Unequal Treatment Costs and Arm Allocation

Why Unequal Costs Change the Allocation Question

With a common control group, each treatment-versus-control comparison uses the same control participants. If treatment arms have different costs per participant, however, equal group sizes need not give the least expensive design that meets the required precision. A costly treatment arm can make it economical to use fewer participants in that arm and more in the shared control, while a less costly arm may be allocated differently.

The calculation in this tutorial is a planning exercise: specify a precision requirement for each treatment-control contrast, then compare allocations that satisfy it. It does not determine which treatment is clinically worthwhile, nor does it replace a power calculation when the planning target is power. The allocation depends on the precision targets, outcome variability, and costs assumed.

Definition: Let \(n_0\) be the number of participants assigned to the common control and \(n_j\) the number assigned to treatment arm \(j\). Let \(c_0\) and \(c_j\) be their respective per-participant costs. Total participant cost is \(C=c_0n_0+\sum_j c_jn_j\). For a continuous outcome with a common variance \(\sigma^2\), the variance of the estimated treatment-minus-control mean difference for arm \(j\) is \(\sigma^2(1/n_0+1/n_j)\), assuming independent observations across groups.

The variance expression assumes that participants are independently sampled or randomized to groups, the outcome variance is the same in all arms, and each group’s observations are independent of those in the other groups. The comparisons are not independent of one another: their estimates share the control observations. That dependence does not change the variance of any one contrast, but it matters for joint inference, which is outside this allocation calculation.

Suppose the design requires the variance of contrast \(j\) to be no greater than \(V_j\). The values \(V_j\) must be set on a scientifically meaningful basis before comparing costs. Write \(v_j=V_j/\sigma^2\). Then the requirement becomes \(1/n_0+1/n_j\leq v_j\). For a given feasible control size \(n_0\), this gives a minimum treatment-arm size. Costs do not appear in that constraint: costs determine which feasible allocation is cheapest, not whether a particular pair of group sizes meets the precision target.

n_0 > \max_j \frac{1}{v_j}, \qquad n_j \geq \frac{1}{v_j-1/n_0}.

The strict condition on \(n_0\) is necessary: if \(1/n_0\geq v_j\), the control contribution alone uses up the entire variance allowance for contrast \(j\), leaving no finite treatment-arm size that can satisfy the requirement. In a real design, sample sizes are integers, so the calculated minimum for each \(n_j\) is rounded upward.

From Precision Constraints to Cost

For any proposed control size, the cheapest way to meet each contrast-specific constraint is to use the smallest feasible treatment size. With positive per-participant costs, adding treatment participants beyond that minimum increases expenditure without helping satisfy the stated constraint. Thus the continuous relaxation of the cost problem is to minimize the cost after substituting the minimum treatment sizes.

C(n_0) = c_0n_0+\sum_j\frac{c_j}{v_j-1/n_0}, \qquad n_0>\max_j\frac{1}{v_j}.

This formula treats group sizes as continuous temporarily, to show the balance underlying the allocation. Its derivative indicates whether a small increase in the control size would increase or decrease total cost. Increasing \(n_0\) costs \(c_0\) per additional control participant, but reduces the number of treatment participants needed to meet each contrast’s precision target.

\frac{dC}{dn_0} = c_0-\sum_j\frac{c_j}{n_0^2(v_j-1/n_0)^2}.

At an interior minimum, this derivative is zero: the marginal cost of adding control participants balances the combined marginal savings from reducing treatment-arm sizes. The derivative increases with \(n_0\), because each treatment-arm savings term decreases as the control becomes larger. Near the smallest feasible control size, at least one treatment requirement becomes arbitrarily large; for very large \(n_0\), control cost dominates. These features give a unique continuous minimum under positive costs and finite, positive targets. The final integer design still requires comparing feasible rounded allocations, because ceilings can shift the least-cost choice.

Worked Example: Unequal Costs With a Common Precision Target

Consider two treatment arms, A and B, and one common control. Assume independent observations and a common outcome variance of \(\sigma^2=1\) squared outcome units. Suppose each treatment-control contrast must have variance at most \(0.04\) squared outcome units. Let per-participant costs be \(c_0=1\) cost unit for control, \(c_A=1\) for A, and \(c_B=4\) for B. Find the least-cost integer allocation.

For either contrast, \(v=0.04\), so at a given \(n_0\) each treatment arm needs at least \(1/(0.04-1/n_0)\) participants. Both arms have the same precision requirement, so they have the same minimum size at any chosen \(n_0\), even though their costs differ. The continuous cost is \(C(n_0)=n_0+5/(0.04-1/n_0)\). Setting its derivative to zero gives

1-\frac{5}{(0.04n_0-1)^2}=0, \qquad n_0=\frac{1+\sqrt{5}}{0.04}\approx80.90.

This continuous result is a guide, not an integer answer. For integer allocation, let both treatment groups have size \(q\). At \(n_0=82\), the required size is

q \geq \frac{1}{0.04-1/82} =\frac{1}{0.027805}\approx35.96, \qquad q=36.

The achieved contrast variance is \(1/82+1/36\approx0.03997\), which is below \(0.04\). Total cost is \(82+36+4(36)=262\) cost units. For comparison, \(n_0=78\) requires \(q=37\), costing \(78+37+4(37)=263\); \(n_0=83\) requires \(q=36\), costing \(83+36+4(36)=263\). Checking feasible integer sizes gives \(n_0=82,n_A=n_B=36\) as the minimum-cost allocation. Equal allocation of 50 participants to each group would meet the target but cost \(50+50+4(50)=300\) cost units. The higher-cost arm makes a larger shared control and smaller treatment groups preferable to that equal allocation.

Worked Allocations With Fixed and Arm-Specific Targets

Worked Example: A Fixed Control Size

Suppose the outcome variance is \(1\) squared outcome units, both contrasts require variance at most \(0.05\), and \(n_0=60\) is fixed. Let costs be \(c_0=2\), \(c_A=1\), and \(c_B=3\) cost units per participant. The treatment requirement for each arm is

n_j\geq\frac{1}{0.05-1/60} =\frac{1}{0.03333}\approx30.00.

Thus \(n_A=n_B=30\) meets the target exactly: \(1/60+1/30=0.05\). Total cost is \(2(60)+1(30)+3(30)=240\) cost units. An allocation of 60 participants to every group also meets the requirement, but costs \(2(60)+1(60)+3(60)=360\) cost units. With the control size held fixed, extra participants in either treatment arm are unnecessary for this variance target.

Worked Example: Different Precision Targets Across Arms

Now suppose \(n_0\) is a design choice, the common outcome variance is \(1\) squared outcome units, and the target variance is \(0.05\) for arm A’s contrast and \(0.04\) for arm B’s contrast. Let costs be \(c_0=2\), \(c_A=4\), and \(c_B=1\) cost units per participant. Compare two feasible allocations.

At \(n_0=50\), the minimum treatment sizes are

n_A=\left\lceil\frac{1}{0.05-1/50}\right\rceil =\lceil33.33\rceil=34, \qquad n_B=\left\lceil\frac{1}{0.04-1/50}\right\rceil =50.

The resulting variances are \(1/50+1/34\approx0.04941\) for A and \(1/50+1/50=0.04000\) for B, so both targets are met. Cost is \(2(50)+4(34)+1(50)=286\) cost units.

At \(n_0=60\), the minimum sizes become \(n_A=\lceil1/(0.05-1/60)\rceil=30\) and \(n_B=\lceil1/(0.04-1/60)\rceil=43\). Their variances are \(1/60+1/30=0.05000\) and \(1/60+1/43\approx0.03992\), respectively. Cost is \(2(60)+4(30)+1(43)=283\) cost units. This second allocation uses more control participants but costs less because it permits a substantial reduction in the expensive A group. It is a lower-cost option than the first, though these two comparisons alone do not establish the global integer minimum.

Choosing and Checking an Allocation

The calculation is most useful when the design team can state a clear target for each treatment-control comparison and obtain defensible estimates of costs and outcome variability. A practical comparison proceeds by generating candidate control sizes, calculating the minimum integer treatment sizes for each arm, checking every precision constraint, and summing costs. If a continuous optimum is available, it helps identify where to search, but it should not substitute for the integer comparison.

Specify assumptions and targets.
State the outcome variance model, the target variance for each contrast, and per-participant costs on a consistent basis.

Find feasible group sizes.
For each candidate \(n_0\), check that it exceeds every \(1/v_j\), then calculate and round each minimum \(n_j\) upward.

Verify and compare.
Substitute integer sizes back into \(1/n_0+1/n_j\leq v_j\) for every arm, calculate total cost, and compare feasible allocations.

A useful diagnostic is to recompute each achieved variance after rounding. Rounding the treatment sizes down can violate a precision target; rounding the continuous control size and all treatment sizes independently does not necessarily give the cheapest feasible integer design. Also inspect the sensitivity of the recommended allocation to uncertain costs and variance estimates. If small plausible changes reverse the preferred design, the apparent cost advantage may not be robust.

Equal allocation is simple and may be justified by other design considerations, but it is not automatically cost-efficient when treatment costs differ. A fixed control-to-treatment ratio is another convenient planning choice; it can be compared against the precision-constrained cost calculation rather than assumed optimal. The cost calculation here omits fixed setup costs, monitoring costs that vary by arm, treatment discontinuation, and any costs incurred at later stages. It also does not account for power, multiplicity, or the joint precision of all contrasts unless those requirements are explicitly incorporated into the targets.

Common Mistake and Limitations

Wrong reasoning: “Arm B costs four times as much, so it should automatically have one-quarter as many participants.” A cost ratio alone does not determine an allocation. Correction: First satisfy each arm’s precision constraint. The shared-control size affects all contrasts, while each arm’s target and outcome variance determine its required treatment size.
Wrong reasoning: “The continuous minimum gives the final group sizes.” Correction: Participants are whole people. Round treatment sizes upward, verify the constraints, and compare the costs of nearby feasible integer allocations.
Wrong reasoning: “A lower total participant cost means the design is better in every respect.” Correction: This objective includes only the specified participant costs and precision constraints. It says nothing by itself about clinical relevance, power, safety, recruitment feasibility, or other design goals.

The shared-control calculation is also inappropriate without modification if outcome variances differ materially between arms, observations are clustered, costs are not proportional to participant numbers, or the target is based on a different sampling model. In such settings, the variance and cost expressions must reflect the actual design. Precision targets should not be chosen merely to make an allocation appear inexpensive.

Key takeaway: With unequal treatment costs, compare feasible allocations by combining every contrast’s precision constraint with the cost of its arm and the common control. The preferred allocation is a property of the full set of targets, variances, and costs—not of any one treatment-arm cost ratio.

Check Your Understanding

Try each question, then check the worked solutions below.

For a variance of \(1\), target contrast variance \(0.04\), and fixed \(n_0=50\), what minimum treatment size meets the target?
With \(n_0=50,n_A=40,n_B=30\), and costs \(c_0=2,c_A=3,c_B=1\), calculate the total cost.
For variance \(1\), target variance \(0.05\), and \(n_0=30,n_T=20\), is the contrast target met? What treatment size is required at that control size?
When \(n_0\) and the precision target are fixed, does a higher treatment-arm cost change the minimum treatment size? Explain.

Solutions. (1) \(n_T\geq1/(0.04-1/50)=1/0.02=50\), so 50 participants are required. (2) Cost is \(2(50)+3(40)+1(30)=250\) cost units. (3) The variance is \(1/30+1/20=0.08333\), which exceeds \(0.05\). The required treatment size is \(1/(0.05-1/30)=60\), so 20 participants are insufficient. (4) No. At fixed \(n_0\), variance target, and outcome variance, the minimum size follows from the precision constraint; cost affects the expenditure and may affect which control size is optimal when \(n_0\) can vary.