Estimating Covariate-Adjusted Allocation Targets

From Fitted Response Rates to Allocation Targets

A covariate-adjusted allocation target is not observed directly. It is calculated from estimated response rates for each treatment arm and a stated allocation objective. The estimates therefore have two roles: they determine the target, and their uncertainty makes the target uncertain as well. Treating the target as a fixed number can conceal that second source of uncertainty.

This tutorial uses a particular, common objective to make the calculation precise: within a covariate profile, choose the arm allocation proportion that minimizes the approximate variance of the estimated difference in binary response rates, for a fixed total sample size. This is a Neyman allocation target. It is a precision-based target, not an ethical rule that automatically favors the arm with the higher response rate. Other objectives can imply different targets.

Definition: Let $Y$ be a binary response, with $Y=1$ denoting response, and let $a\in\{0,1\}$ index the treatment arm. For a baseline covariate profile $z$, let $p_a(z)=P(Y=1\mid A=a,Z=z)$. The Neyman allocation target for arm 1 in that profile is $$ q(z)=\frac{\sqrt{p_1(z)\{1-p_1(z)\}}}{\sqrt{p_0(z)\{1-p_0(z)\}}+\sqrt{p_1(z)\{1-p_1(z)\}}}. $$ The target for arm 0 is $1-q(z)$.

In practice, the rates are estimated from a fitted outcome model. For example, a logistic regression may specify $\operatorname{logit}\{p_a(z)\}=x_a(z)^\mathsf{T}\beta$, where $x_a(z)$ is the model design vector for arm $a$ and profile $z$, and $\beta$ is the parameter vector. With fitted parameters $\hat\beta$, the estimated rates are $\hat p_a(z)=\operatorname{expit}\{x_a(z)^\mathsf{T}\hat\beta\}$, and the estimated target $\hat q(z)$ is obtained by substituting those rates into the definition. The target is consequently a nonlinear function of estimated model parameters.

Why the Target Has This Form

Within one profile, suppose $n_1$ participants are assigned to arm 1 and $n_0$ to arm 0, with $n_0+n_1=n$. If responses are independent Bernoulli outcomes with probabilities $p_0(z)$ and $p_1(z)$, the variance of the difference in the two sample response proportions is $\frac{p_1(z)\{1-p_1(z)\}}{n_1}+\frac{p_0(z)\{1-p_0(z)\}}{n_0}$. The target minimizes this variance for a fixed $n$.

Justification: Write $v_a=p_a(z)\{1-p_a(z)\}$, the Bernoulli variance in arm $a$, and let $r=n_1/n$. The variance is proportional to $\frac{v_1}{r}+\frac{v_0}{1-r}$. Differentiating with respect to $r$ and setting the derivative to zero gives $$ -\frac{v_1}{r^2}+\frac{v_0}{(1-r)^2}=0 \quad\Longrightarrow\quad \frac{r}{1-r}=\sqrt{\frac{v_1}{v_0}}. $$ Solving for $r$ gives $r=\sqrt{v_1}/(\sqrt{v_0}+\sqrt{v_1})$, which is the stated target $q(z)$. The second derivative $2v_1/r^3+2v_0/(1-r)^3$ is positive for $0<r<1$, so this stationary point minimizes the variance.

This calculation assumes that the response probabilities are appropriate for the covariate profile, that responses are independent within arms conditional on the modelled information, and that both arms have positive response variance. It also treats the response rates as known. When they are estimated, the formula gives a plug-in estimate of the target, and uncertainty in the fitted rates must be considered separately.

Worked Example: Calculating a Target From Fitted Logistic Parameters

Suppose a hypothetical trial uses a logistic model with binary covariate $z$, where $z=1$ denotes a specified profile. The fitted coefficients are $\hat\beta_0=-1.3863$, $\hat\beta_Z=0.5390$, $\hat\beta_A=0.8000$, and $\hat\beta_{AZ}=0.2480$, in the model $\operatorname{logit}\{p_a(z)\}=\beta_0+\beta_Zz+\beta_Aa+\beta_{AZ}az$. Estimate the arm 1 target for profile $z=1$.

For arm 0, the fitted linear predictor is $-1.3863+0.5390=-0.8473$. Thus $\hat p_0(1)=1/\{1+\exp(0.8473)\}\approx0.3000$. For arm 1, the predictor is $-1.3863+0.5390+0.8000+0.2480=0.2007$, giving $\hat p_1(1)=1/\{1+\exp(-0.2007)\}\approx0.5500$.

The estimated Bernoulli standard deviations are $\sqrt{0.3000(0.7000)}=0.4583$ in arm 0 and $\sqrt{0.5500(0.4500)}=0.4975$ in arm 1. Therefore, $$ \hat q(1)=\frac{0.4975}{0.4583+0.4975} =\frac{0.4975}{0.9558} \approx 0.5205. $$

The estimated precision-based target is therefore about 52.1% to arm 1 and 47.9% to arm 0 for this profile. The estimate does not mean that arm 1 has a 52.1% response rate; its fitted response rate is 55.0%. The target is an allocation proportion derived from the two rates and the stated variance-minimization objective.

Quantifying Uncertainty in the Estimated Target

A fitted model typically supplies an estimated covariance matrix $\widehat V$ for $\hat\beta$. The delta method uses this matrix to approximate the variance of a smooth function of $\hat\beta$. For a profile $z$, define $g(\beta)=q(z)$ after expressing the two probabilities through the model. Then $\widehat{\operatorname{Var}}\{\hat q(z)\}\approx\nabla g(\hat\beta)^\mathsf{T}\widehat V\nabla g(\hat\beta)$, and the estimated standard error is the square root of this quantity.

The derivatives can be built in two stages. Let $\eta_a=x_a(z)^\mathsf{T}\beta$, $p_a=\operatorname{expit}(\eta_a)$, and $s_a=\sqrt{p_a(1-p_a)}$. Differentiating the target with respect to the rates gives

\frac{\partial q}{\partial p_1} =\frac{s_0}{(s_0+s_1)^2}\frac{1-2p_1}{2s_1}, \qquad \frac{\partial q}{\partial p_0} =-\frac{s_1}{(s_0+s_1)^2}\frac{1-2p_0}{2s_0}.

Since $\partial p_a/\partial\eta_a=p_a(1-p_a)$, the chain rule gives $\partial q/\partial\eta_a=(\partial q/\partial p_a)p_a(1-p_a)$. If the model has shared parameters, the gradient with respect to $\beta$ is formed by applying $\partial\eta_a/\partial\beta=x_a(z)$ and adding the arm-specific contributions. This accounts for covariance between the two fitted response rates; calculating two separate rate standard errors and ignoring their covariance can give a wrong target standard error.

Worked Example: A Delta-Method Interval for the Target

Continue with the hypothetical fitted probabilities $\hat p_0=0.3000$ and $\hat p_1=0.5500$. Suppose the fitted model implies estimated variances $0.0400$ and $0.0500$ for the two linear predictors $\hat\eta_0$ and $\hat\eta_1$, and covariance $0.0100$ between them. These are assumed to be obtained from the model's parameter covariance matrix for the specified profile.

At these fitted rates, $s_0=0.4583$, $s_1=0.4975$, and $s_0+s_1=0.9558$. The derivatives with respect to the probabilities are approximately $\partial q/\partial p_0=-0.2377$ and $\partial q/\partial p_1=-0.05045$. Multiplying by the logistic derivatives gives $\partial q/\partial\eta_0=(-0.2377)(0.3000)(0.7000)=-0.04992$ and $\partial q/\partial\eta_1=(-0.05045)(0.5500)(0.4500)=-0.01249$.

The delta-method variance is therefore $$ (-0.04992)^2(0.0400)+(-0.01249)^2(0.0500) +2(-0.04992)(-0.01249)(0.0100) \approx 0.0001200. $$ The estimated standard error is $\sqrt{0.0001200}=0.01095$. A normal-approximation 95% interval is $0.5205\pm1.96(0.01095)$, or approximately $(0.4990,0.5420)$.

Under the model and large-sample approximation, this interval describes uncertainty in the estimated target for the specified profile. It is not an interval for either arm's response rate. Its endpoints are allocation proportions, not percentages of participants who will respond.

Combining Targets Across Covariate Profiles

A target defined separately for each profile need not equal the target calculated from response rates averaged over profiles. If the intended trial population has profile proportions $w(z)$, where $w(z)\geq0$ and $\sum_z w(z)=1$, the implied overall arm 1 allocation proportion under profile-specific targets is $\sum_z w(z)q(z)$. Its plug-in estimate is $\sum_z\hat w(z)\hat q(z)$ when the profile proportions are also estimated, or $\sum_z w(z)\hat q(z)$ when they are treated as known.

Worked Example: Aggregating Two Profile-Specific Targets

Suppose a hypothetical target population consists of 60% of profile 1 and 40% of profile 2. The fitted response probabilities for profile 1 are $\hat p_0=0.2000$ and $\hat p_1=0.5000$. Its target is $\hat q_1=\sqrt{0.25}/\{\sqrt{0.16}+\sqrt{0.25}\}=0.5000/0.9000=0.5556$. For profile 2, let $\hat p_0=0.5000$ and $\hat p_1=0.6000$. Its target is $\hat q_2=\sqrt{0.24}/\{\sqrt{0.25}+\sqrt{0.24}\}=0.4899/(0.5000+0.4899)=0.4949$.

The overall target implied by applying each profile-specific target is $$ 0.6000(0.5556)+0.4000(0.4949) =0.3334+0.1980 =0.5313. $$ Thus the population-weighted target is approximately 53.1% to arm 1. This is a weighted average of the conditional targets. It is not generally the same as first averaging response rates across profiles and then calculating one target, because the square-root variance transformation is nonlinear.

Assumptions, Checks, and Alternatives

The calculations are useful only if the fitted probabilities and the target criterion are suitable for the design. The fitted outcome model should estimate response under each arm for the relevant baseline profiles. In a randomized trial, treatment assignment supports arm comparisons; a causal interpretation also requires consistent treatment definitions and no interference between participants. Predictions for profiles with little or no information in one arm are largely extrapolations, even if the model returns numerical probabilities. The profile definitions and the population weights used for aggregation must also match the intended trial population.

Several practical checks help reveal a fragile estimate. Examine whether both treatment arms contain observations across the important covariate profiles, whether predicted probabilities are near zero or one, and whether small changes in fitted rates cause large changes in the target. When response probabilities are near the boundaries, the Bernoulli variances are small and the normal approximation for the target may be poor. Sparse data, separation in logistic regression, or a target estimate near 0 or 1 can also make a symmetric delta-method interval misleading. In such cases, resampling the model fit and recalculating the target in each resample can provide a more informative uncertainty assessment, provided the resampling scheme respects the design and the model can be refitted reliably.

The Neyman target is not the only reasonable choice. Equal allocation does not rely on response-rate estimates and can be more stable, while potentially giving up some precision when variances differ. An allocation rule aimed at maximizing expected responses would require a different objective and may favor the arm with a higher fitted response probability; it should not be confused with the variance-minimizing target derived here. In particular, the estimated target alone does not specify how assignment probabilities should be updated during a trial.

Plug-in uncertainty is not optional. A target based on fitted rates is itself estimated, so report its uncertainty when it matters for design decisions.
Do not equate rate with allocation. A fitted response rate of 55% does not imply a 55% allocation target.
Do not collapse profiles without a reason. Profile-specific targets and their population-weighted average preserve distinctions that can disappear in pooled response rates.
Do not treat extrapolation as information. A model can produce a prediction for a profile even when the data provide little support for it.

Key takeaway: An estimated covariate-adjusted allocation target is a function of fitted arm-specific response probabilities and a specified design objective. Its uncertainty follows from uncertainty in the fitted model, and the target should be interpreted in the covariate profiles and population for which it was calculated.

Check Your Understanding

Answer each question, then compare with the solutions below.

For a profile, suppose the two fitted response probabilities are 0.4000 in arm 0 and 0.6000 in arm 1. Calculate the Neyman target for arm 1.
Why can uncertainty in the covariance between the two fitted response rates affect the standard error of the estimated target?
Two profiles have targets 0.60 and 0.45, and occur in proportions 0.25 and 0.75. What is the overall target implied by applying the conditional targets?
Does a Neyman allocation target necessarily allocate more participants to the arm with the higher fitted response rate? Explain.

Solutions. (1) Both Bernoulli variances are $0.4000(0.6000)=0.2400$, so both standard deviations are equal and the arm 1 target is $0.5000$. (2) The target depends jointly on both rates; the delta-method variance includes a covariance term, which may increase or decrease the total variance depending on the derivatives and covariance. (3) The weighted target is $0.25(0.60)+0.75(0.45)=0.4875$, or 48.75% to arm 1. (4) No. The target minimizes the approximate variance of the response-rate difference and depends on both arms' Bernoulli variances, not simply on which arm has the larger response probability.