Bernoulli Response Estimates in a Two-Arm DBCD

From Binary Outcomes to a DBCD Target

In a response-adaptive trial, the allocation rule can depend on estimates of how often participants respond under each treatment. With binary outcomes, each observed outcome is either a response or a nonresponse, so the arm-specific response estimate is a sample proportion. The important calculation is straightforward; the care lies in using the right denominator and keeping three quantities distinct: the estimated response rates, the target allocation shares, and the next-patient assignment probability.

This tutorial uses a deliberately specified target map for illustration. It is not a universal ethical or scientific rule. A real trial must choose and justify its map before applying it, using the trial’s objectives and constraints. The DBCD then uses the map’s output as the target it tries to track.

Definition: For arm $k$, let $n_k$ be the number of participants with an observed binary outcome and let $Y_{ki}=1$ denote a response and $Y_{ki}=0$ a nonresponse. The arm-specific Bernoulli response estimate is $$ \hat p_k=\frac{\sum_{i=1}^{n_k}Y_{ki}}{n_k}. $$ It is the number of observed responses in arm $k$ divided by the number of participants in that arm whose outcomes are observed.

A convenient working model is that, within each arm $k$, outcomes are independent Bernoulli random variables with a stable response probability $p_k$: $Y_{ki}\sim\operatorname{Bernoulli}(p_k)$. For fixed $n_k>0$, $E(\hat p_k)=p_k$; the same conclusion holds when the sample size and observation mechanism are independent of the outcomes. With outcome-dependent stopping or selection, this expectation need not equal $p_k$. The model is an assumption, not something guaranteed by randomization. In particular, it does not accommodate changing response probabilities over calendar time or unmodelled differences among participants.

For an adaptive update to use these estimates as intended, the relevant outcomes must be available when the update is made. The trial must also define how it handles missing or delayed outcomes. Excluding participants with unobserved responses from the denominator estimates the response proportion among participants with observed outcomes; it does not automatically estimate the response probability for all assigned participants if outcome observation is selective.

Calculate Each Arm’s Rate Separately

The numerator and denominator are both arm-specific. Do not divide an arm’s responses by the total number randomized, and do not pool the arms unless the target map explicitly calls for a pooled quantity. The following hypothetical data show the calculation.

Worked Example: Estimating Two Bernoulli Response Rates

In a hypothetical trial snapshot, 24 participants in arm 1 have observed binary outcomes, of whom 13 responded. In arm 2, 22 have observed outcomes, of whom 8 responded. Assume for this calculation that the outcomes are coded consistently as response $=1$, nonresponse $=0$, and that the observed outcomes are available before the next allocation update.

For arm 1, the response estimate is $$ \hat p_1=\frac{13}{24}=0.541666\ldots\approx 0.5417. $$ Thus the estimated response rate is 54.17% among the 24 arm 1 participants with observed outcomes.

For arm 2, $$ \hat p_2=\frac{8}{22}=0.363636\ldots\approx 0.3636. $$ The estimated response rate is 36.36% among the 22 arm 2 participants with observed outcomes. These are plug-in estimates of the two arm-specific response probabilities under the working Bernoulli model; the different denominators are appropriate because each estimate describes its own arm.

The pooled observed response proportion would be $21/46\approx0.4565$, but it is not a substitute for either $\hat p_1$ or $\hat p_2$. Pooling removes the arm distinction that the target map may need.

Apply the Prespecified Target Map

Suppose, solely for illustration, that the trial has prespecified a two-arm map from response estimates to the target share for arm 1:

\rho_1=0.25+0.5\frac{\hat p_1}{\hat p_1+\hat p_2}, \qquad \rho_2=1-\rho_1.

This map assigns arm 1 a target share between 0.25 and 0.75 whenever both response estimates are nonnegative and their sum is positive. It encodes a particular policy choice: arm 1’s target rises with its relative estimated response rate, but remains away from 0 and 1. The map does not itself specify the next patient’s randomization probability. That probability also depends on the current allocation proportions and the DBCD tuning parameter $\gamma$.

Worked Example: Transforming Response Estimates Into Target Shares

Use the estimates from the preceding hypothetical snapshot and the stated map. To avoid a rounding discrepancy, retain the exact fractions through the ratio: $$ \frac{\hat p_1}{\hat p_1+\hat p_2} =\frac{13/24}{13/24+8/22} =\frac{143/264}{239/264} =\frac{143}{239} \approx 0.5983. $$

Therefore, $$ \rho_1=0.25+0.5\left(\frac{143}{239}\right) \approx 0.5492, \qquad \rho_2=1-\rho_1\approx0.4508. $$ The map’s target is approximately 54.92% for arm 1 and 45.08% for arm 2. These shares are targets, not the proportions currently enrolled and not the next-patient assignment probability.

Use the Target in the Two-Arm DBCD Rule

Write the current allocation proportions as $\pi_1=n_1/(n_1+n_2)$ and $\pi_2=n_2/(n_1+n_2)$, where $n_1,n_2>0$ are the numbers currently allocated to the two arms. For a target $(\rho_1,\rho_2)$, the two-arm DBCD assignment probability for the next participant is

P(A_{\mathrm{next}}=1) = \frac{\rho_1(\rho_1/\pi_1)^\gamma} {\rho_1(\rho_1/\pi_1)^\gamma+\rho_2(\rho_2/\pi_2)^\gamma}, \qquad P(A_{\mathrm{next}}=2)=1-P(A_{\mathrm{next}}=1).

Here $A_{\mathrm{next}}$ is the next treatment assignment, and $\gamma\geq0$ controls the strength of correction for the difference between current proportions and targets. The rates affect this calculation through the target map. They do not replace $\pi_1,\pi_2$, which describe current allocation.

The ratio form makes the restoring direction transparent: $$ \frac{P(A_{\mathrm{next}}=1)}{P(A_{\mathrm{next}}=2)} = \frac{\rho_1}{\rho_2} \left(\frac{\rho_1/\pi_1}{\rho_2/\pi_2}\right)^\gamma. $$ If arm 1 is overrepresented relative to its target, then $\pi_1/\rho_1>\pi_2/\rho_2$, so $(\rho_1/\pi_1)/(\rho_2/\pi_2)<1$. For $\gamma>0$, this correction reduces the odds of assigning the next participant to arm 1 compared with the target odds alone. At $\gamma=0$, the assignment probabilities equal the target shares.

Worked Example: Calculating the Next Assignment Probability

Continue with the hypothetical counts $n_1=24$ and $n_2=22$, the target map above, and $\gamma=0.5$. Assume the 46 participants’ allocations and the observed response counts refer to the same trial snapshot. The current proportions are $$ \pi_1=\frac{24}{46}=\frac{12}{23}\approx0.5217, \qquad \pi_2=\frac{22}{46}=\frac{11}{23}\approx0.4783. $$

Using the unrounded target values from the exact rate calculation, the correction factors are approximately $$ \left(\frac{\rho_1}{\pi_1}\right)^{0.5} = \left(\frac{0.5491632}{0.5217391}\right)^{0.5} \approx1.0259, \qquad \left(\frac{\rho_2}{\pi_2}\right)^{0.5} = \left(\frac{0.4508368}{0.4782609}\right)^{0.5} \approx0.9709. $$

The corresponding unnormalized weights are approximately $$ w_1=0.5491632(1.0259)\approx0.5634, \qquad w_2=0.4508368(0.9709)\approx0.4377. $$ Retaining unrounded values through normalization gives $$ P(A_{\mathrm{next}}=1)=\frac{w_1}{w_1+w_2}\approx0.5628, \qquad P(A_{\mathrm{next}}=2)\approx0.4372. $$ Thus the next participant has approximately a 56.28% assignment probability to arm 1. The result is not 54.92%, because the DBCD corrects the target for the current allocation proportions.

A Second Snapshot: Keep the Quantities Distinct

Worked Example: A Different Response Pattern and Allocation

In another hypothetical snapshot, arm 1 has 7 responses among 10 observed outcomes, and arm 2 has 6 responses among 12 observed outcomes. Exactly 10 participants have been allocated to arm 1 and 12 to arm 2, and all have observed outcomes. Outcomes are coded as binary responses, both arms have at least one allocation, and the same illustrative target map is used.

The arm-specific estimates are $$ \hat p_1=\frac{7}{10}=0.7000, \qquad \hat p_2=\frac{6}{12}=0.5000. $$ The target is $$ \rho_1=0.25+0.5\frac{0.7000}{0.7000+0.5000} =0.25+0.5(0.5833\ldots) \approx0.5417, \qquad \rho_2\approx0.4583. $$

The current allocation proportions are $\pi_1=10/22=0.4545$ and $\pi_2=12/22=0.5455$. With $\gamma=0.5$, the correction factors are $$ \left(\frac{0.5416667}{0.4545455}\right)^{0.5}\approx1.0916, \qquad \left(\frac{0.4583333}{0.5454545}\right)^{0.5} =\sqrt{\frac{121}{144}} =\frac{11}{12}\approx0.9167. $$

The weights are approximately $w_1=0.5416667(1.0916)=0.5913$ and $w_2=0.4583333(0.9167)=0.4201$. Hence $$ P(A_{\mathrm{next}}=1)\approx\frac{0.5913}{0.5913+0.4201}=0.5846. $$ Arm 1’s next-patient probability exceeds its target share because arm 1 is currently underrepresented relative to that target. The comparison is between $0.4545$, its current share, and $0.5417$, its target—not between the response estimate $0.7000$ and the assignment probability.

Assumptions, Diagnostics, and Alternatives

Before using a calculated rate in the target map, check the data and the update conditions:

Verify coding and denominators. Confirm which value is a response, count responses within each arm, and use the number of observed outcomes in that arm as the denominator.
Check outcome availability. If outcomes are delayed, a rate based only on completed outcomes may lag behind current enrollment. If missingness differs by arm or response status, the observed-outcome proportion can be misleading for the intended response probability.
Check that each rate is defined. If an arm has no observed outcomes, its sample proportion is undefined. The trial’s initialization and update protocol must supply a valid way to proceed; setting an undefined rate to zero is not a calculation justified by the data.
Check the map’s domain. In the illustrative map, $\hat p_1+\hat p_2=0$ makes the ratio undefined. A trial must prespecify how its target map handles such a state rather than improvising a target after seeing the outcomes.
Assess instability. With small denominators, one additional response can substantially change $\hat p_k$, and hence the target. Record the counts as well as the rates; a decimal alone can conceal how little information it represents.

These estimates are not a substitute for a fixed target chosen without response data: a fixed target does not move as estimated outcomes change. Nor should the two arm-specific rates be replaced by a pooled proportion when the target depends on treatment-specific response. More elaborate approaches can model covariates or continuous outcomes, but they require a stated model and additional assumptions; a simple Bernoulli proportion should not be presented as if it adjusted for those features.

Common Mistake

Common mistake: “Arm 1’s response estimate is 0.5417, so the next participant should have a 54.17% chance of receiving arm 1.” This treats an estimated response rate as an allocation probability. The correction is to first apply the prespecified map to obtain $(\rho_1,\rho_2)$, then use the current allocation proportions and $\gamma$ in the DBCD rule. In the first worked example, the target for arm 1 is about 0.5492 and the next-patient probability is about 0.5628; neither is the arm 1 response estimate of 0.5417.

Key takeaway: For binary outcomes, estimate each arm’s response probability with its own response count divided by its own observed-outcome count. Apply the prespecified target map to those estimates, then use the resulting target—not the rates themselves—in the DBCD allocation rule.

Check Your Understanding

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

In a hypothetical arm with 18 observed outcomes and 11 responses, what is the estimated response rate? Give the calculation and interpret it.
Why is it generally incorrect to divide arm 1’s response count by the total number of participants in both arms when estimating arm 1’s response rate?
If the target share for arm 1 is 0.60 but its current allocation proportion is 0.70, what direction should the DBCD correction move the next-patient probability when $\gamma>0$?
What must be addressed if one arm has no observed outcomes or if the two estimated rates make the target map’s denominator zero?

Solutions. (1) $\hat p_1=11/18=0.6111$, so the estimated response rate is 61.11% among the 18 participants with observed outcomes. (2) The arm-specific estimate uses the number assigned to that arm with observed outcomes as its denominator; dividing by the total across arms does not estimate the response proportion within arm 1. (3) Arm 1 is overrepresented relative to its target, so for $\gamma>0$ the correction reduces its next-patient probability relative to the target odds. (4) The estimate or map output is undefined in that state; the trial needs a prespecified initialization or map rule for it, rather than silently inserting zero or choosing a target after observing the data.