Worked Example: Worsening Prognosis During Adaptive Enrollment

Why a Worked Calculation Matters

When prognosis worsens during enrollment, later participants may have systematically different outcomes from earlier participants, regardless of treatment. If an adaptive algorithm also changes the allocation probabilities over that same period, the two treatment arms can contain different proportions of early and late enrollees. The crude difference between arm averages then combines treatment contrasts with differences in enrollment time.

The calculations below use a deliberately small, hypothetical trial. The outcome is a score for which higher values indicate worse prognosis. Treatment A becomes more favored as enrollment proceeds, while average outcomes worsen in both arms. The example illustrates a realized imbalance; it does not imply that adaptive randomization necessarily creates bias in expectation.

Setup and Assumptions

For the basic calculation, outcomes must be measured on the same scale in both arms, treatment assignment must correspond to the treatment actually analyzed, and enrollment time must be recorded consistently. To interpret a time-adjusted contrast causally, randomization must make treatment assignment independent of potential outcomes conditional on the information used by the assignment mechanism, including relevant time or trial history. Each treatment must also have a positive probability of assignment in the time blocks receiving positive weight; otherwise the corresponding within-block contrast is not estimable from the trial.

A simple time-block adjustment additionally assumes that the blocks adequately capture the relevant prognostic change for the intended comparison. If the treatment effect differs across time, the standardized contrast remains interpretable as a weighted average of block-specific contrasts, but it is not a single common effect. Missing outcomes, changes in outcome measurement, or important changes within blocks can require further modeling and sensitivity analysis.

Crude and Standardized Contrasts

The crude mean in arm \(a\) is a weighted average of its block-specific means, with weights given by that arm’s observed enrollment distribution:

For time standardization, choose a target distribution \(w_k\) and apply it to both arms:

A common choice is the trial’s pooled enrollment distribution, \(w_k=(n_{Ak}+n_{Bk})/N\), where \(N\) is the total number enrolled. This describes a contrast standardized to the trial’s overall mix of enrollment times. Other target weights answer different questions and should be chosen for a substantive reason, not because they produce a preferred result.

Why Standardization Works

The key difference between the calculations is their weighting. The crude contrast uses arm-specific weights \(n_{Ak}/n_A\) and \(n_{Bk}/n_B\); the standardized contrast deliberately uses the same \(w_k\) for each arm. If the block-specific contrast is \(\Delta_k=\bar{Y}_{Ak}-\bar{Y}_{Bk}\), then by definition:

This identity shows exactly what time standardization does: it combines the within-block contrasts under a common target distribution. For the crude contrast, write each block mean as \(\bar{Y}_{Bk}+\Delta_k\). Substituting into the arm-specific averages gives

The first term reflects differences in time composition applied to the B-arm block means; the second averages treatment differences using A’s time distribution. When prognosis changes over time and arm distributions differ, the first term need not be zero. That is how a crude contrast can disagree with a common-weight contrast, even if random assignment was used.

In the worked trial, the common treatment difference is \(-1\) in every block, so any valid common set of block weights gives \(-1\). More generally, the weights matter when the block-specific differences vary. A regression model with block indicators and a common treatment coefficient produces a related time-adjusted comparison under its model assumptions. It should not be treated as automatically identical to every standardization: the estimand, weights, and treatment-by-time specification need to be made explicit.

Diagnostics, Alternatives, and Limitations

Before interpreting an adjusted contrast, inspect the data by enrollment time. A table of arm counts and means, like the one above, reveals whether both treatments appear in each block, whether prognosis changes, and whether the allocation share shifts. Sparse or empty treatment-by-time cells are warnings: a standardized contrast requiring a missing cell mean cannot be calculated directly without extrapolation. A fitted model may still return a number, but that number depends on the model’s assumptions rather than direct within-block information.

When the time trend is not well represented by a few blocks, a prespecified continuous-time regression or flexible trend adjustment may be more appropriate. Regression can use more detail than broad categorization but depends on the chosen functional form and on adequate overlap. Inverse-probability weighting instead uses known assignment probabilities to reweight observations toward a target allocation distribution; very small probabilities can make weights unstable. These methods address related imbalance, but they are not interchangeable without specifying the estimand and assumptions. A direct block-standardized contrast is particularly transparent when the blocks and target weights are substantively meaningful.

This calculation does not by itself provide an uncertainty interval or a hypothesis test. Those require an uncertainty estimator suited to the randomization scheme, the adaptive assignment mechanism, and the adjustment procedure. Nor does time adjustment correct unmeasured prognostic changes within blocks, outcome-dependent missingness, or an incorrectly recorded enrollment time. If treatment effects vary across time, reporting only one adjusted number can conceal that variation; display the block-specific contrasts and state the target weights.