Plugging Response Estimates into a Prespecified Target Map

From Estimated Responses to Allocation Shares

A response-adaptive design may use the observed responses to estimate treatment-specific response parameters, then use a prespecified target map to determine the allocation shares it aims to track. These are two distinct calculations. The estimates enter the map; the map produces target shares; and the DBCD rule combines those shares with the current allocation proportions to determine the next patient's randomization probabilities.

The target map is part of the design, not a formula to select after seeing which allocation looks attractive. Its purpose might be to encode a specified allocation objective, but the numerical exercise here is narrower: given response estimates and a supplied map, calculate the resulting target shares accurately and then use them in the allocation rule.

Definition: Let $\hat{\theta}(n)$ denote the vector of response-parameter estimates available after $n$ patients, and let $g$ be a prespecified target map. If the map returns positive raw weights $w_j(\hat{\theta}(n))$ for $j=1,\ldots,K$, the plug-in target share for arm $j$ is $$ \hat{\rho}_j(n)=\frac{w_j(\hat{\theta}(n))}{\sum_{\ell=1}^{K}w_\ell(\hat{\theta}(n))}. $$ Thus $\hat{\rho}_j(n)\geq 0$ and $\sum_{j=1}^{K}\hat{\rho}_j(n)=1$. If the supplied map already returns shares that sum to one, no second normalization is needed.

The notation $\hat{\rho}_j(n)$ emphasizes that the target can change as the response estimates change. It is a target computed from the information available at time $n$, not a guarantee about the eventual allocation. Write $N_j(n)$ for the number assigned to arm $j$ by that time and $x_j(n)=N_j(n)/n$ for its current allocation proportion. When every $x_j(n)$ is positive, the usual multi-arm DBCD update with tuning parameter $\gamma\geq 0$ is $$ p_j(n+1)=\frac{\hat{\rho}_j(n)\left\{\hat{\rho}_j(n)/x_j(n)\right\}^{\gamma}}{\sum_{\ell=1}^{K}\hat{\rho}_\ell(n)\left\{\hat{\rho}_\ell(n)/x_\ell(n)\right\}^{\gamma}}. $$ Here $p_j(n+1)$ is the conditional probability of assigning the next patient to arm $j$, given the information available at time $n$. The target share and the next-patient probability are generally not the same.

What the Calculation Assumes

A plug-in calculation is meaningful only when the inputs and map are defined. The response estimates must correspond to the parameters expected by the supplied map, and the map must be evaluated using those estimates on its stated scale. For example, if the map takes response probabilities, supplying percentages such as 60 rather than proportions such as 0.60 changes its input and may produce an invalid result.

For the DBCD expression above, the target shares must be nonnegative with a positive total before normalization, and the current proportions used in the ratio correction must be positive. If the map returns a zero share or the design has an arm with no assignments, the stated calculation can involve a zero or undefined ratio. Initialization and boundary handling are design specifications; one must not silently divide by zero or replace a zero by an arbitrary small number.

The calculation treats the estimates available at time $n$ as inputs to a deterministic map. That does not mean the estimates are known without error: they depend on the observed responses and are random across possible trial histories. Nor does a numerical target calculation establish that the map is scientifically appropriate; it only evaluates the map the design has prespecified.

Worked Example: Two-Arm Target From Estimated Response Rates

Suppose a hypothetical trial has observed binary responses for two arms. Arm A has 12 responses among 20 patients, and arm B has 8 among 20. Use the sample response proportions as estimates and suppose the supplied target map assigns raw weight $\sqrt{\hat{\theta}_j}$ to arm $j$. Calculate the target shares and, if the current allocation proportions are $x_A=x_B=0.50$, the next-patient probabilities for $\gamma=1$.

The estimated response proportions are $\hat{\theta}_A=12/20=0.60$ and $\hat{\theta}_B=8/20=0.40$. Both are valid inputs on the probability scale. The raw map weights are $\sqrt{0.60}\approx0.7746$ and $\sqrt{0.40}\approx0.6325$, with total $1.4071$. Therefore,

\hat{\rho}_A=\frac{0.7746}{1.4071}\approx0.5505,\qquad \hat{\rho}_B=\frac{0.6325}{1.4071}\approx0.4495.

These shares are positive and sum to one, apart from rounding. For $\gamma=1$, the unnormalized DBCD weights are $\hat{\rho}_j^2/x_j$. Since both current proportions equal 0.50, the weights are approximately $0.5505^2/0.50=0.6061$ and $0.4495^2/0.50=0.4041$. Normalizing these weights gives $p_A(n+1)\approx0.6000$ and $p_B(n+1)\approx0.4000$. The next patient is therefore assigned using probabilities 0.6000 and 0.4000, not the target shares 0.5505 and 0.4495.

Why Normalization and Substitution Matter

A target map may be written in terms of raw weights rather than shares. Normalization converts those weights to a composition: each arm's value is divided by the sum across arms. This step is essential because raw weights need not add to one. It also provides an immediate arithmetic check: the normalized entries must be nonnegative and sum to one.

The DBCD update then applies a separate correction based on the current proportions. The factor $\{\hat{\rho}_j(n)/x_j(n)\}^{\gamma}$ increases the relative weight of an arm whose current proportion is below its target share and decreases the relative weight of one above its target share. This is a within-update correction, followed by another normalization across arms.

Worked Example: Three Estimated Responses and a Prespecified Map

In a hypothetical three-arm study, estimated response probabilities are $0.35$, $0.40$, and $0.45$, obtained from 7 of 20, 12 of 30, and 18 of 40 observed responses, respectively. The supplied map returns raw weight $1+\hat{\theta}_j$ for each arm. Calculate the target shares, then calculate DBCD probabilities when the current proportions are $(0.40,0.35,0.25)$ and $\gamma=1$.

The raw weights are $1.35$, $1.40$, and $1.45$. Their total is $4.20$, so the target shares are

(\hat{\rho}_1,\hat{\rho}_2,\hat{\rho}_3) =\left(\frac{1.35}{4.20},\frac{1.40}{4.20},\frac{1.45}{4.20}\right) \approx(0.3214,0.3333,0.3452).

The shares are positive and sum to 1.0000 to the displayed precision. The three current proportions also sum to one, and each is positive, so the specified ratio correction is defined. With $\gamma=1$, the DBCD weights are

\frac{0.3214^2}{0.40}\approx0.2583,\qquad \frac{0.3333^2}{0.35}\approx0.3175,\qquad \frac{0.3452^2}{0.25}\approx0.4768.

Their sum is approximately $1.0525$. Dividing each weight by that total gives next-patient probabilities approximately $(0.2454,0.3016,0.4530)$, which sum to 1.0000 after rounding. Arm 3 has a larger next-patient probability than its target share because its current allocation proportion, 0.25, is below its target share, approximately 0.3452. The probabilities are not simply copied from the target map.

Detailed Normalization Check

A useful audit is to retain full precision through the normalization and round only the final reported values. For example, suppose a supplied map evaluated at the response estimates returns the raw weights $578/2025$, $750/2025$, and $845/2025$. The common denominator cancels in normalization, but the sum must still include all three numerators:

\sum_{j=1}^{3}w_j=\frac{578+750+845}{2025}=\frac{2173}{2025}.

Thus the target shares are

(\hat{\rho}_A,\hat{\rho}_B,\hat{\rho}_C) =\left(\frac{578}{2173},\frac{750}{2173},\frac{845}{2173}\right) \approx(0.2660,0.3451,0.3889).

The fractions are the clearest check: their numerators sum to 2173, so the shares sum exactly to one. Rounding the first and third shares too early can conceal an arithmetic error; the correct four-decimal values are 0.2660, 0.3451, and 0.3889.

Worked Example: An Estimated-Parameter Map With Raw Weights

For a purely hypothetical illustration, let estimated response parameters be $\hat{\theta}_A=\sqrt{578/2025}$, $\hat{\theta}_B=\sqrt{750/2025}$, and $\hat{\theta}_C=\sqrt{845/2025}$. Suppose the prespecified map is $w_j(\hat{\theta})=\hat{\theta}_j^2$. These are positive estimates between zero and one, and the map therefore returns the three raw weights in the preceding calculation. Evaluate the map and report the allocation target.

Substitution gives $w_A=(\sqrt{578/2025})^2=578/2025$, $w_B=750/2025$, and $w_C=845/2025$. The sum is $2173/2025$, so dividing each raw weight by the sum yields $(578/2173,750/2173,845/2173)$, or approximately $(0.2660,0.3451,0.3889)$. These are target shares. They become next-patient probabilities only in the special case that the subsequent DBCD correction and normalization produce those same values.

Checks, Alternatives, and Limitations

Several checks catch common failures before an update is used:

Confirm that every response estimate is on the scale expected by the map, and that the map has been evaluated at the current estimates rather than at stale or rounded inputs.
Check that the raw weights have a positive sum. After normalization, verify nonnegativity and a sum of one, allowing only small discrepancies due to rounding.
For the DBCD step, verify that the current proportions sum to one and that every denominator $x_j(n)$ used in the ratio is positive.
Keep target shares, raw target weights, DBCD weights, and next-patient probabilities labelled separately. They are not interchangeable quantities.

A common mistake is to report the normalized target shares as the probabilities for the next patient. That reasoning skips the DBCD ratio correction. The correction depends on the current allocation state, so the next-patient probabilities can differ substantially from the target even when the map has been evaluated correctly. A second mistake is to normalize twice when the map already returns shares; doing so can change the intended target if the purported shares were incorrectly treated as raw weights.

One alternative is a fixed allocation rule that assigns using prespecified probabilities without plugging in response estimates. Such a rule does not adapt its target to estimated responses. A DBCD instead uses the plug-in target together with the current allocation proportions. The plug-in target is not itself an allocation mechanism: the DBCD correction supplies the step-by-step randomization probabilities.

The calculation also has limits. A correct evaluation does not establish that the estimates are stable or that the target map is suitable for the trial's scientific objective. Sparse data can make response estimates variable, and consequential map outputs can therefore move as new responses arrive. A numerical result should be reported with enough digits to verify the arithmetic, but excessive precision does not remove uncertainty in the estimates or justify interpreting a target share as a realized long-run proportion.

Estimate.
Calculate the response-parameter estimates from the available data using the prespecified estimator.

Evaluate and normalize.
Substitute the estimates into the supplied target map, sum its raw outputs, and normalize if they are not already shares.

Apply the allocation rule.
Use the target shares, current proportions, and $\gamma$ to calculate DBCD weights, then normalize those weights to obtain next-patient probabilities.

Check Your Understanding

Answer each question, then compare your work with the brief solutions below.

A target map returns raw weights 2, 3, and 5. What are the target shares?
A map expects response proportions, but an observed estimate is 18 responses among 30 patients. What value should be supplied as the estimate?
Why can the DBCD next-patient probability for an arm differ from its plug-in target share?
For raw weights $578/2025$, $750/2025$, and $845/2025$, what is the common denominator of the normalized shares?
What condition on current allocation proportions is needed to evaluate the displayed DBCD ratio correction directly?

Solutions. (1) Divide by the total 10, giving shares $(0.20,0.30,0.50)$. (2) Supply $18/30=0.60$, not 18 or 60. (3) The correction weights each target share according to its ratio to the current allocation proportion, then normalizes across arms. (4) The normalized shares have denominator $578+750+845=2173$, giving $(578/2173,750/2173,845/2173)$. (5) Each $x_j(n)$ appearing in the denominator must be positive; the current proportions must also represent a valid allocation composition summing to one.