How Adaptive Feedback Links Treatment Assignment to Enrollment Time

Why Feedback Can Connect Assignment to Enrollment Time

In a response-adaptive trial, information from participants who have already enrolled can change the probability used to assign later participants. The assignment of a participant is randomized at that decision, but the probability used for the randomization can depend on the trial’s accumulated history. As that history changes over calendar time, the distribution of treatment assignments can change too.

This creates an important distinction: randomization at each decision does not mean that treatment assignment is independent of enrollment time or of earlier assignments. Earlier assignments can affect earlier responses; those responses can affect later allocation probabilities; and those probabilities govern later random assignments. The resulting relationship is a feature of the adaptive design, not by itself evidence of an error in the randomization procedure.

The earlier material on calendar-time drift concerns how outcomes may change as a trial proceeds. Here the focus is narrower: how the feedback mechanism itself can make assigned treatment related to when a participant enrolled. That relationship matters because calendar time can also track changes in prognosis, clinical practice, or the enrolled population.

History-Conditional Randomization

Definition: Index participants by enrollment order $i=1,\ldots,n$. Let $A_i=1$ if participant $i$ is assigned to the treatment of interest and $A_i=0$ otherwise. Let $H_i$ denote the information available immediately before participant $i$'s assignment, including any responses already observed and any other variables used by the allocation rule. The conditional treatment allocation probability is $$ q_i=P(A_i=1\mid H_i). $$ For a Bernoulli assignment draw with this probability, $E(A_i\mid H_i)=q_i$. In an adaptive design, $H_i$, and therefore $q_i$, can vary across enrollment decisions.

The timing in this definition is essential. $H_i$ contains information available before assignment $i$, not responses that will only become available afterward. If the algorithm uses delayed responses, the history includes the responses available at that decision, not every outcome that has occurred but remains pending. In an implemented trial, the assigned probability should be the probability actually used at that decision, based on the recorded history.

Because $q_i$ is determined by the pre-assignment history, it is generally a random variable across possible trial paths. The conditional statement $E(A_i\mid H_i)=q_i$ describes the randomization at decision $i$. Taking expectations over possible histories gives the unconditional identity

E(A_i)=E\{E(A_i\mid H_i)\}=E(q_i).

This identity does not say that $A_i$ is independent of earlier assignments, earlier responses, or calendar time. It also does not justify conditioning on the entire realized sequence of allocation probabilities. Later probabilities in that sequence may reveal information about earlier assignments because the algorithm used earlier responses to set them.

Worked Example: A Feedback Rule Changes Later Probabilities

Suppose a hypothetical trial assigns each new participant to treatment with probability 0.50 at the first enrollment. At each later enrollment, it uses probability 0.80 if the most recently observed response was a success and probability 0.20 if it was a failure. Consider this possible trial path: the first assignment is treatment, its response is a success, the second assignment is treatment and its response is a success, the third assignment is control and its response is a failure, and the fourth assignment is control.

At enrollment 1, $q_1=0.50$. The observed success makes $q_2=0.80$. The second success makes $q_3=0.80$, and the third participant’s failure makes $q_4=0.20$. Thus the probabilities along this particular path are $0.50, 0.80, 0.80, 0.20$, while the assignments are $1,1,0,0$.

Each assignment is still drawn according to the probability available at its decision. For example, conditional on the history before enrollment 3, $P(A_3=1\mid H_3)=0.80$; the fact that the realized assignment was control does not contradict that probability. The path illustrates how earlier feedback can change later assignment probabilities, not that the algorithm determines assignments without randomization.

Why the Entire Probability Sequence Is Not a Valid Conditioning Set

A tempting but generally incorrect calculation is to treat a realized sequence of adaptive probabilities as if it had been fixed in advance. The problem is that the sequence can encode information about the assignments and responses that produced it. The safe conditional calculation uses the history available at the particular decision; the unconditional calculation averages over possible histories.

Worked Example: A Later Probability Reveals an Earlier Assignment

Consider a hypothetical two-participant trial. The first participant is assigned treatment with $q_1=0.50$. Suppose, only to make the dependence transparent, that the first participant’s response $Y_1$ equals $A_1$. The second participant’s treatment probability is $q_2=0.80$ if $Y_1=1$, and $q_2=0.20$ if $Y_1=0$.

Before the first assignment, $P(A_1=1)=0.50$. If $A_1=1$, then $Y_1=1$ and $q_2=0.80$; if $A_1=0$, then $Y_1=0$ and $q_2=0.20$. Consequently, $$ E(q_2)=0.50(0.80)+0.50(0.20)=0.50, $$ and hence $E(A_2)=0.50$. This is the unconditional probability averaged across the two possible first-participant histories.

But conditional on observing $q_2=0.80$, this example implies $A_1=1$, so $E(A_1\mid q_2=0.80)=1$, not $0.50$. Conditioning on the whole realized probability sequence would therefore not preserve the first assignment’s original probability. The correct decision-specific statement is $E(A_1\mid H_1)=q_1=0.50$; the correct unconditional statement is $E(A_1)=E(q_1)=0.50$.

The assignments can also be dependent. Here $P(A_1=1,A_2=1)=0.50(0.80)=0.40$, so $$ \operatorname{Cov}(A_1,A_2)=0.40-(0.50)(0.50)=0.15. $$ Thus both assignments are randomized according to their rules, yet they are not independent of one another.

Connecting Enrollment Position to Assignment

Let $t_i$ be participant $i$'s enrollment time, or use the enrollment index $i$ when only ordering is relevant. A simple way to describe a linear assignment pattern over enrollment is the weighted sum $S=\sum_{i=1}^n (i-\bar{i})A_i$, where $\bar{i}$ is the average enrollment index. A positive value means treatment assignments are relatively concentrated later in the sequence; a negative value means they are relatively concentrated earlier.

The expectation of this summary follows from the history-conditional randomization rule. The derivation uses no conditioning on the realized full sequence of probabilities:

Use the decision-specific expectation.
For each $i$, $E(A_i\mid H_i)=q_i$, so the law of iterated expectation gives $E(A_i)=E(q_i)$.

Take the expectation of the weighted sum.
Since enrollment indices and their weights are fixed, linearity of expectation gives $E(S)=\sum_{i=1}^n(i-\bar{i})E(A_i)$.

Substitute the marginal allocation probabilities.
Therefore $E(S)=\sum_{i=1}^n(i-\bar{i})E(q_i)$. If the expected treatment probability is higher later in enrollment, this expected weighted sum will tend to be positive.

This is a statement about an expectation across possible trial paths. One realized trial can show a positive or negative assignment-time pattern even when $E(S)=0$, because random assignments fluctuate. Conversely, a systematic change in $E(q_i)$ across enrollment positions can produce a nonzero expected pattern. The realized probabilities $q_i$ are themselves history-dependent; the expression above averages each probability over the histories that could have occurred at that decision.

Worked Example: Quantifying a Realized Assignment-Time Pattern

Suppose a hypothetical four-participant trial has this realized sequence of assignment probabilities and assignments: $q_i=(0.25,0.40,0.60,0.75)$ and $A_i=(0,0,1,1)$. The probabilities are the values recorded at each decision after applying the feedback rule to the history then available. Describe the assignment pattern by the least-squares slope of $A_i$ on enrollment index $i$.

The mean enrollment index is $\bar{i}=(1+2+3+4)/4=2.5$, and the mean assignment indicator is $\bar{A}=(0+0+1+1)/4=0.5$. The slope is $$ \frac{\sum_{i=1}^4(i-\bar{i})(A_i-\bar{A})}{\sum_{i=1}^4(i-\bar{i})^2} = \frac{(-1.5)(-0.5)+(-0.5)(-0.5)+(0.5)(0.5)+(1.5)(0.5)} {(-1.5)^2+(-0.5)^2+(0.5)^2+(1.5)^2} = \frac{2}{5}=0.400. $$

In this realized sequence, the fitted treatment-assignment proportion increases by 0.400 per enrollment position. Since assignment is binary, the slope is a descriptive summary, not a claim that each participant’s probability rose by exactly 0.400. Nor does the calculation establish that the feedback rule caused a systematic trend across repetitions: it describes one path. The recorded probabilities help explain how the algorithm operated, while the assignments show what happened on this path.

Reading the Pattern and Checking the Process

An association between treatment assignment and enrollment time can arise through several stages: responses are observed; those responses affect the next allocation probability; and subsequent assignments are drawn using those updated probabilities. If outcomes or prognosis also change over time, treatment groups may then differ in their distribution across enrollment periods. That possibility motivates careful analysis, but the assignment-time association alone does not establish outcome bias or a treatment effect. Those require further assumptions and analysis.

A basic audit should align each assignment with the information available immediately before it. For each enrollment, record the enrollment time, the assigned treatment, the probability used, the allocation-rule version, and which responses had become available by that time. Compare this record with the implemented algorithm. Pay particular attention to whether pending responses were accidentally treated as observed, timestamps were misordered, or the recorded probability came from a later rather than contemporaneous history.

Interpret near-zero and near-one probabilities carefully. They can make the assignment pattern strongly dependent on accumulated outcomes and can leave little chance of receiving one treatment at some stages. A probability outside the protocol’s allowed range, a probability inconsistent with the recorded history, or unexplained changes in the allocation rule indicate an implementation or data-recording concern. A few unusual assignments alone do not demonstrate such a problem: randomization allows outcomes that are unlikely under the specified probabilities.

The main alternatives differ in how they handle allocation over time. Fixed-probability randomization uses a probability that does not update with response feedback, so it does not create this particular feedback pathway, though chance imbalance across time remains possible. Time-blocked or stratified randomization seeks balance across prespecified periods or strata, but does not use accumulating responses to change probabilities. Response-adaptive randomization uses outcome information to alter later allocation probabilities; this can serve the design’s allocation goals while making the history and timing of each decision essential to interpretation.

Approach	How allocation probabilities change	Implication for assignment and time
Fixed-probability randomization	Remain constant by design	No response-feedback pathway, though realized time imbalance can occur by chance
Time-blocked or stratified randomization	Follow prespecified time periods or strata	Targets balance within defined periods; does not adapt to observed responses
Response-adaptive randomization	Can change with observed response history	Later assignments can be linked to earlier outcomes and enrollment time through the update rule

Common Mistake and Limits

Common mistake: “Every assignment was randomized, so treatment assignment must be independent of enrollment time and earlier assignments.” The correction is that randomization is conditional on the information used at each decision: $E(A_i\mid H_i)=q_i$. If feedback changes $q_i$, then assignment distributions can change over enrollment, and assignments can be dependent across participants. Randomization at each step does not imply independence across the trial.

The relationship is design- and history-dependent. It need not be monotone: feedback can raise a probability at one stage and lower it at another. A realized upward pattern does not prove that the expected allocation probability increases over time, and an absence of an obvious pattern in a small trial does not prove that feedback had no effect. The allocation record can describe what probabilities were used, but it cannot alone establish why responses changed or whether a treatment effect varies over time. These conclusions require additional outcome analysis.

The practical limit is therefore one of interpretation: describe the assignment-time relationship in the context of the actual adaptive rule and recorded histories. Do not replace that sequential account with the claim that treatment was simply independent of time because each allocation involved a random draw.

Check Your Understanding

Answer each question, then compare with the solutions below.

At enrollment $i$, what does $q_i$ mean, and what information belongs in $H_i$?
If $E(A_i\mid H_i)=q_i$, what is the corresponding unconditional expectation?
In the two-participant example, why is it incorrect to conclude $E(A_1\mid q_2=0.80)=0.50$?
A trial has a positive realized slope of assignment on enrollment index. What does this establish, and what does it not establish?
Name two records that help check whether the feedback mechanism was implemented as intended.

Solutions. (1) $q_i$ is the probability of treatment assignment immediately before decision $i$, conditional on $H_i$, the information available before that assignment. (2) Taking expectations gives $E(A_i)=E(q_i)$. (3) In that example $q_2=0.80$ occurs only when $A_1=1$, so conditioning on $q_2=0.80$ reveals the first assignment; the conditional expectation is 1. (4) It establishes a positive descriptive association in that realized sequence, not a systematic expected trend across repetitions, a causal effect of enrollment time, or an outcome bias. (5) Examples include the probability used at each decision, the responses available at that time, enrollment and response timestamps, and the allocation-rule version.