Why Calendar-Time Accrual Matters
An allocation probability describes how a participant is randomized at a particular point in a trial. It does not describe how many participants will be randomized at that probability. If enrollment is slow during one period and rapid during another, the allocation probability used during the rapid period can contribute far more participants to the final treatment counts. Consequently, a calendar-time summary that gives every period equal weight can misrepresent the assignments and outcomes that actually enter the trial.
This tutorial focuses on that connection: the number of participants accrued in each period weights the allocation probabilities and the accumulating outcome data. The examples use prespecified periods and binary outcomes, but the bookkeeping principle applies more broadly. It does not imply that a high-accrual period should automatically receive greater scientific importance; it means that more enrolled participants contribute more observations to a patient-level analysis.
A period is a convenient way to summarize the randomization rule, not a substitute for the participant-level process. The period boundaries might be interim analyses, months, or operationally defined enrollment windows. The essential requirements are that the period counts and the probabilities refer to the same participants, and that the probability used for a period is the one actually in force when its assignments are made.
Expected Counts: The Role of the Accrual Schedule
First consider a schedule fixed before enrollment. Assume each period has a specified number of participants and each participant is independently randomized to A with the stated probability. If \(X_{A,t}\) is the number assigned to A in period \(t\), then \(X_{A,t}\) has a binomial distribution with size \(N_t\) and probability \(q_{A,t}\). Linearity of expectation gives the total expected count.
The expected overall proportion assigned to A is therefore \(\sum_t N_tq_{A,t}/\sum_tN_t\), when the total accrual count is positive. This is an accrual-weighted average of the period probabilities. It differs from the simple arithmetic mean of those probabilities unless the periods have equal accrual or the probabilities happen to make the two averages coincide. Expected counts need not be integers: they describe averages over repetitions of the randomization, not a possible realized count in a single trial.
Worked Example: Two Periods With Different Accrual
Suppose a hypothetical two-arm trial plans to randomize 8 participants in period 1 and 32 in period 2. The allocation probability for A is \(0.50\) in the first period and \(0.70\) in the second. Assume both period counts are fixed in advance and the probabilities do not change within a period.
The expected number assigned to A in period 1 is \(8(0.50)=4\). In period 2 it is \(32(0.70)=22.4\). Thus the total expected A count is \(4+22.4=26.4\), and the expected proportion assigned to A overall is \(26.4/(8+32)=0.66\).
The unweighted mean of the two probabilities is \((0.50+0.70)/2=0.60\), which would imply 24 A assignments out of 40 if mistakenly used as the overall proportion. The discrepancy occurs because period 2 supplies four times as many participants as period 1. The calculation does not predict that this particular trial will assign exactly 26.4 participants to A; it gives the expected count under the stated randomization schedule.
Adaptive Probabilities and Conditional Expectations
In an adaptive trial, the allocation probability for a later period can depend on outcomes observed earlier. Let \(\mathcal{F}_{t-1}\) denote the information available immediately before the assignments in period \(t\), including the data used by the allocation rule. If \(N_t\) is fixed or known at that point and all participants in the period have the same probability \(q_{A,t}\), then, conditional on that information, the expected A count in the period is \(N_tq_{A,t}\). This is a period-by-period statement; it does not justify conditioning on every allocation probability eventually realized in the trial.
When \(N_t\) is fixed in advance, the expression becomes \(\sum_tN_tE(q_{A,t})\). If \(q_{A,t}\) is already determined by information available before period \(t\), then \(E(X_{A,t}\mid\mathcal{F}_{t-1})=N_tq_{A,t}\). Taking expectations gives the unconditional expected contribution from that period. If the period count is random or not known before assignments, use the more general participant-level representation: sum the assignment indicators for participants who are actually randomized, and take the expectation of that sum. In general, one cannot replace \(E(N_tq_{A,t})\) by \(E(N_t)E(q_{A,t})\) unless the relevant dependence conditions justify it.
A related error is to condition on the entire realized allocation schedule and then reuse the period-by-period expected-count formula as though that schedule were fixed independently of earlier assignments. In an adaptive design, later probabilities can reveal information about earlier outcomes and assignments. For example, suppose one participant in period 1 has probability \(0.5\) of receiving A, A always produces a response, and B never does. If period 2 assigns to A with probability 1 after a response and 0 otherwise, learning that the realized period-2 probability was 1 also reveals that the first participant received A. Conditional on that realized schedule, the first-period A count is 1, not \(0.5\). The correct unconditional expected total is \(E[\sum_tN_tq_{A,t}]\); conditional calculations must respect what the conditioning information reveals.
Worked Example: An Update Followed by Rapid Accrual
Consider a hypothetical binary-outcome trial with equal randomization in period 1 and 10 participants. Assume the trial observes 5 assignments to each arm, with 4 responses among the 5 A participants and 2 among the 5 B participants. For illustration, place independent \(\mathrm{Beta}(1,1)\) priors on the response probabilities \(\theta_A\) and \(\theta_B\). The posterior distributions are \(\mathrm{Beta}(5,2)\) for A and \(\mathrm{Beta}(3,4)\) for B, with posterior means \(5/7=0.7143\) and \(3/7=0.4286\).
Suppose the prespecified allocation rule for period 2 sets \(q_{A,2}=m_A/(m_A+m_B)\), where \(m_A\) and \(m_B\) are those posterior means. This is an illustrative rule, not a claim that this formula is optimal. It gives \(q_{A,2}=0.7143/(0.7143+0.4286)=0.625\). If 30 participants are randomized in period 2, the expected period-2 A count, conditional on the period-1 information, is \(30(0.625)=18.75\).
Given the stated period-1 assignments, the conditional expected total A count over both periods is \(5+18.75=23.75\) out of 40. The corresponding conditional expected overall proportion is \(23.75/40=0.5938\). The unweighted average of the period allocation probabilities, \((0.50+0.625)/2=0.5625\), is not the expected overall proportion: the second period contributes three times as many participants. This conditional calculation is valid because the period-1 assignments and outcomes are known at the update, while the period-2 expected count is taken over the upcoming assignments.
Accrual Also Weights the Evidence Used for Updating
For a binary response model with a stable response probability within each arm, patient-level Bernoulli observations contribute one success or failure each to the likelihood. With a \(\mathrm{Beta}(a,b)\) prior, \(s\) observed responses and \(f\) observed nonresponses yield a \(\mathrm{Beta}(a+s,b+f)\) posterior. The calendar period of an observation does not by itself change that update. However, periods with more enrolled participants usually contribute more observations, and therefore more successes and failures, to the posterior.
Worked Example: Pooling Period Data for a Binary Update
Suppose a hypothetical trial records the following completed outcomes. During a lower-accrual first period, each arm has 8 participants: A has 5 responses and B has 3. During a higher-accrual second period, each arm has 24 participants: A has 15 responses and B has 12. Assume all listed outcomes are available for the update, and use independent \(\mathrm{Beta}(1,1)\) priors with a common response probability per arm across periods.
For A, there are \(5+15=20\) responses and \((8-5)+(24-15)=12\) nonresponses. The posterior is \(\mathrm{Beta}(21,13)\), with mean \(21/(21+13)=21/34=0.6176\). For B, there are \(3+12=15\) responses and \((8-3)+(24-12)=17\) nonresponses. The posterior is \(\mathrm{Beta}(16,18)\), with mean \(16/34=0.4706\).
Under the illustrative rule \(q_A=m_A/(m_A+m_B)\), these posterior means give a next-period probability \((21/34)/[(21/34)+(16/34)]=21/37=0.5676\) for A. The calculation gives the high-accrual period its proper influence because it contributes 24 observations per arm, rather than the 8 per arm in the first period.
For comparison, the period-specific response rates are \(5/8=0.625\) and \(15/24=0.625\) for A, and \(3/8=0.375\) and \(12/24=0.500\) for B. Averaging B’s two rates equally gives \(0.4375\), not its pooled patient-level rate \(15/32=0.4688\). An equal-period average answers a different question: it gives each calendar period equal weight rather than each participant. It is not the likelihood-based update under the stated common-probability model.
Assumptions, Diagnostics, and Alternatives
These calculations rely on accurately recorded enrollment periods, assignment probabilities, and outcome availability. For the simple binomial count formula, the number assigned in a period must be known before that period’s assignments, and the stated probability must apply to those assignments. If assignment probabilities change participant by participant within a period, calculate expected counts using the individual probabilities instead of a single period probability. If responses are not yet available at an interim update, only outcomes available under the prespecified update rule should enter; enrollment and completed follow-up are not interchangeable counts.
The pooled Beta update above additionally assumes binary outcomes, the specified Beta prior, and a common response probability within each arm across the pooled periods. If response probabilities differ systematically by calendar period, a common-probability posterior can obscure that structure. A model with period effects or a period-stratified analysis may be more appropriate; the choice depends on the scientific model and the planned analysis. In either case, period labels and denominators should be retained so the data can be checked and the model’s assumptions assessed.
Useful checks include verifying that period counts sum to total enrollment, that arm-specific successes do not exceed their denominators, and that every allocation probability is between 0 and 1. Recalculate the weighted overall allocation probability from the period counts and probabilities, then compare it with the actual arm counts as a descriptive check—not as a requirement that realized counts equal their expectations. For outcome updates, verify the success and failure totals arm by arm. A large difference between an unweighted mean of period rates and the pooled rate is a warning that period sizes differ; it does not, by itself, establish that either analysis is scientifically correct.
| Approach | What it summarizes | Main caution |
|---|---|---|
| Fixed allocation probability | A probability applied uniformly across enrollment | Does not represent a rule that changes using accumulating data |
| Accrual-weighted allocation summary | The expected overall assignment proportion from period probabilities and counts | Must use the appropriate expectation when counts or probabilities are adaptive or random |
| Patient-level pooled outcome update | Successes and failures across all eligible participants under a common-response model | Relies on the common-response assumption and the chosen likelihood and prior |
| Equal-weight average of period rates | The arithmetic mean of the period-specific rates | Does not generally equal the participant-level pooled rate |
A common mistake is to say, “The allocation probabilities were 0.50 and 0.625, so the overall allocation probability was their average, 0.5625.” That reasoning treats a period with 10 participants as equivalent to a period with 30. The correction is to count participants: use \(N_tq_{A,t}\) for each period when the schedule is fixed or use the appropriate conditional or unconditional expectation when it is adaptive. Another mistake is to condition on the full realized adaptive schedule and still treat its probabilities as if they had been fixed in advance. The schedule may encode information about earlier assignments and outcomes, so that conditioning can change the expected earlier counts.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A fixed schedule has 12 participants at \(q_A=0.40\) and 28 at \(q_A=0.65\). What is the expected overall A proportion? Why is the equal-period average not the answer?
- In an adaptive design, what condition makes \(E(X_{A,t}\mid\mathcal{F}_{t-1})=N_tq_{A,t}\) valid for a period?
- For an arm with a \(\mathrm{Beta}(2,3)\) prior, 7 responses and 5 nonresponses are observed. State the posterior distribution and posterior mean.
- Why can conditioning on the whole realized allocation schedule invalidate a period-by-period expected-count calculation?
Solutions. (1) The expected A count is \(12(0.40)+28(0.65)=4.8+18.2=23\), so the expected overall proportion is \(23/40=0.575\). The equal-period average, \(0.525\), gives the two periods equal weight despite their unequal enrollment. (2) The period count must be fixed or known before its assignments, and \(q_{A,t}\) must be the probability in force based on information available before those assignments; if probabilities vary within the period, use the individual probabilities instead. (3) The posterior is \(\mathrm{Beta}(2+7,3+5)=\mathrm{Beta}(9,8)\), with mean \(9/17=0.5294\). (4) Later realized probabilities may depend on earlier outcomes and assignments, so observing the full schedule can reveal information about earlier assignments. The unconditional expected total is \(E[\sum_tN_tq_{A,t}]\), while a conditional expectation given the full schedule must account for what that schedule reveals.