Confounding Calendar Time With Covariate Profiles

Why Calendar Time Belongs in the Allocation Record

An allocation pattern can appear to depend on patient profile when it actually reflects when different profiles enrolled. The reverse can also happen: a changing mix of profiles can make the overall proportion assigned to one treatment rise or fall even though the allocation rule for each profile has not changed. When assignment probabilities adapt to accumulating responses, calendar time is especially important because later assignments may use information unavailable earlier.

A pooled table that reports treatment assignments by profile, but discards enrollment order, cannot reliably distinguish these explanations. A chronological record lets an analyst compare patients enrolled at similar times and with similar profiles, while also showing what response information was available before each assignment. The aim is descriptive diagnosis of the allocation mechanism—not to infer treatment effects from allocation patterns.

Definition: For participant \(i\), let \(t_i\) denote the chronological assignment order, \(Z_i\) the recorded baseline covariate profile, \(A_i\) the treatment assignment, and \(H_{i-1}\) the trial history observed before that assignment. Define \(\pi_i=P(A_i=1\mid H_{i-1},Z_i,t_i)\), the probability of assignment to treatment 1 under the design at that point. A chronological allocation table records these quantities, or useful summaries of them, in enrollment order. The profile \(Z_i\), history \(H_{i-1}\), and probability \(\pi_i\) must be those available or defined before participant \(i\)'s assignment.

The probability \(\pi_i\) is a design record; the realized assignment \(A_i\) is a random outcome of the assignment procedure. They are related but not interchangeable. A participant assigned to treatment 1 does not prove that the probability was high, and a short run of assignments to treatment 1 can occur even when the recorded probabilities are constant. When available, the actual probabilities used by the allocation system are more informative about its rule than assignment frequencies alone.

What a Chronological Table Should Show

At minimum, retain each participant's assignment order, baseline profile, assigned treatment, and the probability of each treatment that was in force at assignment. For response-adaptive allocation, include a concise record of the response information available to the algorithm at that time, such as the number of outcomes accrued or the model-based allocation probabilities it produced. Also record relevant protocol or algorithm changes. These additions help distinguish a response-driven update from a change made for another reason.

For summary, group assignment order into prespecified periods or enrollment cohorts, and cross-classify those periods by profile. For each period-profile cell, report the number enrolled, the number assigned to treatment 1, the observed assignment proportion, and the mean recorded probability of treatment 1. Keeping the probability average and the observed proportion separate matters: one describes the assignment rule, the other describes the realized assignments.

Period	Profile	Enrolled	Assigned to treatment 1	Observed proportion	Mean recorded probability
Early	Profile X	...	...	...	...
Early	Profile Y	...	...	...	...
Later	Profile X	...	...	...	...
Later	Profile Y	...	...	...	...

The periods should be chosen to match meaningful points in the trial's chronology—for example, before and after a documented probability update—not selected after inspecting which grouping creates the strongest pattern. If many profiles or short periods produce sparse cells, show the participant-level sequence as well; a cell average can conceal sharp changes within a period.

Worked Example: Profile Mix Changes Under a Stable Profile Rule

Consider a hypothetical two-treatment trial in which treatment 1 has a fixed assignment probability of 0.70 for the high-risk profile and 0.30 for the low-risk profile. The rule does not change over time. In an early cohort, 80% of participants are high-risk; in a later cohort, 20% are high-risk. What change in the overall probability of treatment 1 should be expected?

For the early cohort, the profile weights are 0.80 high-risk and 0.20 low-risk. Its average assignment probability is

\bar{\pi}_{\text{early}} = (0.80)(0.70)+(0.20)(0.30) = 0.56+0.06 =0.62.

For the later cohort, the weights are 0.20 high-risk and 0.80 low-risk:

\bar{\pi}_{\text{later}} = (0.20)(0.70)+(0.80)(0.30) = 0.14+0.24 =0.38.

Thus the overall probability falls by 0.24, from 0.62 to 0.38, despite no change in either profile-specific probability. The shift is entirely due to profile composition. It is not evidence that the design adapted to responses or changed its rule. In actual data, realized assignment proportions will fluctuate around these averages.

Separating Composition From Within-Profile Change

Let \(g\) index two periods and \(z\) index profiles. Write \(w_{gz}\) for the proportion of participants in profile \(z\) during period \(g\), and \(q_{gz}\) for the period-profile mean probability of treatment 1. Then the period's overall mean probability is \(q_g=\sum_z w_{gz}q_{gz}\). The same identity can be used descriptively with observed assignment proportions in place of mean probabilities, but those proportions also contain random assignment variation.

An exact decomposition compares an early period \(E\) with a later period \(L\):

q_L-q_E = \sum_z (w_{Lz}-w_{Ez})q_{Ez} + \sum_z w_{Lz}(q_{Lz}-q_{Ez}).

The first sum is the composition component: it asks how the overall rate would change if the profile mix moved to its later-period values while the early period's profile-specific probabilities stayed fixed. The second sum is the within-profile component: it measures changes in profile-specific probabilities, weighted by the later-period mix. Their sum is the observed difference in the overall mean probability. The allocation of the total between components depends on the reference period used; the identity does not establish why any probability changed.

Justification: Since the later-period average is \(q_L=\sum_z w_{Lz}q_{Lz}\), add and subtract \(\sum_z w_{Lz}q_{Ez}\). The result is \(q_L=\sum_z w_{Lz}q_{Ez}+\sum_z w_{Lz}(q_{Lz}-q_{Ez})\). Subtract \(q_E=\sum_z w_{Ez}q_{Ez}\) to obtain the stated decomposition. No assumption about a causal effect of profile or time is needed for this arithmetic identity; interpreting the components as design behavior does require correctly recorded profiles, periods, and probabilities.

Worked Example: A Shift With Both Composition and Within-Profile Change

In a hypothetical trial, the early period includes 8 low-risk and 2 high-risk participants. Their mean recorded probabilities of treatment 1 are 0.40 and 0.50, respectively. The later period includes 2 low-risk and 8 high-risk participants, with corresponding mean probabilities 0.50 and 0.75. These probabilities are averages of the probabilities logged before assignment, not proportions of participants who happened to receive treatment 1.

The early overall mean probability is

$$ q_E=(0.80)(0.40)+(0.20)(0.50)=0.32+0.10=0.42. $$

The later overall mean probability is

$$ q_L=(0.20)(0.50)+(0.80)(0.75)=0.10+0.60=0.70. $$

The overall increase is \(0.70-0.42=0.28\). Using early-period probabilities as the reference, the composition component is

$$ (0.20-0.80)(0.40)+(0.80-0.20)(0.50) =(-0.60)(0.40)+(0.60)(0.50) =-0.24+0.30=0.06. $$

The within-profile component is

$$ (0.20)(0.50-0.40)+(0.80)(0.75-0.50) =(0.20)(0.10)+(0.80)(0.25) =0.02+0.20=0.22. $$

The components sum to \(0.06+0.22=0.28\), the total change. Both profile mix and profile-specific probabilities changed. The table shows the latter, but it cannot by itself establish that responses caused the changes: the probability logs and the history available to the allocation algorithm are needed to assess that explanation.

When a Profile Pattern Is Really a Time Pattern

A pooled comparison by profile can be misleading in the opposite direction. Suppose a response-adaptive rule raises the probability of treatment 1 over calendar time, applying the same probability to every profile at each point in time. If one profile tends to enroll early and another tends to enroll later, their pooled assignment proportions can differ even though there is no profile-specific difference at a given time.

Worked Example: Apparent Profile Dependence From Enrollment Timing

In a hypothetical trial, the logged probability of treatment 1 is 0.30 for every participant in the early period and 0.70 for every participant in the later period. Profile X includes 8 early and 2 later participants; profile Y includes 2 early and 8 later participants. Compute each profile's average logged probability.

For profile X, 80% enrolled early and 20% later:

\bar{\pi}_X=(0.80)(0.30)+(0.20)(0.70)=0.24+0.14=0.38.

For profile Y, 20% enrolled early and 80% later:

\bar{\pi}_Y=(0.20)(0.30)+(0.80)(0.70)=0.06+0.56=0.62.

The pooled averages differ by 0.24, but within either period the probability is identical across profiles. The chronological table reveals a time pattern, not evidence of a profile-specific allocation rule. The time pattern could reflect response adaptation, a planned change, or another feature of the algorithm; chronology alone does not distinguish those causes.

How to Diagnose the Pattern

Read the allocation record in sequence, then compare like with like. For each profile, ask whether the recorded probability changes across periods. Within each period, ask whether it differs across profiles. Compare those probability patterns with the realized assignments, which may be noisy, and check whether probability changes line up with the outcomes or other history actually supplied to the algorithm.

Verify the chronology.
Confirm that assignment order and time periods are correct, profiles were measured before assignment, and logged probabilities correspond to the rule used at each assignment.

Compare profile mixes.
Report the proportion in each profile by period. Large changes mean that pooled allocation proportions may shift even under a stable profile-specific rule.

Compare probabilities within cells.
Inspect the mean recorded probability by period and profile. A difference across periods within profiles is a time-related change; a difference across profiles within periods is profile-related.

Check the recorded history and realized assignments.
Determine whether probability updates followed response information available before assignment. Treat assignment proportions as noisy realizations, not as the allocation probabilities themselves.

A model that includes both profile and time terms can be a supplementary summary, but it is not a substitute for checking the recorded mechanism. If probabilities are functions of accumulated responses, a simple time trend may summarize when updates occurred without identifying what caused them. Conversely, a model that omits time can attribute a time pattern to profile when profile composition differs over enrollment. The allocation log, algorithm specification, and chronology are central to interpretation.

Common Pitfalls and Limits

Calling a pooled difference covariate-driven. The wrong reasoning is that a higher treatment 1 proportion in one profile proves the algorithm favored that profile. The correction is to compare profiles within periods and inspect their recorded probabilities; profiles may simply have enrolled at different times.
Calling every time trend response adaptation. A probability changing with enrollment order is not sufficient evidence that responses drove it. Check the algorithm inputs, outcome availability, and any planned or operational changes.
Equating assignment frequency with assignment probability. A small cell can have a very different observed proportion from its logged probabilities by chance. Report both, and include cell sizes.
Overinterpreting sparse cross-classifications. If a profile occurs only in one period, its profile and time patterns cannot be separated from these data alone. Do not claim a within-period comparison that the table cannot support.

A chronological table is a diagnostic, not a cure for incomplete records, poorly defined profiles, or a misimplemented allocation rule. If probabilities were not logged and the algorithm cannot be reconstructed, realized assignments alone may not reveal whether changes were response-adaptive. Also, the decomposition is descriptive: it does not establish that calendar time or profile caused a probability change, and it does not assess treatment efficacy.

Approach	What it shows well	Main limitation
Pooled allocation table by profile	Simple overall differences in observed assignment proportions	Confounds profile composition with enrollment timing
Chronological period-by-profile table	Separates changing profile mix from within-cell probability patterns	Can be unstable or uninformative in sparse cells
Time-and-profile model summary	Compresses patterns across many periods or covariates	Depends on model form and cannot replace the allocation log or identify the reason for updates by itself

Key takeaway: To distinguish composition from adaptation, keep the allocation sequence, profile, assigned treatment, and pre-assignment probability together. A change in pooled assignment rates is not enough: compare probabilities within profile and time, then use the recorded response history to judge whether an update was actually response-driven.

Check Your Understanding

Answer each question using the allocation-record principles above. Solutions are included below.

A stable rule assigns treatment 1 with probability 0.60 to profile A and 0.20 to profile B. The share of profile A falls from 0.75 to 0.25. What happens to the overall mean probability?
Why should an allocation table report both the observed assignment proportion and the mean recorded probability?
Suppose profile X enrolls mostly early and profile Y mostly late, while treatment 1 probabilities rise over time equally for both profiles. What would a pooled profile comparison risk suggesting?
What additional evidence is needed before attributing a calendar-time probability change to response adaptation?

Solutions. (1) The early mean is \((0.75)(0.60)+(0.25)(0.20)=0.50\). The later mean is \((0.25)(0.60)+(0.75)(0.20)=0.30\). It falls by 0.20 due to composition; the profile-specific rule is unchanged. (2) The observed proportion describes realized assignments and varies randomly; the mean recorded probability describes the design probabilities used. (3) It may suggest a profile effect even though the probabilities are equal across profiles within each period; the profiles have different enrollment-time mixes. (4) Inspect the allocation algorithm's inputs and the response information available before each assignment, along with probability logs and records of planned or operational changes. A time trend alone does not identify its cause.