The limited uptake of restricted randomization in C-RCTs suggests a need for investigators to become better versed in these options so that they may work with statisticians to consider advantages and limitations for their particular trial. Therefore, the advantages and limitations of some of the major strategies as they relate to C-RCTs are described below and summarized in Table
1. This is not a complete catalogue of allocation techniques, but rather an introduction to the array of options available to investigators. Since most investigators using restricted randomization techniques have relied on stratification and/or matching, emphasis is placed on other allocation strategies that are particularly promising for C-RCTs. Of these, minimization represents the prototypical option for when clusters are recruited and allocated sequentially, while covariate-constrained randomization represents an ideal option for when blocks of clusters (or all clusters) are recruited prior to allocation.
Table 1
Allocation techniques for covariate balance in C-RCTs: advantages and limitations
Simple/Complete randomization | No need for baseline data; most transparent, accepted | Higher risk for imbalance |
Restricted randomization | | |
Matching | Improves face validity; May balance effectively for many covariates (only if a good match is found) | Loss to follow-up is doubled (pair instead of single loss); challenges with analysis; difficult to estimate/report ICC; reduced degrees of freedom limits power |
Stratification | May be used in combination with other allocation techniques | Can balance for few covariates on its own |
Minimization | Can balance effectively for many covariates | Less transparent, possibly less well-understood by audience; continuous covariates may need to be split into categories; potential for selection bias/predictability |
Covariate-constrained randomization | Balances most effectively for many covariates; limits risk of selection bias | Requires access to baseline data; possibly less well-understood by audience; potential for over-constraint; requires additional statistical support; allocation must occur after recruitment |
Matching
Many investigators conducting community intervention trials believe that matching is a useful mechanism for creating comparable groups at baseline [
29]. Matching provides ‘face validity’ regarding balance between allocation arms [
30] and is thought to be particularly useful when there are few clusters [
31]. For example, one C-RCT matched six pairs of communities by ensuring that they shared geographical characteristics, as well as baseline rates of disease (Table
2) [
32].
Table 2
Examples of restricted randomization descriptions from C-RCTs published in high impact journals
Stratification | ‘To ensure balance between the 2 study arms, family physician practices underwent stratified randomization on the basis of the mean age (< 65 v. ≥ 65 years) and annual rates of emergency department visits (< 200 v. ≥ 200) of their clientele. Stratified randomization was achieved by a separate randomization procedure performed within each of the strata’ [ 33]. |
Minimization | ‘We randomized practices to intervention and control groups using a minimization programme, stratifying by partnership size, training practice status, hospital admission rate for asthma, employment of practice nurse, and whether the practice nurse was trained in asthma care’ [ 34]. |
Covariate-constrained randomization | ‘A balanced randomization procedure ensured that the intervention and control hospitals were balanced with respect to the rates of prophylactic use of oxytocin and episiotomy, the presence or absence of residency programs, the country and region where the hospital was located, and the annual number of births at the hospital. Of 184,756 possible ways of assigning hospitals to the intervention and control groups with acceptable balance, one sequence was randomly selected to determine the composition of the two groups’ [ 35]. |
However, pair-wise matching has a number of challenges in C-RCTs [
36,
37]. First, loss of follow-up from one cluster removes also its match from analysis - this is also true in I-RCTs, but a loss of a pair of clusters could be catastrophic for a trial with a small number of clusters. Furthermore, in C-RCTs, achieving a ‘good’ match that increases power is more difficult as the intra-cluster correlation (ICC) decreases (because as the amount of variability between groups decreases, it becomes harder for the matching process to remove substantial variability) [
38]. In the process of developing matches, one creates the important disadvantage in C-RCTs of making it difficult to properly calculate the ICC [
30], which should be reported to provide guidance to future investigators planning appropriately powered trials. Relatedly, matching may complicate the analysis of the C-RCT, especially when it is desirable to investigate the impact of individual-level factors on the likelihood of the outcome [
39]. When the correlation between matched pairs is poor [
40], or when there is a desire to determine the impact of baseline covariates on the intervention effect, investigators have pursued ‘breaking the matching’ in analysis; this approach may increase risk for type 1 error [
37].
Stratification
Stratified randomization in C-RCTs has the same major limitation as in I-RCTs; the number of strata must be few to avoid unequal allocation (for example, due to incomplete blocks). This is because as the number of strata increase, the risk of incomplete filling of blocks also increase, thereby increasing the risk for imbalances in prognostic variables [
41,
42]. Simulations suggest that the when the total number of strata approach half the total number of units to be allocated (that is, clusters), stratification becomes ineffective [
43] others have recommended limiting the number of strata to less than one-quarter the number of units to be allocated [
42]. Since there usually exist prognostically important covariates at both the cluster and individual levels, many C-RCTs may require active balancing for more covariates than stratification could accommodate. For example, if there are two covariates composed of four and two levels, respectively (for example, region: north, south, east, west; and sex: male, female) the result is eight total strata, suggesting that least 16 (and ideally 32) clusters would be necessary to safely achieve balance.
Note also that balancing for individual-level covariates would require calculating the cluster-level mean (or median) for the variable of interest prior to stratifying. For example, one C-RCT testing an intervention directed at family physicians aimed to reduce visits to the repeat emergency department by patients (Table
2). It stratified the family physicians by a cluster-level covariate (older versus younger physicians) and by a cluster-level mean of a participant-level covariate (high
vs. low rates of emergency department visits) [
33]. In some instances, one could imagine stratifying clusters for political or practical reasons (for example, by geographical location). In such a scenario, there may be a need for additional balancing techniques when allocating within strata [
44]; this should be planned with careful statistical support.
Minimization
Taves described minimization in 1974 [
45] while Pocock and Simon independently reported its potential benefits in 1975 [
41]. Scott and colleagues provide an excellent review of minimization discussing the benefits and limitations of minimization for I-RCTs [
46]. In general, this technique randomly assigns the first participants, then accounts for the covariates of participants previously enrolled and assigns each new participant to the group that provides better balance. As shown in Table
3, if the seventh patient to be allocated to a trial has a high rate at baseline for the outcome of interest (for example, blood pressure) and a moderate rate for a covariate (for example, age), the computer algorithm will account for the characteristics of the six patients already allocated and assign the seventh patient to the arm that improves overall balance in those covariates.
Table 3
Example of minimization (adapted from Scott et al. 2002 )[
46]
Baseline rate |
High | 2 | 2 |
Moderate | 2 | 3 |
Low | 1 | 1 |
Covariate rate |
High | 2 | 3 |
Moderate | 3 | 1 |
Low | 1 | 1 |
Minimization improves covariate balance compared to both simple randomization and stratification; the difference is greater in smaller trials, but this advantage of minimization has been shown to hold for I-RCTs until the sample exceeds 1,000 patients [
46]. Simulations indicate that increasing the number of covariates in minimization does not substantially increase imbalance (in comparison to stratification) [
47]. The number of covariates to be included in the minimization algorithm is primarily limited by statistical concerns since it is recommended that all covariates minimized should be included in statistical analysis [
48]. The ability of minimization to balance more covariates has led to the suggestion by some commentators that it is the ‘platinum standard’ of allocation techniques [
49].
The pharmaceutical industry [
50] and other commentators [
51] warn against the use of minimization mainly due to higher risk of selection bias that comes with predictability of deterministic assignment. The extent of this risk is debated [
52] and must be weighed against the advantages of greater covariate balance. A random component may be added to the minimization procedure so that as imbalance grows the odds of allocation to the arm that reduces imbalance also grow, but are never equal to one [
53]. This may have the advantage of reducing predictability of allocation. Many authors have suggested additional variations on the general minimization approach either to further improve balance or to reduce risk of selection bias [
11,
54‐
56]. For instance, Begg and Iglewicz [
57] (and later Atkinson [
58]) applied optimum design theory (minimizing the variance in the model relating the covariates to the outcome) and allowed for balancing of continuous variables obviating the need to categorize continuous covariates as high and low. It is unclear whether the theoretical advantages of these more complex techniques translate into practical benefit in typical trials [
59].
Another concern may be that by forcing balance in known prognostic covariates, an investigator could (unknowingly) cause imbalance in unmeasured factors. However, it has been suggested that the balance for unmeasured variables can be no worse due to minimization because whenever the unmeasured factor is correlated with the minimized covariate, the balance for this factor will actually be improved and whenever the unmeasured factor is not correlated at all with the minimized covariate then its distribution would be unaffected [
60]. Although it cannot be proven empirically without measuring the unmeasurable, this explains why balance of non-targeted variables should not made worse by using minimization [
61].
The ability of minimization to balance many covariates within a small trial should make it a particularly good allocation technique for C-RCTs. However, to actively balance numerous covariates requires access to data at the time of recruitment; cluster-level means (or medians) would be used to minimize participant-level covariates, such as the practice-level mean of patient blood pressure values [
62]. Fortunately, many C-RCTs take place in the context of medical systems with administrative data or access to historical records [
24]. Despite its promising features and the availability of free software to implement it [
63], minimization was used in only 2% of 300 randomly selected C-RCTs published from 2000 to 2008 [
4]. (Although it is possible that this is an underestimate if minimization is misreported as stratification, this is similar to the estimated overall proportion of trials that use minimization [
48].) One C-RCT using minimization was published in the
BMJ in 2004 (Table
2). It allocated 44 GP practices (clusters) minimizing imbalance across four covariates with 54 total strata [
34]. This trial tested a nurse outreach model aiming to support primary care providers in caring for patients with asthma. It was important to achieve balance across multiple cluster and individual level variables that might confound the effect of the intervention on asthma emergency visits. If the investigators had used traditional stratification, the trial would have been at high risk of imbalance due to over-stratification.
Covariate-constrained randomization
If data are available for the important cluster and/or individual-level covariates of participants prior to the allocation procedure, more complex techniques may be used to ensure acceptable balance. For example, Moulton [
64] described a procedure in which a statistical program could be used to enumerate all the possible allocations of participating clusters when clusters and their covariates are known in advance. Next, the investigators would narrow this list of allocations down to the ones that met prespecified criteria for balance across baseline covariates. Finally, the actual allocation would be chosen randomly from this constrained list, thereby achieving an acceptable allocation while retaining randomness in the selection process. As seen in Table
4, if there were only four clusters recruited, these could be allocated into two arms in six different ways. In two of the possible allocations (A, F), the difference in the baseline performance is very large. In a trial with more clusters, the possible allocations increases exponentially, and it is possible to remove unacceptable allocations from the list and chose randomly from any remaining allocations with acceptable balance.
Table 4
Example of covariate-constrained randomization (adapted from Moulton 2004 )[
64]
Allocation | Intervention | Control | Difference |
A | 25 | 50 | 60 | 75 | 30 |
B | 25 | 60 | 50 | 75 | 20 |
C | 25 | 75 | 50 | 60 | 5 |
D | 50 | 60 | 25 | 75 | 5 |
E | 50 | 75 | 25 | 60 | 20 |
F | 60 | 75 | 25 | 50 | 30 |
|
Covariate rate
|
Allocation | Intervention | Control | Difference |
A | 80 | 60 | 75 | 70 | 2.5 |
B | 80 | 75 | 60 | 70 | 12.5 |
C | 80 | 70 | 60 | 75 | 7.5 |
D | 60 | 75 | 80 | 70 | 7.5 |
E | 60 | 70 | 80 | 75 | 12.5 |
F | 75 | 70 | 80 | 60 | 2.5 |
This approach has been shown in simulation studies to provide even better balance than minimization resulting in increased power, especially for trials with few units allocated as is common in C-RCTs [
65,
66]. This may be partially explained by the fact that covariate-constrained randomization can balance continuous covariates without loss of power from categorization of these variables (for example, high, medium, low) as occurs in minimization. However, when there are very few clusters as in the example illustrated in Table
4, the parameters for assessing balance may need to be widened so that the actual allocation to be utilized can be randomly selected from a larger set. Over-constraint due to strict balancing requirements that result in very few eligible allocations may force certain clusters together. For example, in Table
4, if the caliper for balance in the baseline rate or the confounder rate was set at a mean difference of 10, only two allocations would remain and both feature the highest and lowest ranking clusters together in one arm. This is not desirable since it means that the allocation is no longer truly random and this situation may invite skepticism regarding active manipulation by the investigator [
64]. In addition to requiring added statistical support during the process of recruitment and allocation, the main drawback of this approach is that to acquire the necessary data, recruitment of numerous clusters must be completed prior to any cluster allocation. Investigators can allocate blocks of clusters as they are enrolled, though the first block should have at least eight units and the subsequent ones at least six [
67].
Covariate-constrained randomization was used in only 2% of 300 randomly selected C-RCTs published from 2000 to 2008 [
4]. However, the availability ready-made algorithms to implement this approach [
67,
68] may make the process more accessible. Like minimization, it is possible to apply more complex formulae when conducting this procedure. Rather than setting parameters regarding covariates to be within 10% of each other, investigators can balance with the goal of minimizing variance or decreasing the effects of adjusted analyses in a cluster-trial, as proposed by Raab and Butcher [
20]. This particular approach was used in a study published recently in the
New England Journal of Medicine[
35], indicating the growing acceptability of this allocation technique by editors (and readers).