Covariate-Adjusted Allocation Under Treatment Heterogeneity

Why Treatment Heterogeneity Changes Allocation

A treatment can have different effects for different patient profiles. For example, a binary baseline covariate may identify patients for whom the treatment improves response and another group for whom the benefit is small or reversed. If an allocation rule uses only a pooled treatment effect, it can hide that pattern: a positive average effect does not imply positive benefit in every profile.

The aim here is to make the profile-specific effect explicit and show how it enters a conditional allocation target. Such a target is a design choice based on estimated responses, not a universal optimum. The earlier material on conditional allocation targets and model-based updating motivates the setting; the key additional feature here is a treatment-by-covariate interaction.

Definition: Let \(A=1\) denote assignment to the experimental treatment and \(A=0\) assignment to control. Let \(Z\in\{0,1\}\) be a binary baseline covariate, and let \(Y\) be a response for which larger values are preferred. Define the profile-specific mean response as \(\mu_a(z)=E(Y\mid A=a,Z=z)\). The profile-specific treatment benefit on the mean-response scale is \(\Delta(z)=\mu_1(z)-\mu_0(z)\). Treatment heterogeneity across the two profiles is present on this scale when \(\Delta(1)\ne\Delta(0)\).

For a binary success outcome, \(\mu_a(z)\) is a success probability and \(\Delta(z)\) is a risk difference. Positive values favor treatment 1; negative values favor control. The scale matters: heterogeneity in risk differences is not identical to heterogeneity in odds ratios or another effect measure.

Modeling an Interaction and Estimating Profile Effects

A simple mean model that represents profile-specific effects is

E(Y\mid A,Z)=\beta_0+\beta_A A+\beta_Z Z+\beta_{AZ}AZ.

Under this identity-link model, the control mean at \(Z=0\) is \(\beta_0\), and the treatment mean there is \(\beta_0+\beta_A\), so \(\Delta(0)=\beta_A\). At \(Z=1\), the control mean is \(\beta_0+\beta_Z\), and the treatment mean is \(\beta_0+\beta_Z+\beta_A+\beta_{AZ}\), so

\Delta(1)=\beta_A+\beta_{AZ}, \qquad \Delta(1)-\Delta(0)=\beta_{AZ}.

Thus, \(\beta_{AZ}\) is the difference between the two risk differences when the model uses the identity link. For binary outcomes, this model should only be used when its fitted means are sensible probabilities between 0 and 1. A saturated estimate avoids imposing a common treatment effect: estimate each treatment-by-profile mean by its observed cell mean, then subtract within each profile. Regression models can instead use other links, but their interaction coefficients have scale-specific interpretations.

Assumptions for the basic calculation: \(Z\) is measured before assignment, so the profile is known when the allocation decision is made. Assignment probabilities are positive for both treatments in each profile, allowing responses to be observed under both treatments over time. The response definition and follow-up are comparable across treatment groups, and the outcomes used for updating are available before the next relevant allocation. A model-based target additionally relies on an adequate model for the conditional means; the saturated cell-mean calculation does not require a no-interaction assumption.

In an adaptive trial, these means are unknown. At a given allocation update, write their estimates as \(\hat{\mu}_a(z)\), and calculate \(\hat{\Delta}(z)=\hat{\mu}_1(z)-\hat{\mu}_0(z)\). The estimates are based on accumulated responses, so early profile-specific differences can be noisy. The allocation rule must state how estimates are converted into assignment probabilities rather than treating an estimated effect as a probability itself.

From Profile-Specific Benefit to a Target

One illustrative rule favors treatment 1 as its estimated benefit increases, while keeping both treatments available. Let \(\operatorname{expit}(u)=1/(1+\exp(-u))\), let \(0<\varepsilon<1/2\) be a minimum allocation probability, and let \(\kappa>0\) control how rapidly allocation responds to estimated benefit. Define the target probability of treatment 1 for profile \(z\) by

q_1(z)=\varepsilon+(1-2\varepsilon)\operatorname{expit}\!\left(\kappa\hat{\Delta}(z)\right), \qquad q_0(z)=1-q_1(z).

Here \(\kappa\) sets the sensitivity of the rule to the effect scale, and \(\varepsilon\) ensures that neither treatment’s target probability falls below \(\varepsilon\). The function is increasing in \(\hat{\Delta}(z)\): a larger estimated benefit raises \(q_1(z)\). When the estimated benefit is zero, \(\operatorname{expit}(0)=1/2\), so \(q_1(z)=1/2\). These targets are conditional on profile; they do not require that every patient receive the same allocation probability.

Worked Example: Opposite Effects Across Two Profiles

Suppose a hypothetical trial has a binary success outcome. At an update, the observed data are as follows. The cell means are sample proportions, and each cell has 40 participants.

Profile \(Z\)	Treatment 1 successes	Control successes	Estimated benefit
0	28/40 = 0.700	22/40 = 0.550	0.150
1	20/40 = 0.500	26/40 = 0.650	-0.150

For \(Z=0\), \(\hat{\Delta}(0)=28/40-22/40=0.700-0.550=0.150\), a 15-percentage-point estimated benefit. For \(Z=1\), \(\hat{\Delta}(1)=20/40-26/40=0.500-0.650=-0.150\), a 15-percentage-point estimated disadvantage. The estimated interaction contrast is \(-0.150-0.150=-0.300\), or a 30-percentage-point difference between the profile-specific risk differences.

Now use the illustrative rule with \(\varepsilon=0.10\) and \(\kappa=4\). For profile 0,

q_1(0)=0.10+0.80\operatorname{expit}(4\times0.150) =0.10+\frac{0.80}{1+\exp(-0.60)} \approx0.6165, \qquad q_0(0)\approx0.3835.

For profile 1, the sign reverses:

q_1(1)=0.10+0.80\operatorname{expit}(4\times-0.150) =0.10+\frac{0.80}{1+\exp(0.60)} \approx0.3835, \qquad q_0(1)\approx0.6165.

The rule therefore targets more treatment 1 in profile 0 and more control in profile 1, while retaining at least 10% allocation to either arm in both profiles. This is a hypothetical design calculation, not evidence that either treatment is truly superior in a population.

Why the Interaction Must Be Kept in the Calculation

The identity-link model provides a short justification for using profile-specific targets. If the interaction is omitted, the model asserts \(\beta_{AZ}=0\), which forces \(\Delta(1)=\Delta(0)=\beta_A\). That restriction can be inappropriate when the treatment effect differs by profile. In the example, the pooled response proportions are both \(48/80=0.600\), giving a pooled risk difference of zero, even though the two profile-specific differences are \(0.150\) and \(-0.150\). A pooled summary alone would miss the opposing pattern.

A conditional rule applies the same mapping to each profile’s own estimated effect. The rule’s direction follows from the derivative

\frac{d q_1(z)}{d\hat{\Delta}(z)} =(1-2\varepsilon)\kappa \operatorname{expit}\!\left(\kappa\hat{\Delta}(z)\right) \left[1-\operatorname{expit}\!\left(\kappa\hat{\Delta}(z)\right)\right]>0,

because \(1-2\varepsilon>0\), \(\kappa>0\), and the expit function is strictly between 0 and 1. Therefore, increasing the estimated benefit cannot decrease the target probability for treatment 1. Also, \(\operatorname{expit}(u)+\operatorname{expit}(-u)=1\), so the rule gives complementary treatment targets when the estimated effects are equal in magnitude and opposite in sign.

Worked Example: Interactions on the Logistic Scale

Suppose a hypothetical fitted logistic model for success is

\operatorname{logit}P(Y=1\mid A,Z) =-0.4+0.5A+0.8Z-1.1AZ.

At \(Z=0\), the control log odds are \(-0.4\) and the treatment log odds are \(-0.4+0.5=0.1\). The corresponding success probabilities are \(\operatorname{expit}(-0.4)\approx0.4013\) and \(\operatorname{expit}(0.1)\approx0.5250\). Thus the estimated risk difference is \(0.5250-0.4013=0.1237\).

At \(Z=1\), the control log odds are \(-0.4+0.8=0.4\). The treatment log odds are \(-0.4+0.5+0.8-1.1=-0.2\). The probabilities are \(\operatorname{expit}(0.4)\approx0.5987\) and \(\operatorname{expit}(-0.2)\approx0.4502\), giving a risk difference of \(0.4502-0.5987=-0.1485\). The fitted model therefore predicts a positive risk difference in profile 0 and a negative one in profile 1.

The interaction coefficient \(-1.1\) is not itself the difference between these risk differences. In this logistic model, the treatment log-odds contrast is \(0.5\) at \(Z=0\) and \(0.5-1.1=-0.6\) at \(Z=1\); the treatment odds ratios are \(\exp(0.5)\approx1.649\) and \(\exp(-0.6)\approx0.5488\). The interaction coefficient is the difference in log odds ratios. To construct targets on the risk-difference scale, first compute the profile-specific probabilities and subtract them, as done above.

Worked Example: Sparse Profile Data and an Uncertain Target

Consider another hypothetical update with 10 participants in each treatment-by-profile cell. In profile 0, treatment 1 has 6 successes and control has 5; in profile 1, treatment 1 has 7 successes and control has 4. The estimated benefits are \(0.600-0.500=0.100\) for profile 0 and \(0.700-0.400=0.300\) for profile 1.

With \(\varepsilon=0.10\) and \(\kappa=4\), the targets are

q_1(0)=0.10+0.80\operatorname{expit}(0.40)\approx0.5790, \qquad q_1(1)=0.10+0.80\operatorname{expit}(1.20)\approx0.7148.

The larger estimate in profile 1 produces the larger treatment target. But the estimates are imprecise. For independent binomial cell proportions, an approximate standard error for a difference is

SE\{\hat{\Delta}(z)\} \approx \sqrt{\frac{\hat{p}_1(z)[1-\hat{p}_1(z)]}{n_1(z)} +\frac{\hat{p}_0(z)[1-\hat{p}_0(z)]}{n_0(z)}}.

For profile 1, this is \(\sqrt{0.7(0.3)/10+0.4(0.6)/10}=\sqrt{0.045}\approx0.2121\), comparable in size to the estimated benefit of 0.300. The computed target is therefore a direct consequence of the rule and current estimate; it should not be read as a precise estimate of a lasting treatment advantage.

Diagnostics, Alternatives, and Limitations

Before relying on interaction-aware targets, check whether the profile-specific estimates are supported by the design and data. Useful checks include:

Cell counts and response availability. Very small treatment-by-profile cells make sample proportions volatile. If outcomes are delayed, the most recently assigned participants may not yet contribute responses.
Target extremes. Targets repeatedly near the exploration bounds can signal a large estimated effect, a noisy estimate, or an overly sensitive mapping. The bounds prevent deterministic allocation but do not make the estimated effect reliable.
Model and scale consistency. Confirm that the interaction is included when warranted, that the response scale matches the target interpretation, and that fitted probabilities remain valid if an identity-link model is used.
Profile definition and timing. The covariate must be measured before assignment and coded consistently. A profile that is rare or changes definition during the trial may not support stable updates.

The main alternatives answer different design aims. A pooled response-adaptive rule uses one overall effect and can overlook opposing profile effects. An interaction-aware rule estimates separate effects and can tailor allocation, but it needs sufficient response information in each profile. A fixed balanced allocation within profiles prioritizes treatment balance and does not chase early estimated benefits. Minimization can improve covariate balance, as covered in the earlier material on balancing prognosis, but covariate balance alone does not estimate or act on treatment heterogeneity.

Approach	What drives allocation?	Main trade-off
Pooled response adaptation	Overall estimated treatment response	Simpler, but can conceal opposing profile effects
Interaction-aware adaptation	Estimated treatment effect within each profile	Tailored targets, but more sensitive to sparse cells and model error
Fixed balance within profiles	Predetermined allocation ratio in each profile	Does not chase noisy response estimates, but does not adapt to estimated benefit
Covariate balancing or minimization	Current imbalance in baseline covariates	Addresses balance rather than profile-specific treatment response

No target formula eliminates uncertainty or guarantees that the treatment with the higher estimated response is truly better for a profile. Adaptive allocation also changes which responses are observed under each treatment over time, so estimates must be based on the design’s actual assignment and updating process. Exploration, careful model specification, and regular review of profile-specific data are protections, not substitutes for adequate information.

Common Mistake

A common error is to fit or calculate one pooled treatment effect, then assume that the resulting allocation target applies equally to every patient. That reasoning silently imposes no treatment-by-covariate interaction. The correction is to estimate \(\mu_1(z)\) and \(\mu_0(z)\) within each profile, compute \(\hat{\Delta}(z)\), and apply the stated target rule separately for each \(z\). A second error is to treat a logistic interaction coefficient as a risk-difference interaction; compute profile-specific predicted probabilities first when the allocation rule is defined on a risk-difference scale.

Key takeaway: Treatment heterogeneity is about how the treatment contrast changes across profiles. Estimate the contrast on a clearly stated scale within each profile, then map those estimates to conditional allocation targets while preserving exploration and acknowledging uncertainty.

Check Your Understanding

Answer each question using the definitions and calculations above. Brief solutions follow.

If \(\hat{\Delta}(0)=0.08\) and \(\hat{\Delta}(1)=-0.12\), what is the estimated interaction contrast on the risk-difference scale?
With \(\varepsilon=0.10\), what target probability does the stated rule assign to treatment 1 when \(\hat{\Delta}(z)=0\)? Explain why.
In the logistic model of the worked example, what does the interaction coefficient represent, and why is it not the risk-difference interaction?
Why can a pooled risk difference of zero coexist with meaningful profile-specific estimated effects?
Name one reason to retain a positive exploration floor when profile-specific effects are uncertain.

Solutions. (1) The contrast is \(\hat{\Delta}(1)-\hat{\Delta}(0)=-0.12-0.08=-0.20\), or a 20-percentage-point smaller risk difference in profile 1. (2) It is \(0.10+0.80\operatorname{expit}(0)=0.10+0.80(0.5)=0.50\). (3) It is the difference between treatment log-odds contrasts across profiles; risk differences require subtracting profile-specific predicted probabilities, which depend on baseline risk. (4) Profile-specific effects can have opposite signs and cancel when averaged over profiles. (5) Continued assignment to both treatments supplies responses in each profile and limits the chance that early noisy estimates make allocation nearly deterministic.