Block Randomization

Block Randomization and the Choice of Block Size in Trials

Block randomisation divides a trial's allocation sequence into short runs, and each run contains a fixed number of assignments to every arm in a random order, so the arms can never drift far apart in size during recruitment. The usual version is the permuted block design. With two arms and a block of four, every block holds two assignments to A and two to B in one of six possible orders. For health economics the method matters because the treatment effects, resource use and quality-adjusted life years that feed a trial-based cost-effectiveness analysis come from the groups it creates. Choosing a block size trades balance against predictability. The wider principle of randomisation and the separate safeguard of allocation concealment have their own pages.

How permuted blocks keep arm sizes close

The ICH E9 guideline on statistical principles for clinical trials accepts unrestricted randomisation but notes that some advantages can generally be gained by randomising subjects in blocks. Blocks help keep the groups comparable when subject characteristics change over time, for example after a change in recruitment policy, and give a better guarantee that the groups will be of nearly equal size. The CONSORT 2025 explanation and elaboration describes restricted randomisation as any randomised approach that is not simple randomisation, with blocked randomisation as the most common form.

Altman and Bland explain the mechanics in their BMJ statistics note. The analyst lists the possible orders within a block, picks blocks at random to build the sequence, and the numbers in the two groups can never differ by more than half the block length. Block size is normally a multiple of the number of treatments, and large blocks control balance less well. The number of distinct orders available for one block is as follows.

$$N = \frac{b!}{\prod_{j=1}^{k} n_j!}$$

where $N$ is the number of distinct orders, $b$ is the block size, $k$ is the number of arms and $n_j$ is the number of assignments to arm $j$ in each block, so that the $n_j$ sum to $b$. For a 1:1 trial with blocks of four, $N = 4!/(2!\,2!) = 24/4 = 6$. Unequal allocation works the same way, so a 2:1 ratio needs a block size divisible by three, and a block of six with four assignments to A and two to B has $N = 6!/(4!\,2!) = 720/48 = 15$ orders. The CONSORT 2025 elaboration quotes a trial that used a block size of 6 with a 2:1 ratio.

Block size as tighter balance against a more guessable sequence

Block size sets the main trade-off in the method. ICH E9 asks for block lengths short enough to limit possible imbalance but long enough to avoid predictability towards the end of a block. It adds that investigators and other relevant staff should generally be blind to the block length, that two or more block lengths randomly selected for each block can achieve the same purpose, and that details which make the sequence predictable, such as block length, should not appear in the trial protocol.

Efird describes the mechanism of the risk. When groups are unmasked, the treatment that has so far occurred least often in the current block is likely to come next. The CONSORT 2025 elaboration notes that a person running the trial could deduce some future allocations after discovering the block size, that this is much more likely in unblinded trials, and that blinding, larger blocks and randomly varied block sizes can ameliorate the problem. It also reports that in a study of 179 open, unblinded randomised trials, small block sizes were associated with subversion. ICH E9 observes that predictability theoretically does not matter in a double-blind trial, although the pharmacological effects of drugs may allow intelligent guesswork.

Worked example of what a known block of four gives away (illustrative)

Consider an open-label two-arm trial with 1:1 allocation and a fixed block of four that the recruiting clinician has worked out. The clinician always predicts the arm with more unused places left in the current block and guesses when the two are level. The six possible orders are equally likely, and each position can be scored step by step.

1. First position. Nothing has been revealed, so the prediction is correct with probability 1/2.
2. Second position. One assignment is known and the remaining three places hold two of the other arm, so predicting that arm is correct with probability 2/3.
3. Third position. In two of the six orders (AABB and BBAA) the first two assignments match, so the last two are certain. In the other four orders one place of each arm remains, giving 1/2. The expected score is (1/3 x 1) + (2/3 x 1/2) = 2/3.
4. Fourth position. The last assignment is always determined, so the score is 1.

The expected number of correct predictions per block is 1/2 + 2/3 + 2/3 + 1 = 2.83 (exactly seventeen sixths) out of 4, or 70.8% against 50% under simple randomisation. Allocations known with certainty average 1 + 1/3 = 1.33 per block, one third of all assignments. Repeating the calculation by enumeration for other fixed 1:1 block sizes gives the following results.

The balance side of the trade-off is just as concrete. In an illustrative pilot trial of 40 people, simple randomisation produces a split of 15:25 or worse with probability of about 15.4% (binomial, n = 40, p = 0.5), and 14:26 or worse with probability of about 8.1%. With blocks of four the arms never differ by more than two at any point and finish at exactly 20:20. Larger blocks cut certain deductions only slowly, which is why ICH E9 and CONSORT point to hidden or randomly varied sizes. A clinician who can predict the next arm can steer which patients enter it, and the way such selection bias flows into an incremental cost-effectiveness ratio is set out on the allocation concealment page.

Stratified block randomisation

Stratification applies blocking within subgroups. Altman and Bland describe it as producing a separate block randomisation list for each stratum, and stress that allocation must use blocks within each stratum because simple randomisation there gives no control of balance, which defeats the purpose. The CONSORT 2025 elaboration makes the same point, stating that stratification without some form of restriction is ineffective, and adds that stratification by centre is common in multicentre trials.

The number of strata grows multiplicatively. In Altman and Bland's breast cancer example, two tumour size groups, three node groups and menopausal status give 2 x 3 x 2 = 12 strata, which may exceed what is practical, and rare combinations may not reach their intended balance. ICH E9 states that more than two or three stratification factors are rarely necessary, are less successful at achieving balance and are logistically troublesome. It suggests a separate scheme for each centre or allocating several whole blocks to each centre. The EMA guideline on adjustment for baseline covariates warns that as strata multiply, empty or thinly occupied strata become more likely, and points to dynamic allocation for balancing several factors at once, with deterministic schemes to be avoided.

Incomplete blocks also add up. Because each stratum can stop part way through a block, the half-block bound applies per stratum, so with 12 strata and blocks of four the overall difference between arms could in principle reach 12 x 2 = 24 even though every stratum list is balanced.

Sequence generation, concealment and blinding

Block randomisation is a rule for generating the allocation sequence. It says nothing about who can see that sequence. ICH E9 requires that the next subject receives the next free number in the appropriate schedule (within the stratum, if stratified), that the number is allocated only once entry to the randomised part of the trial is confirmed, and that the schedule is filed securely by the sponsor or an independent party. Efird concludes that the best protection against selection bias is to blind both the order of blocks and their size, and notes that random block sizes help only when assignments could otherwise be determined with certainty. Varied block sizes therefore reduce deduction but do not replace concealment, the methods for which are covered on the allocation concealment page.

Analysis and the economic evaluation built on the trial

Blocking and stratification change how a trial should be analysed. The EMA covariate guideline states that factors used as the basis of stratification should normally be included as covariates or stratification variables in the primary outcome model, except where stratification was purely administrative, such as the block in block randomisation. Kahan and Morris explain that balanced randomisation introduces correlation between treatment groups, so ignoring balancing factors gives P values that are too large and confidence intervals that are too wide. In their review of 258 trials published in 2010 in four general medical journals, stratified permuted blocks were the most common balancing method (85 trials), yet only 42 of the 163 trials that balanced on any factor (26%) adjusted for all of them in their primary analysis.

The same randomised groups supply the incremental costs and QALYs in a trial-based economic evaluation, so the reasoning carries over, and cost and QALY regressions that include the centre and stratification factors respect the design in the same way as the clinical model. Efird notes that statistical power is maximised with equal group sizes, a point that also bears on the precision of incremental cost estimates in small trials (see statistical power). For appraisal, CONSORT 2025 item 17b asks trial reports to give the type of randomisation and details of any restriction, such as stratification, blocking and block size. Its elaboration asks for how blocks were generated, the size or sizes, whether size was fixed or randomly varied, and whether trialists became aware of the block size. An analyst using a randomised controlled trial in an HTA submission can read this item to judge how open the sequence was to prediction.

Limits and common misreadings

Several misreadings recur in trial reports and in the economic models that draw on them. Most come from treating balanced arm counts as proof of a well-run allocation, or from assuming that one safeguard covers for another.

• Equal numbers are not equal patients. Blocks control how many people enter each arm. Comparability of prognosis comes from randomisation itself, and chance imbalance in baseline characteristics remains possible, especially in small strata. Balanced allocation also does nothing about later loss to follow-up or non-adherence, which can still bias incremental costs and QALYs.
• Random block sizes are not concealment. They make deduction harder but leave a visible schedule open to anyone with access to it.
• Identical group sizes do not prove blocking. The CONSORT 2025 elaboration notes that many reports show near-identical group sizes without mentioning restriction, which may reflect under-reporting, though non-random manipulation is also suspected.
• Exact balance holds only at block boundaries. Efird points out that an interim analysis or early termination can fall mid-block, leaving temporary inequality.
• Blocking and cluster allocation answer different questions. Cluster randomisation changes what is allocated (groups rather than individuals), and a cluster trial can still allocate its clusters in blocks.
• Dynamic allocation is a different method. Minimisation balances several factors at once by tracking current imbalance, whereas block randomisation balances arm counts within each list.

Sources

• Altman DG, Bland JM. How to randomise. BMJ. 1999;319(7211):703-704. https://doi.org/10.1136/bmj.319.7211.703
• Efird J. Blocked randomization with randomly selected block sizes. International Journal of Environmental Research and Public Health. 2011;8(1):15-20. https://doi.org/10.3390/ijerph8010015
• European Medicines Agency, Committee for Medicinal Products for Human Use. Guideline on adjustment for baseline covariates in clinical trials (EMA/CHMP/295050/2013). Adopted 26 February 2015, effective 1 September 2015, accessed 5 October 2026. https://www.ema.europa.eu/en/adjustment-baseline-covariates-clinical-trials-scientific-guideline
• Hopewell S, Chan AW, Collins GS, Hróbjartsson A, Moher D, Schulz KF, et al. CONSORT 2025 explanation and elaboration: updated guideline for reporting randomised trials. BMJ. 2025;389:e081124. https://doi.org/10.1136/bmj-2024-081124
• International Council for Harmonisation. ICH E9: Statistical principles for clinical trials (CPMP/ICH/363/96). Step 4, 5 February 1998, section 2.3.2, accessed 5 October 2026. https://database.ich.org/sites/default/files/E9_Guideline.pdf
• Kahan BC, Morris TP. Reporting and analysis of trials using stratified randomisation in leading medical journals: review and reanalysis. BMJ. 2012;345:e5840. https://doi.org/10.1136/bmj.e5840