Unit of Allocation

Unit of Allocation in Randomised Trials and Cost-Effectiveness Analysis

The unit of allocation is whatever the random allocation in a trial assigns to a study arm. In most randomised controlled trials it is the individual patient, but it can be a general practice, a hospital ward or a village, and within one person it can be an eye or a treatment period. It matters for health economics because service and public health interventions are often allocated by practice or area, and the effects, costs and net benefits from such trials are only as precise as the number of units randomised. Treating patients as independent when practices were allocated makes the evidence look more certain than it is, and that false certainty passes into reimbursement decisions.

Levels at which trials allocate interventions

Chapter 6 of the Cochrane Handbook states the governing principle, that the analysis must take into account the level at which randomisation occurred, and that in most circumstances the number of observations analysed should match the number of units randomised. The level can sit above, at or below the individual person, and each level brings its own correlation between observations.

The Cochrane Handbook calls the first row a cluster-randomised trial. Clustering can also arise in an individually randomised trial when one surgeon, nurse or therapist treats many participants, and chapter 23 suggests reviewers consider inflating the variance with a design effect in that case too.

Why a trial allocates groups instead of patients

Chapter 23 of the Cochrane Handbook gives three reasons for allocating groups. The intervention may have a group effect, such as the herd immunity of a vaccine; allocating individuals within one setting may cause contamination, as when clinicians trained for one arm treat patients in both; or group allocation may be more convenient. Gomes and colleagues add group-level interventions such as incentives for health providers.

The decision problem often points to the right unit. A commissioner funding a pharmacist service adopts it practice by practice, so the practice is the natural unit for the trial and its economic evaluation. The price is fewer independent units, since two practices with 500 patients each give two units of replication, not 1,000. Sample size and recruitment bias under group allocation belong to cluster randomisation.

Matching the unit of analysis to the unit of allocation

A unit-of-analysis error occurs when the analysis treats as independent a smaller unit than the one allocated. Whiting-O'Keefe and colleagues described the problem in 1984, finding that 20 of 28 health care experiments (71%) on health professionals' performance used patient observations to draw conclusions about provider behaviour or outcomes determined jointly by patients and providers. Divine and colleagues found the same error in 38 of 54 studies (70%) of physicians' patient care behaviour published from 1980 to 1990.

The Cochrane Handbook explains the consequence for a cluster trial analysed as if individuals had been randomised. Confidence intervals become artificially narrow and P values artificially small, which can produce false-positive conclusions, and in a meta-analysis the study receives too much weight. The error can run the other way, since analysing a crossover trial as a parallel-group trial gives intervals that are too wide, a conservative error.

Two analyses respect a group unit. One analyses a summary for each cluster, so the sample size is the number of clusters, at some loss of precision when cluster sizes vary. The other analyses individuals with a model that allows for clustering, such as a mixed effects model or generalised estimating equations. When a published trial ignored the clustering, the Handbook gives an approximate correction based on the design effect.

$$DE = 1 + (M - 1) \times ICC, \qquad n_{\text{eff}} = \frac{n}{DE}, \qquad SE_{\text{adj}} = SE \times \sqrt{DE}$$

where $DE$ is the design effect, $M$ is the average cluster size, $ICC$ is the intraclass correlation coefficient, $n$ is the number of participants in an arm, $n_{\text{eff}}$ is its effective sample size, $SE$ is the standard error from an analysis that ignored clustering and $SE_{\text{adj}}$ is the standard error allowing for it. For a binary outcome both participants and events are divided by the design effect. Inflating the standard error is equivalent and avoids rounding to whole numbers.

Chapter 6 of the Handbook lists further mismatches, such as counting treatment cycles instead of women in subfertility trials or entering a shared comparator group twice in one meta-analysis.

Unit of allocation in a trial-based cost-effectiveness analysis

An economic evaluation alongside a cluster trial faces two correlations at once. Costs and outcomes are each correlated within clusters, and they are correlated with each other at both the individual and the cluster level. Gomes and colleagues set out a bivariate hierarchical model with a random effect for each cluster in the equations for cost and effect.

$$c_{ij} = \beta^{c}_{0} + \beta^{c}_{1} t_{j} + u^{c}_{j} + \varepsilon^{c}_{ij}, \qquad e_{ij} = \beta^{e}_{0} + \beta^{e}_{1} t_{j} + u^{e}_{j} + \varepsilon^{e}_{ij}$$

where $c_{ij}$ and $e_{ij}$ are the cost and health outcome of patient $i$ in cluster $j$, $t_{j}$ is 1 if cluster $j$ was allocated to the intervention and 0 otherwise, $\beta^{c}_{1}$ and $\beta^{e}_{1}$ are the incremental cost and effect, $u^{c}_{j}$ and $u^{e}_{j}$ are cluster random effects with correlation $\psi$, and $\varepsilon^{c}_{ij}$ and $\varepsilon^{e}_{ij}$ are individual errors with correlation $\rho$. The incremental net benefit at a willingness to pay $\lambda$ per unit of effect is $\lambda \beta^{e}_{1} - \beta^{c}_{1}$.

In their case study of 40 clinics with intracluster correlations of about 0.01, ignoring clustering left the mean incremental net benefit similar but understated the uncertainty and the expected value of further research. Ng and colleagues found the deviance information criterion the most useful guide to choosing among such multilevel models.

Worked example with a practice-randomised trial reported per patient

All numbers in this example are illustrative. A trial randomised 12 general practices to a pharmacist-led medicines review and 12 to usual care, with 25 older patients per practice, so 300 patients per arm. Over 12 months, 45 intervention patients and 66 usual care patients were admitted to hospital. The report analysed patients as independent, and an intraclass correlation of 0.03 for admission is taken from comparable practice-based trials.

1. Estimate the effect as reported. The admission risks are 45 / 300 = 0.150 and 66 / 300 = 0.220, so the risk ratio is 0.150 / 0.220 = 0.682 and its natural logarithm is -0.383.

$$SE = \sqrt{\frac{1}{45} - \frac{1}{300} + \frac{1}{66} - \frac{1}{300}} = \sqrt{0.030707} = 0.175$$

where 0.175 is the standard error of the log risk ratio with patients treated as independent. The 95% confidence interval for the risk ratio is 0.484 to 0.961 and the two-sided P value is 0.029.

2. Allow for the practice as the unit of allocation. With 25 patients per practice the design effect is 1 + 24 × 0.03 = 1.72.

$$SE_{\text{adj}} = 0.175 \times \sqrt{1.72} = 0.1752 \times 1.3115 = 0.230$$

where 0.230 is the standard error allowing for clustering. The 95% interval widens to 0.435 to 1.070, which includes no effect, and the P value rises to 0.096.

3. Check through the effective sample size. Dividing by the design effect gives 300 / 1.72 = 174.4 patients per arm, with 26.2 and 38.4 admissions. The standard error is again 0.230, so the 600 patients carry about as much information as 349 independent ones.

4. Carry the effect into an economic model. Suppose an admission costs GBP 2,500 and the review GBP 40 per patient. At the usual care risk of 0.220 the point estimate avoids 70 admissions per 1,000 patients and saves GBP 175 per patient, or GBP 135 net. The review saves money when the risk ratio is below a threshold.

$$RR^{*} = 1 - \frac{40}{0.220 \times 2500} = 1 - 0.0727 = 0.927$$

where $RR^{*}$ is the break-even risk ratio. Holding baseline risk and unit costs fixed and sampling the log risk ratio from a normal distribution in a probabilistic sensitivity analysis, the probability of a saving is 0.960 with the naive standard error and 0.910 with the adjusted one.

5. Interpret. The point estimate is unchanged, but the chance that adopting the review loses money more than doubles, from 0.040 to 0.090. The per-patient analysis would show stronger evidence, and a smaller expected value of perfect information, than 24 randomised practices can support.

Assumptions behind the adjustment

The design effect correction is approximate. The intraclass correlation is often unreported and borrowed from similar studies. The Cochrane Handbook notes that values below 0.05 are typical, that even small values can widen intervals substantially when clusters are large, and that larger natural units tend to have smaller values.

The formula assumes similar cluster sizes and a common design effect in both arms, and the Handbook asks for sensitivity analyses when the intracluster correlation is borrowed. Costs and health outcomes can have different values, which Gomes and colleagues varied separately in their simulation, so one correction applied to net benefit is rough and a multilevel analysis is preferred when patient-level data exist.

Interpretation for HTA and reimbursement decisions

The unit of allocation shapes what a trial estimates as well as how precisely. The Cochrane Handbook notes that an intervention applied to a whole community may work better than one given to randomised individuals within it, for example by reducing re-infection, so cluster and individually randomised trials may estimate different effects.

Most cluster trials are pragmatic, and the Handbook notes that they aim to support high-level decisions about health care, such as adopting a service across practices or wards. The number of clusters, not patients, limits statistical power, and an economic analysis that ignores the unit of allocation will usually push the probability of cost-effectiveness too far towards 0 or 1 and understate the value of further research.

Unit of allocation, unit of analysis and unit of observation

The unit of analysis is what the statistical analysis treats as independent, and it should match the unit of allocation unless the model allows for the correlation. The unit of observation is where outcomes are measured. A multicentre trial is not a cluster trial, since Gomes and colleagues note that patients within a centre are randomised to different treatments, whereas a cluster trial allocates the whole centre. The cluster effect is the correlation that grouping creates, whatever the unit of allocation.

Common misreadings of the unit of allocation

Most errors come from counting what is easy to count. The result is spurious precision.

1. Counting patients when practices were randomised. A trial of 24 practices has 24 units of replication, however many patients it records.
2. Counting eyes, teeth, cycles or visits as people. Measurements within a person are correlated, and the denominator should be the people randomised.
3. Correcting the clinical analysis but not the economic one. Costs and QALYs carry their own clustering, so a model should not take the unadjusted standard error.

Sources

• Higgins JPT, Eldridge S, Li T. Chapter 23: Including variants on randomized trials. In: Higgins JPT, Thomas J, Chandler J, et al., editors. Cochrane Handbook for Systematic Reviews of Interventions version 6.5. Cochrane; 2024. Sections 23.1 and 23.2. https://www.cochrane.org/authors/handbooks-and-manuals/handbook/current/chapter-23
• Higgins JPT, Li T, Deeks JJ (editors). Chapter 6: Choosing effect measures and computing estimates of effect. In: Higgins JPT, Thomas J, Chandler J, et al., editors. Cochrane Handbook for Systematic Reviews of Interventions version 6.5. Cochrane; 2024. Section 6.2. https://www.cochrane.org/authors/handbooks-and-manuals/handbook/current/chapter-06
• Whiting-O'Keefe QE, Henke C, Simborg DW. Choosing the correct unit of analysis in medical care experiments. Medical Care. 1984;22(12):1101-1114. https://doi.org/10.1097/00005650-198412000-00005
• Divine GW, Brown JT, Frazier LM. The unit of analysis error in studies about physicians' patient care behavior. Journal of General Internal Medicine. 1992;7(6):623-629. https://doi.org/10.1007/BF02599201
• Gomes M, Ng ES, Grieve R, Nixon R, Carpenter J, Thompson SG. Developing appropriate methods for cost-effectiveness analysis of cluster randomized trials. Medical Decision Making. 2012;32(2):350-361. https://doi.org/10.1177/0272989X11418372
• Ng ES, Diaz-Ordaz K, Grieve R, Nixon RM, Thompson SG, Carpenter JR. Multilevel models for cost-effectiveness analyses that use cluster randomised trial data: an approach to model choice. Statistical Methods in Medical Research. 2016;25(5):2036-2052. https://doi.org/10.1177/0962280213511719