Allocation Floors and Ceilings to Preserve Temporal Overlap

Why Bound Adaptive Allocation?

A response-adaptive rule can favor an arm as outcome information accumulates. If its allocation probabilities become very small for another arm, that arm may receive few or no participants late in enrollment. This can weaken comparisons among participants enrolled at similar calendar times, particularly when prognosis or treatment effects change over time. Earlier material on positivity under strong temporal allocation shifts motivates this concern; here the focus is on a practical design control: keeping every arm’s assignment probability within prespecified lower and upper bounds.

A floor limits how small an assignment probability can become, while a ceiling limits how large it can become. Strictly positive floors for every arm preserve a positive chance of assigning participants to each arm at each decision point, but the bounds do not force any particular realized number of assignments. Their value and cost therefore need to be assessed before enrollment, in relation to the trial’s adaptive objective and the amount of representation desired over relevant periods.

Definition: Suppose there are \(K\) treatment arms. Immediately before assigning participant \(t\), let \(\pi_{tk}=P(A_t=k\mid\mathcal H_{t-1})\) be the conditional probability of assigning that participant to arm \(k\), given the information \(\mathcal H_{t-1}\) available at that decision. An arm-specific allocation floor \(l_k\) and ceiling \(u_k\) constrain the rule by requiring \(l_k\leq\pi_{tk}\leq u_k\) for every arm \(k\) and every assignment decision \(t\). The probabilities must also satisfy \(\sum_{k=1}^K\pi_{tk}=1\).

The history \(\mathcal H_{t-1}\) can include responses already observed and other information the design permits the allocation rule to use. The bounds apply to the probability used at the moment of randomization, conditional on that history—not to the proportion assigned to an arm in the completed trial or a particular enrollment period. Unless stated otherwise, assume the bounds are fixed across assignment decisions and satisfy \(0\leq l_k\leq u_k\leq1\). Time-specific bounds can also be used, but the feasibility check must then hold separately at each decision time.

Feasibility and a Constrained Allocation

Bounds are useful only if there is at least one probability vector satisfying them. The following condition is both necessary and sufficient.

Theorem: If \(0\leq l_k\leq u_k\leq1\) for every arm \(k\), there exists a vector \((\pi_1,\ldots,\pi_K)\) such that \(\sum_{k=1}^K\pi_k=1\) and \(l_k\leq\pi_k\leq u_k\) for every \(k\) if and only if \(\sum_{k=1}^K l_k\leq1\leq\sum_{k=1}^K u_k\).

Necessity follows by summing the componentwise inequalities: any valid vector must have \(\sum_k l_k\leq\sum_k\pi_k=1\leq\sum_k u_k\). For sufficiency, write \(L=\sum_k l_k\) and \(U=\sum_k u_k\). If \(L=U\), the assumed inequalities imply \(L=U=1\), so the vector of lower bounds itself is feasible. If \(L<U\), define \(\lambda=(1-L)/(U-L)\). The inequalities \(L\leq1\leq U\) imply \(0\leq\lambda\leq1\). Set \(\pi_k=l_k+\lambda(u_k-l_k)\). Each component lies between \(l_k\) and \(u_k\), and

\sum_{k=1}^K\pi_k =L+\lambda(U-L) =L+(1-L) =1.

Thus a valid allocation vector exists. In particular, bounds outside the probability range cannot be treated as harmless: the probability-range restriction is part of the theorem’s hypothesis.

A response-adaptive algorithm may first compute a target vector \(q_t=(q_{t1},\ldots,q_{tK})\) from accrued response information. The target is assumed to be a probability vector: its entries are nonnegative and sum to one. If it violates the bounds, one principled adjustment is to select the feasible vector closest to it in squared Euclidean distance:

\boldsymbol{\pi}_t =\underset{\boldsymbol{\pi}}{\operatorname{argmin}} \sum_{k=1}^K(\pi_k-q_{tk})^2 \quad\text{subject to}\quad \sum_{k=1}^K\pi_k=1,\qquad l_k\leq\pi_k\leq u_k.

When the feasible set is nonempty, this optimization has a solution; the strictly convex objective makes that solution unique. This is one explicit way to define how constraints alter an adaptive target. Other prespecified rules can be used, but they should be stated clearly and should always return a vector satisfying the bounds and the sum-to-one requirement. Simply clipping entries and then renormalizing is not generally safe: renormalization can move an entry outside its bounds.

Worked Example: A Binary Target Hits Its Bounds

Suppose a two-arm response-adaptive rule proposes \(q_t=(0.92,0.08)\). The design requires both arms to have probabilities between \(0.25\) and \(0.75\). The bounds are feasible because \(0.25+0.25\leq1\leq0.75+0.75\).

With two arms, the probabilities must sum to one, so constraining the first arm to \([0.25,0.75]\) also constrains the second arm to \([0.25,0.75]\). The target assigns too much probability to arm 1 and too little to arm 2. The nearest feasible vector is \((0.75,0.25)\): it is on the boundary of each interval, and moving either component inward would move the vector farther from \((0.92,0.08)\) while preserving the sum.

At this decision, arm 1 still has the larger assignment probability, but the original target’s \(0.92\) probability is reduced to \(0.75\). The floor protects arm 2 from being assigned with probability below \(0.25\); the ceiling limits how strongly the rule can favor arm 1.

Worked Example: Projecting a Three-Arm Target

Suppose the adaptive target is \(q_t=(0.75,0.20,0.05)\), and every arm has bounds \(0.15\leq\pi_{tk}\leq0.60\). The bounds are feasible: their lower bounds sum to \(0.45\), and their upper bounds sum to \(1.80\).

For this squared-distance projection, the solution has the form \(\pi_{tk}=\min\{u_k,\max\{l_k,q_{tk}-\tau\}\}\), where the common value \(\tau\) is chosen so the probabilities sum to one. Here \(\tau=-0.05\) gives

\begin{aligned} \pi_{t1}&=\min\{0.60,\max\{0.15,0.75-(-0.05)\}\}=0.60,\\ \pi_{t2}&=\min\{0.60,\max\{0.15,0.20-(-0.05)\}\}=0.25,\\ \pi_{t3}&=\min\{0.60,\max\{0.15,0.05-(-0.05)\}\}=0.15. \end{aligned}

The entries sum to \(0.60+0.25+0.15=1.00\), and each is within its bounds. Arm 1’s target exceeded its ceiling, while arm 3’s target was below its floor. The middle arm absorbs the remaining probability needed to make a valid allocation. This illustrates that constraining one component can change more than that component: the final vector must still sum to one.

What a Floor Means for Temporal Representation

A floor provides a useful probability statement about representation within an enrollment window. Suppose arm \(k\) has \(\pi_{tk}\geq l_k\) at each of \(m\) consecutive assignment decisions. Conditional on the history before each decision, the probability of not assigning the next participant to arm \(k\) is at most \(1-l_k\). Applying this bound sequentially gives

P(\text{no participant in the window is assigned to arm }k) \leq(1-l_k)^m.

This inequality does not require assignments to have identical probabilities or to be independent: the probabilities may adapt to the observed history, provided the conditional floor holds at every decision. It does not, however, say that an arm must appear in every window. It bounds the probability of its absence. The expected number assigned to arm \(k\) in the window is at least \(m l_k\), by summing the lower bounds on the conditional assignment probabilities, but that expectation is not a guaranteed count.

Worked Example: Interpreting a Floor Over a Short Window

In a hypothetical three-arm trial, suppose every arm has a floor of \(0.10\), and consider a window of 12 consecutive participants. For any one arm, the probability that it receives no participant in that window is at most

(1-0.10)^{12}=0.90^{12}\approx0.2824.

Therefore the probability that this arm is represented at least once in the window is at least \(1-0.90^{12}=0.717570463519\ldots\), or approximately \(71.76\%\). Its expected number of assignments is at least \(12(0.10)=1.20\), but the realized number could still be zero. The bound is for each arm separately. It does not imply that all three arms will appear in the window with probability at least \(71.76\%\); a lower bound for simultaneous representation would require accounting for the possibility that any of the three arms is absent.

Designers can work backward from a desired absence probability. If the goal is to make the probability of no assignments to an arm in any specified window of \(m\) participants at most \(\delta\), a sufficient floor is

(1-l_k)^m\leq\delta \quad\Longleftrightarrow\quad l_k\geq1-\delta^{1/m}.

This is a design calculation, not a promise about the realized trial. The window length matters: a floor that gives a modest chance of representation in a short window can give a much stronger bound over a longer one. The chosen floor must also be compatible with all other arms’ bounds.

Choosing Bounds and Understanding Their Cost

Bounds encode a trade-off. Larger floors protect representation more strongly but restrict how far the allocation can move toward an arm favored by the adaptive target. Ceilings similarly prevent concentration on one arm. If the lower bounds nearly sum to one, little probability remains to redistribute; if the upper bounds sum to one, the allocation may be pinned to the upper limits. A useful design process is to specify the time window in which overlap matters, set a tolerable absence probability for that window, calculate candidate floors, and then check overall feasibility with the ceilings.

The bounds should be fixed in the design protocol or otherwise determined by a prespecified rule. They are not a substitute for deciding which response-adaptive target is scientifically appropriate. Nor do they guarantee balance in observed participant characteristics, equal sample sizes, or comparable outcomes. They constrain assignment probabilities, not the random assignments themselves.

Approach	What it provides	Main limitation
Fixed allocation	Constant assignment probabilities over enrollment, if the rule is implemented as specified	Does not adapt allocation probabilities to accruing response information
Unconstrained response adaptation	Allows the target allocation to move freely as the rule updates	May assign an arm a very small probability, weakening representation at that time
Response adaptation with floors and ceilings	Retains bounded conditional assignment probabilities while allowing some adaptation	Can pull the implemented probabilities away from the adaptive target and still gives no deterministic count guarantee

Bounds also do not remove calendar-time differences in outcomes. If prognosis changes during enrollment, keeping all assignment probabilities positive preserves an opportunity for concurrent comparisons, but it does not make participants enrolled at different times exchangeable or ensure that a particular outcome analysis is unbiased. The allocation probabilities should be recorded as implemented; analyzing a trial as though it used its unconstrained target would describe the wrong randomization rule.

Diagnostics and Common Pitfalls

Before enrollment, verify that each bound lies in \([0,1]\), that \(l_k\leq u_k\), and that the lower bounds sum to no more than one while the upper bounds sum to at least one. Check these conditions at every time point if bounds vary over time. During enrollment, review the recorded conditional probabilities against the specified limits and confirm that each implemented vector sums to one. Repeated values at a floor or ceiling can be an informative sign that the adaptation is being constrained often; it is not by itself evidence of an implementation error. Compare the constrained probabilities with the target probabilities to see how much adaptation the bounds are suppressing.

Mistake: treating a floor as a minimum observed count. A floor of \(0.10\) does not guarantee that 10% of participants in every period go to that arm. It guarantees a conditional probability of at least \(0.10\) at each assignment decision; the count remains random.
Mistake: checking only that lower bounds sum to at most one. This is necessary but not sufficient. The upper bounds must also sum to at least one, and every interval must be a valid probability interval with \(0\leq l_k\leq u_k\leq1\).
Mistake: clipping and renormalizing without rechecking. Renormalization can violate a floor or ceiling. Use a specified bounded-allocation procedure and verify its output satisfies all constraints.
Mistake: assuming positive assignment probabilities solve every temporal problem. Positive probabilities permit assignments to each arm; they do not ensure enough realized observations in every period or remove outcome drift.

A floor and ceiling are therefore best understood as controls on the range of adaptation. They preserve a minimum conditional chance of assignment and cap the maximum chance, while leaving random variation in realized enrollment. When a design requires a guaranteed minimum number of participants in each arm in every period, probability bounds alone are not that guarantee.

Check Your Understanding

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

Three arms have lower bounds \((0.20,0.20,0.20)\) and upper bounds \((0.50,0.70,0.80)\). Are the bounds feasible?
An arm has a conditional allocation floor of \(0.15\) for each of eight assignments. Give an upper bound on the probability it receives no assignments.
Explain why an expected count of at least 1.6 assignments in a window of 16 participants does not guarantee at least one assignment.
A valid adaptive target is clipped to the specified intervals, but its components then sum to \(0.94\). Is it a valid allocation vector? What must be done?

Solutions. (1) Yes. The lower bounds sum to \(0.60\leq1\), the upper bounds sum to \(2.00\geq1\), and each lower bound is no greater than its corresponding upper bound, with all bounds in \([0,1]\). (2) The probability is at most \((1-0.15)^8=0.85^8\approx0.2725\). (3) An expectation is an average over possible random assignments, not a lower bound on every realized count; zero assignments can occur even when the expected count exceeds one. (4) No. Allocation probabilities must sum exactly to one. The vector must be adjusted using a prespecified procedure that restores the sum while retaining every floor and ceiling; its feasibility should be checked before using the rule.