Covariate-Adjusted Allocation With Multiple Treatments

Why Three-Treatment Allocation Needs Care

When a trial offers three treatments, an incoming participant's covariate profile can inform the probability of assignment to each one. A profile-specific rule may favor treatments with better estimated responses for that profile, while still assigning a positive probability to every treatment. The calculation has two distinct parts: first construct and normalize a target across the three treatments; then, if desired, mix that target with an exploration distribution.

These steps matter because a response estimate is not itself an allocation probability. For example, three estimated response rates need not sum to one, and simply using them as assignment probabilities would not define a valid randomization rule. A valid rule must produce nonnegative probabilities that sum to one for every profile at every assignment decision.

The examples use a binary response, coded as success or failure, and assume that success is favorable. The same framework can be used with other response measures only after defining an appropriate score and explaining why larger scores should receive more allocation weight.

Definition: Let $x$ denote an incoming participant's covariate profile, and let $k\in\{A,B,C\}$ index the three treatments. Write $\hat p_k(x)$ for the estimated probability of success under treatment $k$ for profile $x$. For a prespecified power parameter $\gamma>0$, define the profile-specific target $$ q_k(x)=\frac{\hat p_k(x)^\gamma}{\sum_{j\in\{A,B,C\}}\hat p_j(x)^\gamma}. $$ With exploration fraction $\varepsilon$, where $0\leq\varepsilon\leq1$, define the assignment probability $$ \pi_k(x)=(1-\varepsilon)q_k(x)+\frac{\varepsilon}{3}. $$

Assumptions Behind the Rule

The allocation formula is a design rule, not a guarantee that a treatment is truly better. Its usefulness depends on how the response estimates are produced and on whether the rule matches the trial's aims. At minimum, the team must specify the response definition, the covariates used to define profiles, the estimation model, the value of $\gamma$, and the exploration fraction before applying the rule.

For the binary-response examples, suppose that responses are recorded consistently, treatment assignment is randomized using the stated probabilities, and the response model gives meaningful estimates of success probability for the profiles and treatments being considered. A model may borrow information across profiles, but its predictions should be calibrated well enough for allocation decisions. In particular, estimates should not be treated as reliable merely because a fitted model returns a number for every treatment-profile combination. Sparse data, poor overlap, or model misspecification can make those numbers unstable or misleading.

The power parameter controls how strongly the target responds to differences in estimated success. With $\gamma=1$, target probabilities are proportional to the estimated rates. Larger values put relatively more target weight on the highest estimated rate; smaller positive values make the target less concentrated. This choice does not correct biased estimates or establish that the most-favored treatment is best. It specifies how estimates influence assignment.

From Estimated Responses to Valid Probabilities

The normalization in $q_k(x)$ converts nonnegative treatment scores into a probability vector. The exploration mixture then reserves part of the assignment probability for uniform randomization. This is useful when response estimates are uncertain: even a treatment with a lower current estimate remains available for learning.

Result: If all three estimated scores $\hat p_k(x)^\gamma$ are nonnegative and their sum is positive, then $q_k(x)\geq0$ and $\sum_k q_k(x)=1$. Consequently, for $0\leq\varepsilon\leq1$, the mixed probabilities satisfy $\sum_k\pi_k(x)=1$ and $\pi_k(x)\geq\varepsilon/3$.

The justification is direct. Summing the target probabilities gives the sum of the three numerators divided by that same sum, which is one. Summing the mixed probabilities gives $(1-\varepsilon)\sum_kq_k(x)+3(\varepsilon/3)=1-\varepsilon+\varepsilon=1$. Since each target probability is nonnegative, each mixed probability is at least $\varepsilon/3$. If $\varepsilon>0$, this guarantees a positive assignment chance to every treatment, provided the rule is actually implemented as specified.

If every estimated rate is zero, the powered scores have zero sum and the target is undefined. The design must specify a fallback before the trial begins, such as using equal target probabilities for that profile until estimates become informative. Adding an arbitrary small number after seeing a problem is not a substitute for a prespecified rule.

Worked Example: Three Unequal Response Estimates

Suppose the current model estimates success probabilities for one profile as $\hat p_A(x)=0.56$, $\hat p_B(x)=0.65$, and $\hat p_C(x)=0.67$. Use $\gamma=2$ and $\varepsilon=0.12$. These are hypothetical estimates; assume the response is measured consistently and the model is considered adequate for this illustrative calculation.

First square each estimate to obtain the allocation scores: $0.56^2=0.3136$, $0.65^2=0.4225$, and $0.67^2=0.4489$. Their sum is $0.3136+0.4225+0.4489=1.1850$. Thus the unrounded targets are each score divided by $1.1850$.

\begin{aligned} q_A(x)&=\frac{0.3136}{1.1850},& q_B(x)&=\frac{0.4225}{1.1850},& q_C(x)&=\frac{0.4489}{1.1850}. \end{aligned}

Now apply $\pi_k(x)=0.88q_k(x)+0.04$, since $\varepsilon/3=0.12/3=0.04$. Calculate from the unrounded targets:

\begin{aligned} \pi_A(x)&=0.88\left(\frac{0.3136}{1.1850}\right)+0.04 \approx 0.2729,\\ \pi_B(x)&=0.88\left(\frac{0.4225}{1.1850}\right)+0.04 \approx 0.3538,\\ \pi_C(x)&=0.88\left(\frac{0.4489}{1.1850}\right)+0.04 \approx 0.3734. \end{aligned}

The probabilities sum to one before rounding. The displayed four-decimal values sum to $1.0001$, a harmless consequence of rounding each probability separately. Treatment C has the largest probability for this profile, but none of the three probabilities is below the exploration floor of 0.04.

Reading the Probabilities by Profile

A probability such as $\pi_B(x)=0.3538$ is conditional on the profile $x$ and on the trial history used to produce the current estimates. It means that, at this assignment decision, treatment B is selected with probability about 35.38% for an eligible participant with that profile, under the specified randomization rule. It does not mean that 35.38% of all participants in the completed trial must receive B, nor does it express certainty about B's true response rate.

Different profiles can have different response estimates and therefore different allocation probabilities. This is an intended feature of profile-specific allocation. For an overall allocation summary, profile frequencies also matter: the probability of receiving a treatment averaged across incoming profiles is a weighted average of the profile-specific probabilities, with weights given by the incoming profile distribution.

Worked Example: A Second Profile Gets a Different Allocation

Consider a second hypothetical profile $z$, with model-estimated success probabilities $0.72$, $0.60$, and $0.48$ for treatments A, B, and C. Assume the same $\gamma=2$ and $\varepsilon=0.12$. The rule's assumptions are the same as in the previous example; here we are illustrating how different predictions change the allocation.

The squared scores are $0.72^2=0.5184$, $0.60^2=0.3600$, and $0.48^2=0.2304$, with total $1.1088$. Therefore,

\begin{aligned} \pi_A(z)&=0.88\left(\frac{0.5184}{1.1088}\right)+0.04\approx0.4514,\\ \pi_B(z)&=0.88\left(\frac{0.3600}{1.1088}\right)+0.04\approx0.3257,\\ \pi_C(z)&=0.88\left(\frac{0.2304}{1.1088}\right)+0.04\approx0.2229. \end{aligned}

The unrounded probabilities sum to one, and the rounded values also sum to one at four decimal places. Treatment A is favored for profile $z$, whereas treatment C has the largest probability in the first example. This illustrates why one overall ranking of treatments need not describe a profile-specific rule.

Worked Example: Combining Two Profile-Specific Probabilities

Suppose, for a simple hypothetical planning calculation, 60% of incoming participants have profile $x$ and 40% have profile $z$. Using the unrounded probabilities from the preceding examples, the average assignment probability for treatment A across these incoming profiles is

0.60\left[0.88\left(\frac{0.3136}{1.1850}\right)+0.04\right] +0.40\left[0.88\left(\frac{0.5184}{1.1088}\right)+0.04\right] \approx 0.3443.

Thus the expected proportion assigned to A is about 34.43% if the profile mix remains 60% $x$ and 40% $z$, and if the displayed rule and estimates apply throughout. This is a weighted average, not a guarantee of the realized proportion in a finite trial. If the profile mix changes, this overall probability changes even when the profile-specific allocation rule stays fixed.

Diagnostics, Alternatives, and Limitations

Before and during use, check the mechanics and the inputs separately. The mechanical checks are that probabilities are finite, nonnegative, sum to one before rounding, and respect the intended exploration floor. Input checks concern whether profile-specific estimates are plausible and supported by data. Compare observed and predicted response rates where data permit, examine whether some treatment-profile combinations have little information, and investigate abrupt probability changes after modest updates. Such changes may indicate unstable estimates or an overly aggressive value of $\gamma$.

The approach differs from equal randomization, which assigns the same probabilities regardless of estimated response, and from a fixed allocation ratio, which does not adapt to profile-specific estimates. Equal or fixed probabilities can be simpler and less sensitive to model error; the adaptive rule can respond to estimated differences but inherits their uncertainty. A larger exploration fraction keeps assignment more spread out, while a smaller one makes probabilities track the target more closely. Neither choice eliminates misspecification, and a positive floor does not ensure enough observations in every profile-treatment combination.

A further limitation is that the rule optimizes a chosen score, not every possible trial objective. In the examples, the score is the estimated success rate raised to a power. This does not account automatically for treatment cost, safety, uncertainty, or a scientific goal of precise treatment comparisons. Those priorities must be incorporated explicitly into design decisions rather than inferred from the allocation probabilities.

Common pitfall: Rounding $q_k(x)$ before applying the mixture can change the reported final probability. For treatment B in the first example, using the rounded target $0.3565$ gives $0.88(0.3565)+0.04=0.35372$, which rounds to $0.3537$. The correct calculation uses the unrounded target: $0.88(0.4225/1.1850)+0.04=0.35375527\ldots$, which rounds to $0.3538$. Retain full precision through the calculation and round only the final probabilities.

Another common error is to interpret an allocation probability as an estimated response probability. They answer different questions: $\hat p_k(x)$ estimates response under treatment $k$ for profile $x$, whereas $\pi_k(x)$ is the chance the design assigns that treatment at the current decision. Keep the two quantities distinct in reports and implementation records.

Check Your Understanding

Answer each question, then compare with the brief solutions.

If three positive treatment scores are $0.20$, $0.30$, and $0.50$, what are their normalized targets?
With three treatments and $\varepsilon=0.15$, what minimum probability does the exploration mixture assign to each treatment?
Why can the four-decimal probabilities in the first worked example sum to 1.0001?
If two profiles occur in proportions 25% and 75%, how should their profile-specific probabilities for treatment A be combined to obtain the average assignment probability for A?
What is wrong with setting assignment probabilities equal to three estimated response rates without normalization?

Solutions. (1) The scores sum to $1.00$, so the targets are $0.20$, $0.30$, and $0.50$. (2) The floor is $\varepsilon/3=0.15/3=0.05$. (3) Each exact probability is rounded independently, so the displayed values need not sum exactly to one. (4) Use $0.25\pi_A(x)+0.75\pi_A(z)$; this weighted average assumes those are the relevant profile proportions. (5) The rates generally do not sum to one, so they are not a valid probability vector until converted by a specified normalization rule.