assigns the treatment with the lowest total. In the case of a tie in the overall total and we're filling the first position, assign A or B randomly. In the case of a tie in the overall total and we're filling the second position, assign a different treatment than the 1st position. 

3rd Position: If positions 1 and 2 are the same treatment, assign the other treatment. If the first 2 positions are different, the algorithm looks at the overall balance and registers which treatment has the lowest total and assigns the treatment with the lowest total. If the first 2 positions are different and there's a tie in the overall total assign A or B randomly. 

4th Position: Assign the treatment to complete the block.

3 treatments (A, B, C) ratio 1:1:1 and a blocksize of 3.

1st Position: The algorithm looks at the overall balance and registers which treatment has the lowest total and assigns the treatment with the lowest total. In the case of a 3-way tie in the overall total assign A, B, or C randomly. If there is a 2-way tie for lowest total, randomly assign the treatment from one of the two treatments tied for the lowest total.

2nd Position: The algorithm looks at the overall balance among all treatment groups that haven’t been assigned to the first position in this block and registers which treatment has the lowest total and assigns the treatment with the lowest total. In the case of a 2-way tie for lowest total, randomly assign the treatment from one of the two treatments tied for the lowest total.

3rd Position: Assign the treatment to complete the block

3 treatments (A, B, C) ratio 1:1:1 and a blocksize of 6.

2nd Position: The algorithm looks at the overall balance and registers which treatment has the lowest total and assigns the treatment with the lowest total. In the case of a 3-way tie in the overall total randomly assign to one of the two treatments that weren’t assigned in position 1. If there is a 2-way tie for lowest total, randomly assign the treatment from one of the two treatments tied for the lowest total.

3rd Position: The algorithm looks at the overall balance among all treatment groups that haven’t been assigned 2 positions in this block and registers which treatment has the lowest total and assigns the treatment with the lowest total. In the case of a 3-way tie in the overall total assign to the treatment that wasn’t assigned in positions 1 and 2, unless both treatments in positions 1 and 2 are the same in which case randomly assign to one of the other two treatments. If there is a 2-way tie for lowest total, randomly assign the treatment from one of the two treatments tied for the lowest total.

4th Position: The algorithm looks at the overall balance among all treatment groups that haven’t been assigned 2 positions in this block and registers which treatment has the lowest total and assigns the treatment with the lowest total. If all 3 treatments in positions 1, 2, and 3 are distinct and there’s a 3-way tie in the overall total randomly assign to any of the 3 treatments. If all 3 treatments in positions 1, 2, and 3 are distinct and there is a 2-way tie for lowest total randomly assign to one of the 2 treatments tied for lowest total. If 2 of the same treatment have been assigned in positions 1, 2, or 3 and there’s a tie for lowest total among the remaining treatments, assign the treatment that wasn’t assigned in positions 1, 2 or 3.

5th Position: If 2 of the 3 treatments have been assigned twice in this block, assign the other treatment. Otherwise, The algorithm looks at the overall balance among the 2 treatment groups that haven’t been assigned 2 positions in this block and registers which treatment has the lowest total and assigns the treatment with the lowest total. If there’s a tie among the 2 remaining treatments randomly assign to one of those treatments.

6th Position: Assign the treatment to complete the block

2 treatments and a blocksize of 3, 2:1 A:B ratio.

1st Position: The algorithm looks at the overall balance (normalized by the ratio) and registers which treatment has the lowest total and assigns the treatment with the lowest total. In the case of a 2-way tie in the overall total assign A or B randomly.

2nd Position: If the assignment in the 1stposition was B, assign A. Otherwise, the algorithm looks at the overall balance (normalized by the ratio) and registers which treatment has the lowest total and assigns the treatment with the lowest total. In the case of a 2-way tie in the overall total assign A or B randomly.

3rd Position: Assign the treatment to complete the block.

2 treatments and a blocksize of 6, 2:1 A:B ratio.

1st and 2nd Positions: The algorithm looks at the overall balance (normalized by the ratio) and registers which treatment has the lowest total and assigns the treatment with the lowest total. In the case of a 2-way tie in the overall total assign A or B randomly.

3rd, 4th, and 5th Positions: If B has been assigned twice, assign A. Otherwise, the algorithm looks at the overall balance (normalized by the ratio) and registers which treatment has the lowest total and assigns the treatment with the lowest total. In the case of a 2-way tie in the overall total assign A or B randomly.

6th Position: Assign the treatment to complete the block.

Articles

A novel adaptive-block method of randomization to maximize the efficiency of overall treatment group balance, while maintaining balance at investigational centers in smaller sized studies, is proposed.

An Adaptive-Block Randomization Method when Stratifying by Investigator in Small-to Medium-Sized Studies

	Most statisticians would agree that dynamic randomization results in superior treatment group balancing over list-based stratified permuted blocked randomization, thereby increasing the precision in a randomized clinical trial. There have been several studies and review articles that provide ample evidence in favor of dynamic randomization.

	However, in our routine interactions with statisticians and clinical trial managers, we continue to encounter the perception that dynamic randomization carries extra risk and complexity. We argue that from the implementation side of dynamic randomization, today’s modern computing platforms, thorough validation techniques, and robust randomization systems alleviate technical risk. From a regulatory perspective, dynamic randomization has been well received at the FDA for many years. We have worked on clinical trials where the FDA has explicitly requested dynamic randomization over list-based permuted blocked randomization. Any perceived regulatory risk when using dynamic randomization may be leftover from attitudes at European regulatory authorities, for example, the Committee on Proprietary Medicinal Products (CPMP). This stems from a European guidance document issued in 2003 that carried the opinion that dynamic randomization was “controversial.”

 However, because of excellent responses to that guidance document,

 and increased familiarity from more frequent use of dynamic randomization over the past 12 years, the perception of this risk has diminished. Thus, the hesitancy to utilize the benefits of dynamic randomization could arise from the perceived complexity. This may be due to a lack of familiarity with the process for choosing the randomization parameters and how to implement the algorithm for randomization in clinical trials.

	The motivation for this article was to dissipate the perceptions of complexity by elucidating the details of our process for choosing the randomization parameters. We also provide guidance for how we implement dynamic randomization for clinical trials in randomization and trial supply management (RTSM) systems.

	We begin by comparing the parameters for stratified permuted block randomization with those required for dynamic randomization. When designing a stratified permuted block randomization, the statistician must determine the stratification factors, their levels, and the block size. To maximize statistical power, the statistician desires to preserve the treatment group ratio(s) among all groups of subjects defined by demographic and disease characteristics that affect the outcome. The statistician must obtain the information about these groups from clinicians so that he or she can define the stratification factors and levels. The person also must obtain a relative hierarchy of importance for each factor for two reasons. Firstly, most methods of dynamic randomization allow for weighting of the stratification factors so that imbalances among factors with greater weight get brought back into balance more quickly. Some algorithms explicitly balance in hierarchical order among the factors.

 Secondly, in permuted block randomization there are practical limits to how many stratification factors can effectively be used before the balancing performance breaks down.

 Given the relative importance of each factor, the statistician may recommend dropping the ones that aren’t as important if there are too many. In most cases, the important factors are correlated and balancing on one factor will achieve balance on other correlated factors.

	In permuted block randomization, choosing the block size is a tradeâoff between the ability to maintain the blind and the efficiency of balancing the treatment groups. The larger the block size, the less chance there is that a pattern can be detected in the treatment group allocation.

 However, the larger the block size, the greater the probability of treatment group imbalances over the entire study as well as within stratification factor levels. In fact, the probability of imbalances is proportional to the number of strata and the block size.

 The convention is to use 2 times the sum of the treatment group ratio. For instance, in a two-treatment group study with a 1:1 ratio, the most common block size chosen is 4. In a two-treatment group study with a 2:1 ratio, the most common block size chosen is 6. Because of the strong inclination among statisticians to follow this convention, it is extremely rare to base the choice of the block size on the results of quantifying the probability of treatment group imbalance for different block sizes.

	When designing a dynamic randomization algorithm, all the same criteria apply for choosing the stratification factors as in permuted blocked randomization. However, there is much greater flexibility in the number of factors that one can effectively include in the dynamic randomization. Unlike permuted block randomization, which balances the 

â1)st factor level, …, dynamic randomization directly balances on the marginal distribution of each factor. This makes it possible to add more factors to the algorithm without greatly reducing the ability to balance among the other factors.

 When planning a dynamic randomization, there is no block size to choose from. One must choose whether the allocation will be deterministic (minimization) or based on a biasedâcoin toss. However, because of the opinion in the ICHâ9 guidance, it is generally advised to use a biasedâcoin. Then the probability of the biased coin has to be chosen, e.g., for two treatment groups: 75/25, 80/20, 85/15, etc. Lastly, if weights will be applied to the stratification factors, the sizes of the weights need to be chosen. The next few paragraphs describe our process for choosing the biasedâcoin probabilities and the weights.

	Similar to choosing the block size in permuted block randomization, choosing the biasedâcoin probability is a tradeâoff between efficiently balancing the treatment groups and the level of randomness in the treatment allocation. The extremes in the biasedâcoin spectrum are fair coin on one end to deterministic on the other. To make the choice, we simulate all possible enrollment patterns into the study under a range of biasedâcoin probabilities and compare the balancing characteristics for each choice. For even treatment group allocation ratios, e.g. 1:1, 1:1:1, 1:1:1:1, we look at the distribution of absolute treatment group differences overall and within each factor level. For 2:1, 3:1, 4:1, etc., we look at treatment group ratios. After calculating the treatment group differences/ratios over all possible enrollment patterns, we may summarize them using a plot. An example is given in Figure 1.

	The plot is meant to illustrate the imbalance measures for the overall study averaged over all the simulations for various biasedâcoin probabilities. In this example, it shows a clear monotonic decrease in the imbalance between the treatment groups. However, one can see the greatest decrease in imbalance when going from a biasedâcoin probability of 0.8 to 0.85. A statistician could use this as a rationale for choosing a biasedâcoin probability of 0.85 since there is a marked decrease in imbalance from 0.8, but no worthwhile decrease at 0.9 or 0.95. The same would be constructed and examined within each stratum.

	The advantage to using MonteâCarlo simulations of enrollment is that we can realistically model all possible randomization order patterns within each stratum and overall. We customize the enrollment parameters to fit the study. For instance, one of the stratification factors may be electrocorticogram (ECOG) status 0â1, vs. 2â3, where 80% of the subjects enrolled will have an ECOG of 2â3. We replicate that breakdown in our MonteâCarlo simulations so that the imbalance statistics calculated from the simulation data accurately reflect what could happen in the proposed trial. When site is a stratification factor, the sponsor usually has a good idea of the number of sites, but not how many subjects each site will enroll. We have a large database of studies with many different numbers of sites and their patterns of enrollment. We use that database, the sample size, and the number of sites to create a realistic site enrollment pattern for use in the MonteâCarlo simulations.

	As previously mentioned, stratification factors typically have a hierarchical level of importance. For instance, it may be more important to maintain treatment group balance within ECOG score levels than gender. In this case, the statistician can specify weights that will correct ECOG stratum imbalances more quickly than imbalances within gender level. The tools for choosing the weights also utilize the simulations previously discussed. Essentially, we try out a few sets of weights and examine the imbalance measures. Figure 2 illustrates how this is done. The example is a 100-subject study with three 2âlevel stratification factors with a 2:1 active:placebo ratio. The plot on the left in Figure 2 shows a dot for each simulation showing the number of randomized subjects within one of the ECOG strata. The Xâaxis indicates the number randomized to Active and the Yâaxis to Placebo. With near perfect balancing, we would see a tight cloud of points around a line with a slope of 0.5.

	The simulations show us that with the parameters of this randomization plan, we are obtaining a slope close to 0.5, and a measure of dispersion of 0.54. The medians show that on average there is an almost perfect 2:1 ratio. The plot on the right in Figure 2 shows the results of applying a weight of 10 to this factor. By adding the weight, the dispersion is reduced by 11.4% to approximately 0.48. The tightening of the ratios around 0.5 is evidenced visually as well. This may provide sufficient evidence that by weighting the ECOG stratification factor, we can effectively improve the consistency of the desired 2:1 ratio.

	Of course, the statistician would want to look at the same plots of the other factors, such as gender, to determine the impact of adding the weight to the ECOG factor on the other factors. As an illustration, this is shown in Figure 3 for male gender. The increase in dispersion within the treatment group ratios within males after applying a weight to ECOG is approximately 520%. The increase in dispersion in treatment group balance among the males by adding the weight to ECOG may outweigh the reduction in dispersion within the ECOG strata. However, the relative importance of achieving balance within gender may be far lower than balance within ECOG score, in which case this scenario may be acceptable. The answer is always driven by the clinical science of the therapeutic area. The statistician has to tune these balancing parameters with the science to maximize the precision of the study.

	In this article, we have provided an overview of our quantifiable techniques for choosing the parameters in dynamic randomization. Prior to the start of a study, a statistician must make plenty of decisions regarding power, analysis methods, handling missing data, etc. Adding more certainty to the randomization plan is of great help to the statistician. As with most statistical techniques, we find that when a statistician has experienced the quantitative process for choosing the randomization parameters for the first time, the mystery greatly dissipates and confidence grows. A statistician is most confident when he or she has quantified evidence from data to support their decisions. Fortunately, we have a sound process, a wealth of experience, and the data from thousands of randomized studies for statisticians to develop dynamic randomizations that will increase the precision of their studies.

	At Bracket, our mission is to bring quality, resourcefulness, and dependability to the medicine development process. Improvements in the ability to implement sound and robust dynamic randomization systems provide us an easy landscape to utilize modern simulation techniques and graphical displays of quantitative data for making decisions about the biasedâcoin probabilities and weights. Future discussions will explore more subtle and flexible methods to biasedâcoin and weighting in dynamic randomization.

, PhD, is Principal Biostatistician at Bracket and Associate Professor at Stanford University

	1. McEntegart, D.J. “The Pursuit of Balance Using Stratified and Dynamic Randomization Techniques: An Overview,” 

	2. Buyse, M., McEntegart, D.J, “Achieving Balance in Clinical Trials: An Unbalanced View from EU Regulators, “ 

	3. Therneau TM. “How many stratification factors are “too many” to use in a randomization plan?” 

	4. Kalish LA, Begg CB. “Treatment allocation methods in clinical trials: A review.” 

	5. CPMP, C. f. (2003). “Points to Consider on Adjustment for Baseline Covariates.” 

http://www.ema.europa.eu/docs/en_GB/document_library/Scientific_guideline/2009/09/WC500003639.pdf

	6. Hodges, D. B. (1957). “Design for the Control of Selection Bias.” 

	7. McEntegart, D. (2008). “Blocked Randomization.” In R. S. D'Agostino, 

(p. DOI:10.1002/9780471462422.eoct301). Hoboken: John Wiley & Sons, Inc.

	8. Ledford D, Busse W, Trzaskoma B, Omachi TA, Rosén K, Chipps BE, Luskin AT, Solari PG.J. “A Randomized Multicenter Study Evaluating Xolair Persistence of Response After LongâTerm Therapy." 

. 2016 Nov 5. pii: S0091â6749(16)31274âX. doi: 10.1016/j.jaci.2016.08.054

An overview of quantifiable techniques when choosing the parameters in dynamic randomization for clinical trial enrollment.

Selecting the Parameters of a Dynamic Randomization

	In a recent visit to a major biotechnology company, I asked a senior statistician how often she’s required to think about randomization design. She chuckled, “about once every two years – not enough to think much about what I’m doing”. She followed with the comment that she mostly relies on the IXRS/RTSM statistician to recommend the randomization parameters for their studies. Having worked with clinical teams on hundreds of randomization designs, my sense is that her statements would represent the experience and attitude of most study statisticians. 

	These sentiments would be especially true for Dynamic Randomization where complex algorithms determine the probabilities of treatment group assignment at each randomization. Static list based randomization would seem to be fairly straightforward. However, there’s more to think about with list based randomization than most would realize. By far, the primary considerations to the statistician are the block size and the stratification. The choice of the block size is an attempt to minimize predictability and at the same time maximize the treatment group balance. There isn’t a large amount of flexibility in choice of the block size. The ICH-E9 guideline specifically recommends that the block size be at least 2

 is the number of subjects per treatment in the allocation ratio. For instance, with 2 treatments and a 1:1 ratio the minimum block size would be 4. For a 2 treatment 3:1 ratio, the minimum block size would be 8. A simple way to remember the minimum block size is to sum the treatment group ratio and multiply it by 2.

	Choice of block size gets tricky when stratification is imposed on the randomization. The more levels of stratification the greater the risk of imbalance in the treatment groups. This is due to the number of incomplete blocks at the end of the randomization. For example, let’s examine the risk of imbalance in a 2 treatment study with a 1:1 ratio and a block size of 4 with a sample size of 100 and the randomization is stratified for 10 sites. In the extreme, if the randomization ends after the first two slots are filled in the AABB permutation within each of the 10 sites, then the final treatment group totals would be 55 in treatment group A and 45 in treatment group B. Although the loss in study power resulting from this imbalance is minimal, there are other considerations where this anomaly could be more serious. For instance, if an interim analysis occurs after 25% of the enrollment completes and the treatment group totals were off by 10 it would handicap the efforts of the data monitoring committee to interpret the results. A 10% imbalance in the treatment group totals could also bring up trial credibility questions to journal or regulatory reviewers. Considering that the integrity of clinical research is at risk on many fronts, methods to minimize treatment group imbalance are important tools for reducing negative perceptions about the integrity of the study. 

	A straightforward remedy to the situation presented above is to NOT over-stratify the randomization. Or, when stratification on several factors is warranted, we recommend use of Dynamic Randomization which has been shown to have optimal treatment group balancing results. However, in my experience 80% of trials in our database of over 2000 randomized studies used blocked randomization. Even with straightforward site stratification the imbalances can be problematic, as illustrated in the example above. Thus, it’s worth thinking more creatively about blocked randomization and considering some novel approaches.

	We decided to look more closely at the set of permutations for a given treatment group ratio/block size combination and study the effects on predictability and imbalance if we remove the permutations most likely to cause imbalance. The term “predictability” refers to the probability an investigator could predict the next assigned treatment. For a more complete description of what we mean by predictability please see Shum, Hamilton, and Lo (2016), or Blackwell and Hodges (1957) for a detailed explanation. To avoid getting too technical, let’s look at a simple example to illustrate what we’re proposing. Take a 2 treatment study with a 2:1 ratio of treatment groups A to B. The minimum block size would be 6. There are 15 permutations in this set, shown in Figure 1.

	The first and last permutations (1 and 15) are the most likely to create imbalance in treatment group totals. For instance, using the example we looked at above if the randomization ends after the first four slots are filled in the AAAABB permutation within each of the 10 sites, the final treatment group totals would be 80 in treatment group A and 20 in treatment group B, which is a 4:1 ratio instead of the desired 2:1 ratio.  By removing the 1st block we could eliminate that scenario and reduce the potential maximum imbalance of that example to 70 in group A and 30 in group B – a reduction of 42%.

	The previous example occurs with a very low probability, but with such costly consequences it makes most statisticians nervous anyway. To get a better idea of the typical benefit of our proposed method we used Monte Carlo simulated samples and calculated the average reduction in imbalance if we eliminated permutations 1 and 15 from our set of permutations. It turned out that the average gain in balance was almost 9% with only an increase of 4% in predictability. If we also removed blocks 2,3,10, and 14 we achieved an average gain in balance of almost 22%, with only a 10% increase in predictability. We also found dramatic results when we looked at a study design with 2 treatments and a 4:1 ratio (block size of 10). There are 45 treatment group order permutations. When we deleted the 12 that were the most likely to create imbalance we reduced average imbalance by 18.2%, with less than a 6% increase in predictability.

	A recent study with one of our clients illustrates how we can utilize this strategy. The randomization design was 3 treatments with a 1:1:1 ratio (block size of 6) stratified by investigational center. This randomization design has 90 treatment order permutations. Based on the enrollment projections by center we decided we needed 360 randomization slots per center, or 60 blocks. Instead of generating all possible permutations and randomly eliminating 30 blocks per center, we purposefully eliminated the 30 permutations from each stratum (center) that were most likely to cause imbalance, ensuring minimal imbalances in the treatment group totals at the end of enrollment.

	In general, the larger the number of possible permutations in a given design, or the greater the treatment group ratio, the more room there is to eliminate blocks most likely to cause treatment group imbalances. We use Monte Carlo simulations to find the average imbalance with specific block combinations and then calculate their associated predictability. We can then choose the combination of blocks that maximizes balance and minimizes predictability for a given study design. This proposal is novel in the literature. It is a straightforward way to increase the chances of treatment group balances with minimal impact on predictability.

	The conversation with my statistician colleague about the limited amount of time and desire to think about blocking strategies points out the importance of having a consulting resource available. Fortunately, study statisticians can rely on the IXRS/RTSM statistician to identify and recommend smarter ways to “build blocks” for randomizing your study.

, is Principal Biostatistician, Bracket; 

		ICH-E9. (1998). International Conference on Harmonization, E9 document, Guidance on Statistical Principles for Clinical Trials. 

		Shum C., Hamilton, SA., and Lo, KH., (2016). A Comparison of Techniques for Creating Permuted Blocked Randomization Lists. 

		D. Blackwell and J. L. Hodges (1957). Permutation for the Control of Selection Bias. 

While block randomization has been shown to be a more frequently used form of trial randomization design, the use of dynamic randomization has been shown to have optimal treatment group balancing results.

Building Blocks for Better Balance

Two of the most important elements to the integrity of a controlled clinical study are the patient randomization and the treatment blinding. In clinical trials, we are most often comparing groups defined by treatments (e.g., active vs. placebo, or dose levels). Effective randomization helps to balance the known and unknown prognostic factors across comparative groups. Treatment group imbalances within important prognostic variables cast doubt on the primary results at best. More importantly, they can reduce the probability of a positive statistical outcome. For instance, if the outcome variable under study is influenced by age, such as mortality, a chance allocation of one treatment group to older subjects would likely cause an increase in mortality in that group that could bias the treatment effect in the study.

	Statistically, the effect of imbalance can often be adjusted for in a model, if the analysis method is parametric. However, if the covariate adjusted model is not pre-specified, the FDA will not count it as the primary analysis. Often, there is a set of unknown prognostic factors. Many of them are due to different levels of experience and expertise treating the disease under study at the enrolling investigational centers. For example, we may expect better outcomes among patients treated at a high-level academic medical center than at a rural hospital in a developing country. A common technique for mitigating the influence of these unknown prognostic factors on the treatment effect has been to ensure equal treatment group ratios of patients across enrolling centers. Indeed, the ICH E9 Guidance

 states, “It is advisable to have a separate random scheme for each center, i.e., to stratify by center or to allocate several blocks to each center.” In addition, blocking a randomization list also prevents bias coming from temporal trends. For example, if the ratio of a two-treatment-groups study was 1:1, we would desire a 1:1 ratio at each enrolling center. We will refer to this example repeatedly in the next few paragraphs.

	By far, the most common way to proactively balance the known and unknown prognostic factors is to stratify the randomization using a permuted block randomization list.

 At clinical trial specialty services provider Bracket, its experience with approximately 1,500 controlled clinical studies confirms that permuted block randomization was used more than 80% of the time. 

	The first step in creating a permuted blocked randomization list is to determine the block size. In general, the best treatment group balance is achieved with the smallest block size. For example, a study with two treatment groups allocated in a 1:1 ratio could use a block size of 2. However, this block size carries the greatest risk of predicting the next treatment assignment. For instance, a block size of 2 with a 1:1 ratio has two permutations (A:B, B:A). If the block size is known, then it is certain that the second study subject will receive a different treatment than the first. In this case, if there are any reactions to one of the treatments, such as injection site pain, redness, fever, etc., then the next treatment assignment can be readily inferred. Hence, to avoid the potential of unblinding, a block size larger than the sum of the treatment group ratio is used. Typically, two times the sum of the treatment group ratio is used. In the previous example, a block size of 4 would contain six unique permutations of treatment order, which makes prediction of the next randomization assignment much more difficult.

	The next step would be to select the block permutations. If stratification is employed, the list of blocks would be divided equally among the levels of the stratification factor. A different method of assigning the sampled blocks to strata would be to dynamically assign blocks within each stratum as needed. When the first subject is ready to randomize with stratum specific characteristics, a block is allocated to that stratum. The subject is assigned to the first treatment in the block and the remaining slots are assigned as subjects continue to randomize within that stratum. As randomizations continue and no more slots are available in the previously assigned block, a new block is assigned. In this manner, blocks from the stratified randomization list are assigned to strata dynamically. Implementation of this method requires a central randomization system, often referred to as randomization and trial supply management (RTSM), interactive response technologies (IRT), and Interactive Voice/Web Response System (IVRS/IWRS).

	To minimize the ability to guess the next treatment assignment (selection bias), study statisticians desire a relatively equal representation of permutations in their randomization lists, especially within strata. Selection bias was originally discussed at a fairly technical level by Blackwell and Hodges in 1957.

 To illustrate the idea of selection bias, imagine a study with two treatment groups (A, B) where the patients are randomly assigned to one of the groups using a permuted blocked list stratified by enrolling center with a block size of 2. There would be two permutations within this list [A, B] and [B, A]. Suppose, by chance, a stratum was allocated all of one permutation so that the list for that center looked like this: [A, B, A, B, A, B, A, B, …]. Because of the perfectly alternating pattern of treatment allocation, there would be a high probability of guessing the next treatment, thus, a high selection bias (especially in an open-label study). One way to reduce the selection bias would be to increase the block size to four. However, in many studies, maintaining a high level of treatment group balance requires a block size of 2.

	We like the idea of producing balance beyond that simply produced by permuted blocks, and think it is an attractive feature that the possibility of having the same sequences repeated is reduced. At the same time, we realize that this is a controversial suggestion since others may disagree that imposing further balance is desirable. In particular, the 2003 Committee for Proprietary Medicinal Products (CPMP) guidance document warns sponsor companies to avoid the use of dynamic randomization, which is designed to increase treatment group balance by imposing constraints to the randomization.

 Nevertheless, in the remainder of this manuscript we explore ways to reduce severe departures and same sequences of blocks.

	The other method to reduce selection bias is to ensure all the permutations are represented in a relatively uniform distribution within each stratum. With current permuted block methods, a probability exists that a significant departure from a uniform distribution of permutations can occur. In fact, one will often see that a particular permutation appears consecutively for a few blocks. This event is not rare for a randomization list, especially for a list with only a few block patterns. In our example, a list of 24 blocks per stratum, the probability of having three consecutive blocks with the same design is 0.3350 per stratum. The probability of having four consecutive blocks with the same design is 0.0885 per stratum. In a more extreme example, the probability of having 10 consecutive blocks with the same design is 0.00000138434 per stratum. While the probabilities might not seem high, since these probabilities are additive, they increase as the number of blocks and/or strata.

	An informal method to avoid extreme departures is to generate a permuted block list with blocks pre-assigned to strata, and apply an acceptance test. A new list would be generated for any rejected list and the acceptance criteria would be applied to the new list. This process repeats until the list passes the acceptance criteria. A shortfall of this method is that at the end of randomization, typically only a subset of blocks is used within each stratum. However, it is unknown how many blocks will be used, so the acceptance criterion is applied to all the blocks, not just the blocks that were used. Our experience has shown that lists with many strata are easily rejected using this criterion.

	The novel method proposed in this paper to achieve uniform block distribution would be to sample the permutations within stratum without replacement. “With replacement” indicates that samples are independent. “Without replacement” indicates that samples are not independent. To further illustrate using a block size of 4, we would sample from the six possible block permutations without replacement until no more are available, then sample six more, and repeat until we have reached the desired number of blocks within each stratum (see diagram below). This method produces a uniform distribution of permutations within each stratum.

	Therefore, an acceptance test is not required. This method can also be applied to dynamic block assignment and would produce an equivalent distribution of permutations. To our knowledge, the properties of this method of uniform block distribution have not been studied. We conducted a simulation study that compares the properties of the three methods of assigning blocks to strata described above: Stratified permuted blocks sampled with and without replacement, and dynamic block assignment sampled without replacement.

	Prior to the simulations, we theorized that the more restrictions the method imposed on the list, the lower the probability for extreme block pattern distributions, and strings of repeated block patterns. In other words, we expected that the uniform permuted block distribution method would be superior at equal block pattern representation, followed by the pre-assigned permuted blocks method with rejection method, then the pre-assigned permuted blocks, and, finally, the dynamic block assignment method. In our example, the appearance of three of the same block patterns in a group of six where optimally they would all be unique almost doubles the selection bias for that grouping. When we examined the frequency of three of the same block patterns occurring in a group of six, our results show that this novel method we propose can reduce the overall selection bias by almost a third.

	The randomization method studied is a stratified permuted block permutation. We studied the four methods of allocating blocks to strata described in the previous section, i.e.:

 – blocks are assigned with the proper treatment ratio within each block based on enrollment at a particular location.

 – blocks are pre-assigned at random to the stratum.

Pre-assigned permuted blocks with rejection

 – a set number of blocks in the randomization list is assigned to a strata before study start after the list has passed a Fisher’s exact test to the block patterns within each stratum.

	We chose to study permuted blocks with two treatments (for the purposes of this study, “active” and “placebo”), as this is a very common study design we encounter. For each of the four block assignment methods, there were three block ratios examined in the simulation. The ratios were: 2A: 1P; 3A:1P; 1A:1P. Three block sizes were used in the simulation. The block sizes were: 3, 4, and a combination of a block size of 4 and 2. The block sizes were allocated at random. We chose three sample sizes to examine (

=500), as these are very common sample sizes that we encounter. Because the majority of randomization lists are stratified by investigational site, we chose to stratify the randomization by site in this analysis.

	For the above illustration, there were 12 combinations of ratios dependent on sample size and block size for each method. In all, there were 48 combinations of simulation study parameters examined.

	In the simulation experiment, the four methods were studied by a Monte Carlo simulation. Results were reported for each method.

	As mentioned previously, the blocks were distributed based on the order of enrollment. The location of the enrollment site became a stratification factor. For example, when Patient 1 enrolled at a site, Site A, Site A occupied the first block in the list. When Patient 2 enrolled into the study at another site, Site B, Site B occupied the second block in the list. Enrollments at each site progressed through the assigned block until that block was filled with patients before having another block in the list assigned. The same method of block assignment was applied throughout the sample simulation until all patients were assigned with a treatment.

	At the end of the sample simulation, this method produced:

		A list of sites where patients had enrolled

		The corresponding blocks and treatments assigned from those sites

	Since the result from the simulation had sites assigned listed, those assigned sites became a stratification factor when the study concluded.

	Since the blocks were assigned at random, each block permutation assignment to a stratum was independent from the previous block assignment. Therefore, it was possible for a single block permutation to be assigned consecutively two, three, or more times.

	In the pre-assigned permutated block method, each site is a stratification and the blocks were pre-assigned to these sites so that all patients at a particular site would receive treatments in the order they enrolled into the study. Pre-assigned permutated blocks with rejection adds an additional step- the list is rejected if it had a significant disparity of block permutations when compared against the ideal block permutation distribution.

	Based on the treatment ratio within each block and its block size, we determined the applicable block permutations for each study. For instance, for a study with a treatment ratio of 2A:2P and a combination of block sizes of 2 and 4, the block permutations were as follows: AAPP, PPAA, APAP, PAPA, APPA, PAAP, AP and PA The uniform permuted block distribution method ensured that each block permutation showed up an equal number of times in the list before repeating any block permutation. The entire randomization list should have an equal representation of block distribution, given the number of slots in a randomization list.

	Several characteristics were evaluated including how frequently each method departed from an acceptable block distribution, and how far each departed from it. We analyzed the performance of block distribution of each method by slicing block permutation uniqueness into deciles, counts of triple and multiple block permutation evaluations, and an analysis of block distribution using Fisher’s Exact Test. The purpose of using three distinct methods-as opposed to statistical tests-offered different approaches to examining block permutation arrangement and the uniqueness of block permutations per strata.

	The results show that there is very little difference between methods 1-3. However, the uniform permuted block distribution method, as expected, produces a much higher percentage of block permutation uniqueness within the strata.

	Among methods 1-3, Pre-assigned permuted blocks with rejection method generally produces lists with more even distributions of block permutations that are less volatile. Out of the four methods, the dynamic Site assignment method yields the most volatile block permutation distribution. In other words, the permutations in each site have a wider variation in terms of block permutation compilation compared to the other methods.

	In the pre-assigned permuted blocks with rejection method (our informal method of avoiding extreme departure), the standard deviations were smaller, indicating a smaller probability of having an undesirable distribution of block permutations in each site.

	The uniform permuted block distribution method yielded the least spread in terms of p-value (a standard deviation of 0) and the highest p-values (p-value of 1). Combining the previous results, uniform permuted block distribution method ensured an equal representation of block permutations on a pre-populated randomization list. It also guaranteed an equal representation of block permutations for each stratification if a trial stopped at any point before enrollment was completed.

	In the past, study statisticians have been concerned about an unequal representation of block permutations. Before our simulations, we theorized a novel process, uniform permuted block distribution method, which would sample block permutations without replacement, thereby reducing the probability of overrepresentation (or underrepresentation) of block permutations. We compared the four methods of randomization list creation: dynamic site assignment method, pre-assigned permutated blocks method, pre-assigned permuted blocks with rejection method, and our new method, uniform permuted block distribution method. We analyzed the methods by measuring uniqueness, repeated block permutations, and Fisher’s Exact Test. After performing analyses, we concluded that the uniform permuted block distribution method could reduce occurrences of a significant departure from the ideal number of block permutations per stratification.

	We wish to note that, in theory, the uniform permuted block distribution method can be implemented in a dynamic site assignment permutation. The challenge only being the complex programming involved to create the uniform block permutations as new blocks are assigned dynamically to sites as needed.

	1. ICH-E9. (1998). International Conference on Harmonization, E9 document, Guidance on Statistical Principles for Clinical Trials. 

	2. N.W. Scott, G. C. McPherson, C. R. Ramsay, et.al. (2002). The method of minimization for allocation to clinical trials: a review. 

	3. D. Blackwell and J. L. Hodges (1957). Permutation for the Control of Selection Bias. 

	4. CPMP, C. f. (2003). Points to Consider on Adjustment for Baseline Covariates. 

	W. N. Kernan, C. M Viscoli, R. W. Makuch, et.al. (1999). Stratified Randomization for Clinical Trials. 

	McEntegart, D. (2008). Blocked Randomization. In R. S. D'Agostino, 

 (p. DOI:10.1002/9780471462422.eoct301). Hoboken: John Wiley & Sons, Inc. 

Two of the most important elements to the integrity of a controlled clinical trial are the patient randomization and the treatment blinding.
With a focus on block pattern distribution, four methods of randomization list creation are analyzed.