Planning Control Availability for Future Arms

Why Plan Control Availability?

A future treatment arm can only use controls recruited while that arm is open and under conditions that make them eligible for its comparison. Therefore, a platform may have many control participants overall but relatively few controls available to a newly opened arm. Planning ahead means forecasting the control accrual that will occur during the arm’s relevant calendar window, not merely counting all controls already enrolled.

The forecast is useful for choosing an opening date, evaluating a randomization allocation, and identifying when the available concurrent control sample may be smaller than planned. It does not establish that the controls will be comparable in every respect: the earlier material on concurrent control sets and site activation explains why calendar time, eligibility, and site participation matter. Here the emphasis is on predicting how many eligible concurrent controls may accrue.

Define the Control-Accrual Window

Definition: For a planned arm, the control-accrual window is the period during which participants randomized to control can contribute to that arm’s concurrent comparison. Its start is generally the arm’s opening date; its end is the planned analysis cutoff or another prespecified date at which control accrual for that comparison stops. A control is counted only if the participant is eligible for the comparison and was recruited at a participating site during the window.

Let $N_0$ denote the number of eligible control participants accrued in that window. A basic planning quantity is $E(N_0)$, the expected number of controls. For a design that requires at least $m$ controls, the more decision-relevant quantity is $P(N_0\ge m)$. These are different summaries: a forecast can have an adequate mean while still giving a substantial probability of falling short.

The window should reflect the operational plan. If a site activates after the arm opens, it contributes only from its activation date. If the analysis cutoff is fixed, later recruitment does not contribute to the planned count. If the cutoff is uncertain, it is sensible to evaluate multiple plausible cutoffs rather than silently treating one date as certain.

From Recruitment Rates to Expected Controls

Suppose that, while a particular allocation is in force, eligible participants enter the platform at an average rate $\lambda(t)$ participants per unit of calendar time. Let $q_0(t)$ be the probability that a new participant is assigned to the shared control at time $t$. Under a stable allocation rule and a constant total recruitment rate, the expected control accrual over a window of length $T$ is

E(N_0)=\lambda q_0 T.

For example, if 10 eligible participants are expected per month and one in three is assigned to control, the expected control rate is $10(1/3)=3.33$ participants per month. Over four months, expected control accrual is $3.33(4)=13.3$. The expected count need not be an integer; the realized count is an integer.

When rates or allocations vary over time, the corresponding expectation is the accumulated control rate:

E(N_0)=\int_{t_{\mathrm{open}}}^{t_{\mathrm{cutoff}}}\lambda(t)q_0(t)\,dt.

With multiple sites, the total expected count can be written as a sum of site-specific contributions. A site contributes only when it is active, enrolling eligible participants, and assigning participants under the relevant control allocation. Thus a platform-wide average rate should not be applied to months in which many sites are not yet open.

Modelling Uncertainty in the Count

A common planning model treats eligible participant arrivals as a Poisson process: arrivals in non-overlapping time intervals are independent, and the number arriving in an interval of length $T$ has a Poisson distribution with mean $\lambda T$ when the rate is constant. If each arrival is independently assigned to control with probability $q_0$, then control arrivals are also Poisson, with mean $\lambda q_0T$. This assignment result is called Poisson thinning.

Planning result: Under a Poisson arrival model with independent randomization to control at a constant probability $q_0$, $N_0\sim\operatorname{Poisson}(\mu_0)$, where $\mu_0=\lambda q_0T$. Consequently, $$ P(N_0\ge m)=1-\sum_{k=0}^{m-1}e^{-\mu_0}\frac{\mu_0^k}{k!}. $$ The model assumes a suitably stable rate over the window, independent arrivals, and assignment probabilities matching the planned allocation. It is a planning approximation, not a guarantee of realized accrual.

The reasoning behind the mean is direct. Divide the window into short intervals of length $\Delta t$. In one such interval, the expected number of arrivals is approximately $\lambda\Delta t$; the expected number assigned to control is approximately $q_0\lambda\Delta t$. Adding these expectations across the window gives $\lambda q_0T$ for a constant rate, or the integral of $\lambda(t)q_0(t)$ when they change over time. Under the Poisson assumptions, thinning also gives the full count distribution, not just its mean.

If a fixed number $n$ of participants is expected or scheduled and each is independently randomized to control with probability $q_0$, a binomial model, $N_0\sim\operatorname{Binomial}(n,q_0)$, may be more appropriate than a Poisson model. The distinction matters: a fixed total imposes a ceiling of $n$ on the control count, while a Poisson model does not. A Poisson approximation can be useful when arrivals are naturally modelled over time and the expected count is moderate relative to the possible total, but its assumptions should be checked against operations.

Worked Example: Probability of Reaching a Minimum Count

Suppose a future arm is expected to be open for two months before the planned cutoff. Eligible participants arrive at a constant rate of 4 per month, and one half are assigned to control. Assume independent Poisson arrivals and independent randomization at the stated probability. What is the chance of accruing at least 5 eligible controls?

The control rate is $4(0.5)=2$ per month, so the Poisson mean over two months is

\mu_0=4(0.5)(2)=4.

Thus $N_0\sim\operatorname{Poisson}(4)$, and

\begin{aligned} P(N_0\ge5) &=1-P(N_0\le4)\\ &=1-e^{-4}\left(1+4+\frac{4^2}{2!}+\frac{4^3}{3!}+\frac{4^4}{4!}\right)\\ &=1-e^{-4}(34.3333)\\ &\approx 1-0.62884\\ &\approx 0.3712. \end{aligned}

Under this model, there is about a 37.1% chance of accruing at least 5 eligible controls within the two-month window. The expected count is 4, but that fact alone does not describe the chance of reaching the target. In particular, the probability of reaching 5 is less than one half despite a mean close to the target.

How Timing and Allocation Change Availability

Opening an arm earlier generally lengthens the period in which its concurrent controls can accrue, provided the sites and recruitment process are already operating. Delaying an opening may shorten the window, and a calendar period with lower recruitment can further reduce expected controls. The effect is captured by the integrated control rate, not by the number of months alone.

Allocation also matters. If the total arrival rate is fixed, assigning a smaller share to control lowers the expected control count. In a platform with several active arms, the control probability is determined by the randomization weights that apply at that time. It can change when arms enter or leave, so planning should use the allocation actually intended for each period rather than a single average ratio if the rules change.

Worked Example: Comparing Allocation Choices

A platform expects 12 eligible participants per month for four months after a planned arm opens. Compare two possible allocation choices: 1:1 between control and one experimental arm, and 2:1 experimental-to-control. Assume the total arrival rate is unaffected by the allocation and that the rates remain constant.

With 1:1 allocation, the control probability is $q_0=1/2$. The expected control count is

E(N_0)=12\left(\frac12\right)(4)=24.

With 2:1 experimental-to-control allocation, the control probability is $q_0=1/3$. The expected count is

E(N_0)=12\left(\frac13\right)(4)=16.

The 1:1 choice yields 8 more expected controls over the four-month period under these assumptions. If the design needs at least 20 controls, the first choice has an expected count above that threshold, while the second has an expected count below it. Neither comparison alone gives the probability of meeting the target; that requires a distributional model. Nor is the choice determined solely by control count: allocation must also be considered alongside the design’s scientific and operational requirements.

Uncertain Recruitment Rates and Scenario Planning

An accrual-rate estimate is itself uncertain. A single forecast based on the most likely rate can conceal the risk of a slow-recruitment period. One practical approach is to calculate control availability under several plausible scenarios—for example, slower, central, and faster recruitment—and report both expected counts and the probability of reaching the target in each scenario. If a probability distribution over scenarios is justified, the overall probability is the scenario-weighted average of the conditional probabilities.

This differs from simply inserting the average rate into one Poisson model. A mixture of possible rates generally has more variability than a Poisson distribution with the mixture’s average rate, so the chance of meeting a threshold can differ. Scenario probabilities should come from defensible planning information; they should not be chosen just to produce a preferred forecast.

Worked Example: Two Plausible Recruitment Scenarios

A three-month planning window has two plausible recruitment scenarios. With probability 0.6, the eligible participant rate is 4 per month; with probability 0.4, it is 1 per month. In either scenario, one half are assigned to control. Assume a Poisson arrival process within each scenario. Find the expected control count and the probability of accruing at least one control.

The scenario-specific control means are $4(0.5)(3)=6$ and $1(0.5)(3)=1.5$. Therefore, the unconditional expected count is

$$ E(N_0)=0.6(6)+0.4(1.5)=3.6+0.6=4.2. $$

For a Poisson random variable with mean $\mu$, the probability of at least one event is $1-P(N_0=0)=1-e^{-\mu}$. Averaging the scenario-specific probabilities gives

\begin{aligned} P(N_0\ge1) &=0.6(1-e^{-6})+0.4(1-e^{-1.5})\\ &\approx0.6(0.9975)+0.4(0.7769)\\ &\approx0.9093. \end{aligned}

The scenario-weighted probability is about 90.9%. If the average mean 4.2 were instead treated as one fixed Poisson mean, the result would be $1-e^{-4.2}\approx0.9850$, a different answer. The latter calculation ignores uncertainty about which rate scenario applies. The example shows why uncertainty in the recruitment process can matter even when an average expected count is available.

Checking the Forecast and Recognizing Its Limits

A forecast is only as useful as its inputs and assumptions. Review the accrual data by calendar period and site, and compare observed recruitment with the rates used in the plan. Look for changing site participation, pauses, eligibility changes, or substantial variation in monthly counts. These patterns may mean that a constant-rate model is inadequate. Also verify that the assumed allocation probability matches the randomization schedule during the entire window.

When the rate is changing predictably, a time-varying rate model can reflect that pattern. When it is not predictable, scenario analysis or simulation can show the consequences of plausible variation. A binomial calculation may suit a fixed planned total; a Poisson model suits event counts over time under its stated assumptions. Deterministic projections based only on an expected rate are simple and transparent, but they do not quantify the chance of shortfall.

Finally, the forecast concerns availability, not the validity of the comparison. A sufficiently large concurrent control set may still differ from the treatment group in relevant features, and a large total control count does not compensate for a short or mismatched calendar window. Conversely, a low forecast should prompt consideration of feasible timing or allocation changes, not automatic use of controls from outside the arm’s concurrent period.

Common Pitfalls

Counting all platform controls. The wrong reasoning is that every control recruited since platform launch is available to every future arm. Count only controls meeting the arm’s eligibility and calendar-window definition.
Reporting only the expected count. An expected count above a target does not mean the target is likely to be met with high probability. Calculate or simulate the count distribution when the probability of shortfall matters.
Using the wrong allocation probability. Applying a 1:1 control share throughout the window is incorrect if the planned randomization weights change as arms enter or leave. Calculate expected accrual period by period.
Confusing forecast precision with certainty. A numerical result from a Poisson model is conditional on its assumptions and estimated rate. It is not a guarantee of future recruitment.
Treating availability as comparability. Having enough concurrent controls does not by itself show that they are exchangeable with the arm’s participants on other relevant features.

Key takeaway: Define the arm’s eligible concurrent-control window first. Then combine the recruitment rate, site activity, and allocation probability over that window; quantify uncertainty when a minimum control count matters, and revisit the forecast as operational information changes.

Check Your Understanding

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

An arm will be open for 3 months. The eligible arrival rate is 6 participants per month, and the control allocation probability is $1/3$. Under a constant-rate model, what is the expected control count?
For a Poisson control count with mean 2, calculate the probability of at least one control.
Why might a binomial model be preferable to a Poisson model when the total number of participants is fixed?
A platform’s total recruitment rate is 10 per month, but only half the sites are expected to be active during the first month. Why might applying the full rate from the arm’s opening date overstate control availability?

Solutions. (1) The expected count is $6(1/3)(3)=6$ controls. (2) $P(N_0\ge1)=1-P(N_0=0)=1-e^{-2}\approx0.8647$, or about 86.5%. (3) With a fixed total $n$ and independent control assignment probability $q_0$, the count has a binomial distribution and cannot exceed $n$; a Poisson model does not impose that fixed upper limit. (4) The full platform rate assumes all sites are contributing throughout the window. If only half are active initially, the early rate may be lower, so the forecast should reflect the active sites and their expected contributions over time.