Three Quantities That Answer Different Questions
In a doubly adaptive biased coin design (DBCD), an allocation target, the allocation proportions already observed, and the probability used for the next patient are related, but they are not interchangeable. Confusing them can make an adaptive design appear to have missed its objective simply because its finite-sample allocation has not yet reached the target, or can make a one-patient randomization probability sound like a promised treatment share.
Consider two treatments, A and B. At a given point in a trial, the design may specify that its desired allocation shares are 0.60 to A and 0.40 to B. If the first 20 patients include 12 assigned to A, the observed share for A is also 0.60. But the probability of assigning the twenty-first patient to A need not be 0.60. In a DBCD, the next assignment probability can respond to the current imbalance as well as the target.
The target answers, “What share does the design seek?” The observed proportion answers, “What share has occurred so far?” The conditional probability answers, “How likely is a particular arm to receive the next patient, given the history so far?” In each case the arm-specific quantities sum to one across all arms. A target may be fixed or may be updated by the design; the distinction among the three quantities remains the same.
How the Next Assignment Changes the Observed Share
Let \(I_{j,n+1}\) equal 1 if patient \(n+1\) is assigned to arm \(j\), and 0 otherwise. Then \(N_j(n+1)=N_j(n)+I_{j,n+1}\). Since the new denominator is \(n+1\), the updated observed proportion is
The indicator has conditional expectation \(E[I_{j,n+1}\mid\mathcal F_n]=q_{j,n+1}\). Taking the conditional expectation of the update therefore gives
This is a one-step identity: it does not require the next-patient probability to equal the target. It shows that the expected direction of the next change in the observed proportion depends on the difference between the next-patient probability and the current observed proportion. A probability above the current proportion pulls its conditional expectation upward; a probability below it pulls the expectation downward. The size of that expected change is divided by \(n+1\), so one patient has less impact on the overall proportion in a larger trial.
The actual next proportion is random. It can take only the values corresponding to assigning the next patient to one of the arms, whereas the conditional expectation averages over those possible assignments. This distinction is important even when the expected next proportion moves in the direction of the target.
Worked Example: A Next-Patient Probability Above the Target
Suppose a hypothetical two-arm DBCD has a fixed target of 0.60 for A and 0.40 for B. After 20 patients, 12 have received A and 8 have received B. For this history, suppose the design assigns patient 21 to A with probability 0.75 and to B with probability 0.25.
The current observed proportion for A is \(X_A(20)=12/20=0.6000\). The target for A is also 0.6000, but the next-patient probability is \(q_{A,21}=0.7500\). These equalities and inequality refer to different quantities.
If patient 21 is assigned to A, the new observed proportion is \(13/21=0.6190\). Its change from the previous proportion is exactly \(13/21-12/20=8/420\approx0.01905\). If patient 21 is assigned to B, the new A proportion is \(12/21=0.5714\).
The conditional expected A proportion is \(0.75(13/21)+0.25(12/21)=12.75/21=0.6071\). Equivalently, the expected change is \((0.75-0.6000)/21=0.007143\). Thus the expected proportion increases, but the next observed proportion is not guaranteed to be 0.6071 or to equal the target.
Why a DBCD Can Use a Probability Different From Its Target
A DBCD uses the current allocation history to influence the next randomization. In broad terms, if an arm’s observed share is below its target, the design may give it a next-patient probability above that share, helping correct the accumulated shortfall. If its observed share is above its target, the next probability may be lower than that share. This describes the purpose of feedback, not a particular allocation formula: the exact probability depends on the design rule.
Consequently, the next-patient probability may be more extreme than the target share. Assigning a probability of 0.75 to A does not mean that the target is 0.75; it can be a temporary response to the observed allocation history. Nor does a target of 0.60 imply that every patient, or the next patient in particular, must have a 0.60 chance of receiving A.
Worked Example: Correcting an Observed Excess
In a second hypothetical example, the target share for A remains 0.60, but 13 of the first 20 patients have been assigned to A. Suppose the DBCD gives the next patient a 0.30 probability of A and a 0.70 probability of B for this history.
The current A proportion is \(13/20=0.6500\), which exceeds the target. If the next patient receives A, the new share is \(14/21=0.6667\). If the next patient receives B, it is \(13/21=0.6190\). Neither outcome is the target exactly.
The conditional expected A proportion is \(0.30(14/21)+0.70(13/21)=13.30/21=0.6333\). The expected change is also \((0.30-0.6500)/21=-0.01667\). The probability of A is below the current observed share, so the conditional expectation moves downward. Yet it remains above 0.60 after one assignment; feedback adjusts allocation probabilities, not the past assignments.
A Short Justification and Its Assumptions
The update identity relies only on sequential assignment and the definition of the conditional probability; it does not rely on a response model or a special form of the DBCD. For the stated conditional expectation to be meaningful, the history \(\mathcal F_n\) must contain the information available to the design before patient \(n+1\) is assigned, and the probability \(q_{j,n+1}\) must be the probability actually used conditional on that history. Assignments across arms are mutually exclusive, so their indicators sum to one and their probabilities sum to one.
To verify the identity, write \(X_j(n+1)\) as the current count plus the next assignment indicator, divided by \(n+1\). Conditional on the history, \(N_j(n)\) and \(X_j(n)\) are already known, while the conditional mean of the indicator is \(q_{j,n+1}\). Substitution gives the result directly:
Because \(I_{j,n+1}\) is conditionally Bernoulli with probability \(q_{j,n+1}\), the conditional variance of the next observed share is \(q_{j,n+1}(1-q_{j,n+1})/(n+1)^2\). This also makes clear why its conditional expectation is not a promise about the realized allocation: the next assignment is still random unless the probability is 0 or 1.
Worked Example: A Temporary Probability Need Not Equal the Target
Suppose 45 of the first 100 hypothetical patients are assigned to A. The target share for A is 0.55. Assume the design’s current rule gives A probability 0.70 for patient 101. Treat these as supplied design quantities; no particular DBCD formula is being assumed.
The observed proportion is \(45/100=0.4500\), below the target of 0.5500. The probability for the next patient, 0.7000, is above both. If patient 101 receives A, the observed A proportion becomes \(46/101=0.4554\); if patient 101 receives B, it becomes \(45/101=0.4455\).
The conditional expected proportion is \((45+0.70)/101=45.70/101=0.4525\). The expected increase is \((0.70-0.45)/101=0.002475\). The example illustrates the roles of the quantities: the target specifies a desired share, the observed share records accumulated assignments, and the next probability governs one random assignment. The expected share can move toward the target without reaching it immediately.
Monitoring, Alternatives, and Limitations
A useful allocation report keeps the three quantities separate. At selected trial times, report the target share for each arm, the observed counts and proportions, and the next-patient assignment probabilities. In a multi-arm design, check that the target shares sum to one and that the probabilities actually used for the next assignment also sum to one. When the target can change, record its value at the relevant time rather than comparing the observed proportion with an unspecified target.
A large difference between the observed proportion and target is not, by itself, proof that the design has failed: finite-sample allocation is random, and the target need not be attained at each interim point. More informative checks include whether the probabilities were calculated from the intended history, whether they are recorded correctly, whether their values match the documented rule, and whether repeated simulations under the planned design show the intended allocation behavior. A probability that is always set equal to the target would remove the use of current allocation imbalance in the next assignment; it is not automatically a DBCD merely because the target was chosen adaptively.
| Approach or quantity | What it specifies | What it does not guarantee |
|---|---|---|
| Target allocation | The desired treatment shares under the design | The exact finite-sample proportions |
| Observed allocation proportion | The share of assignments already made to an arm | The probability of the next assignment |
| DBCD next-patient probability | The conditional chance of assigning the next patient to an arm | The target share or a particular realized next allocation |
| Fixed-probability randomization | The same assignment probabilities at each step | Feedback responding to current allocation imbalance |
| Fixed-count allocation | A prescribed number of assignments to each arm | The same patient-by-patient probability behavior as a DBCD |
These distinctions are about allocation, not about whether an ethical target is appropriate or how it should be estimated. The target must be specified by the design, and the DBCD rule determines how that target and the allocation history are translated into a next-patient probability. A target that is poorly chosen remains a poor target even if allocation tracks it accurately. Conversely, a sound target does not ensure that a small trial’s observed proportions will closely match it.
Check Your Understanding
Answer each question using the distinction among target shares, observed proportions, and next-patient probabilities. Solutions follow the questions.
- At \(n=10\), 6 patients have received A. The target share for A is 0.60, and the next-patient probability for A is 0.80. State each quantity and calculate the conditional expected A proportion after the next assignment.
- At \(n=20\), 14 patients have received A, and the next-patient probability for A is 0.20. What is the probability of receiving A, and what are the two possible observed A proportions after the next assignment?
- Explain why a next-patient probability of 0.75 does not establish that the target share is 0.75.
- For one arm, show why the conditional expected change in observed proportion is \((q_{j,n+1}-X_j(n))/(n+1)\).
Solutions. (1) The target is 0.6000, the observed proportion is \(6/10=0.6000\), and the next probability is 0.8000. The expected next proportion is \((6+0.80)/11=6.80/11=0.6182\). (2) The next probability for A is 0.20. If the next patient receives A, the proportion is \(15/21=0.7143\); otherwise it is \(14/21=0.6667\). (3) The target is the desired share, whereas 0.75 is a conditional probability for one assignment. A DBCD can use a probability different from the target in response to the allocation history. (4) From \(X_j(n+1)=X_j(n)+(I_{j,n+1}-X_j(n))/(n+1)\), take conditional expectations given \(\mathcal F_n\), and substitute \(E[I_{j,n+1}\mid\mathcal F_n]=q_{j,n+1}\). The result is \((q_{j,n+1}-X_j(n))/(n+1)\) for the conditional expected change.