Treatment-Specific Delay Distributions

Why Model Delays by Treatment Arm?

An adaptive randomization rule can use only information that has become available by the time of an allocation update. If outcome resolution is faster in one arm than another, the rule may have substantially more resolved outcomes from one arm, even when the numbers assigned to the arms are similar. This difference can affect the precision and stability of the information driving adaptation. Modeling delay distributions separately by arm makes that imbalance explicit.

Here, a delay is the elapsed time from a participant’s randomization to the point at which that participant’s outcome is resolved and available for the rule. Let \(A_i\) denote participant \(i\)’s assigned arm, and let \(D_i\) denote the participant’s resolution delay, measured in days unless stated otherwise. The focus is the distribution of \(D_i\) within each arm, not a model that jointly specifies delay and response.

The distribution answers more than “which arm is faster?” It describes the probability of resolution at each follow-up time. Two arms can have similar average delays but different early resolution probabilities, or one arm can be faster early and slower later. Since adaptive rules are often updated on a schedule, \(F_a(t)\) at the relevant elapsed times can be more directly useful than a single average.

How Unequal Delays Change the Available Data

Suppose an allocation update occurs at calendar time \(\tau\), and participant \(i\) was randomized at time \(T_i\). That participant’s outcome is available at the update if \(D_i\leq \tau-T_i\), provided \(\tau\geq T_i\). Thus, participants in the same arm may have different resolution probabilities if they have different amounts of follow-up.

If the participants assigned to arm \(a\) have elapsed follow-up times \(u_1,\ldots,u_{n_a}\), then, under the arm-specific delay model, the expected number with resolved outcomes is

where \(M_a\) is the number resolved by the update. If their resolution indicators are conditionally independent, then \(\operatorname{Var}(M_a\mid u_1,\ldots,u_{n_a})=\sum_{j=1}^{n_a}F_a(u_j)\{1-F_a(u_j)\}\). These expressions quantify how many observations an update can expect to use; they do not mean that a fractional outcome is observed.

For a fixed cohort of \(n_a\) participants all followed for \(t\) days, the expected resolved count simplifies to \(n_aF_a(t)\). Consequently, equal assignment counts do not imply equal numbers of resolved outcomes. In an adaptive trial, the resulting imbalance can change the information available to the rule and contribute to unstable allocation decisions. The size and direction of any distortion in response estimates also depend on whether delay is related to response, as emphasized in the earlier material on informative outcome delays.

Specifying Arm-Specific Delay Models

A model should be chosen to describe the resolution process relevant to the update schedule. One common parametric choice is the exponential model:

This model is simple, but its constant-rate assumption can be restrictive. For example, if many outcomes resolve near a scheduled assessment, the resolution rate may change sharply over time. A Weibull model allows the rate to vary with elapsed time. With shape \(k_a>0\) and scale \(\eta_a>0\), its distribution is

When \(k_a=1\), this reduces to an exponential model with rate \(1/\eta_a\). When \(k_a\ne1\), the resolution rate changes over time. A separate model for each arm permits both the shape and the scale to differ, when the data support estimating them.

A treatment-specific model need not be parametric. One may estimate each arm’s resolution curve empirically, for example with a Kaplan–Meier estimator when some delays are right-censored. The appropriate choice depends on the amount and pattern of follow-up, the purpose of the model, and whether smooth estimates at the update times are needed. An arm-specific model is not automatically better simply because it is more flexible: sparse data can make separate estimates highly uncertain.

Estimating Delay Distributions With Incomplete Follow-Up

At an interim analysis, some participants may still be unresolved. Their delays are right-censored: the study tells us that their delays exceed their observed follow-up times, but not their eventual resolution times. Treating those follow-up times as completed delays would systematically understate the unresolved time and can distort the fitted distribution.

For a simple exponential model in arm \(a\), let \(x_i\) be the observed time for participant \(i\), and let \(\delta_i=1\) if resolution was observed and \(\delta_i=0\) if the delay was right-censored. If censoring is independent of the delay within arm, the likelihood contribution is \(f_a(x_i)\) when \(\delta_i=1\), and \(S_a(x_i)\) when \(\delta_i=0\). The combined likelihood is therefore

Taking the log likelihood and differentiating gives \(\ell(\lambda_a)=d_a\log(\lambda_a)-\lambda_a T_a+\text{constant}\), where \(d_a=\sum_i\delta_i\) is the number of resolved delays and \(T_a=\sum_i x_i\) is total observed time at risk. Setting the derivative \(d_a/\lambda_a-T_a\) to zero yields \(\hat{\lambda}_a=d_a/T_a\). This estimate uses censored observations as evidence that resolution had not occurred by their censoring times.

Checks, Comparisons, and Limits

Useful diagnostics focus on the delay data, not just the fitted rate. Plot estimated arm-specific resolution curves over the time range relevant to allocation updates; compare observed and model-predicted resolution proportions at those times; inspect the timing of resolutions for evidence of scheduled batches or changing rates; and report the number at risk or still unresolved in each arm. Sparse late follow-up can make the tail of a curve particularly uncertain. A model may fit early resolution well while extrapolating poorly to later times.

A central assumption for interpreting arm-specific delay curves is that the observed resolution process is adequately represented by the model within each arm. Independent censoring is needed for the likelihood calculation above: conditional on arm, whether follow-up ends by censoring should not depend on the unobserved resolution time. If participants with particular delays are more likely to leave observation, or if resolution depends on unmodeled features that differ across arms, estimated curves can be misleading.

The choice is not simply between “pooled” and “separate.” A pooled model is defensible if a common delay distribution is credible; a treatment-specific model is useful when arm differences are plausible and estimable. A practical analysis can compare arm-specific curves and quantify uncertainty before deciding whether pooling is reasonable. Regardless of the model, an estimated delay distribution describes resolution timing; it does not itself correct an outcome estimate when resolution is related to the outcome.

The distinction matters for adaptive randomization. If delays differ by arm but are unrelated to response within each arm, unequal maturity primarily means unequal amounts of response information at an update. If resolution also depends on response, the resolved outcomes can be a selected subset, and the adaptive rule may be affected in ways that delay curves alone cannot resolve. Modeling the joint relationship between response and resolution is a separate task.

• Mistake: Compare only the average delay and conclude which arm contributes more information at every update. Correction: Compare \(F_a(t)\) at the actual follow-up ages; delay curves may cross.
• Mistake: Count a censored follow-up time as if it were the completed delay. Correction: Record that the delay exceeds the censoring time and use a method that accounts for right-censoring.
• Mistake: Interpret an arm-specific delay difference as evidence of a response difference. Correction: Resolution timing and response are distinct quantities; a delay model alone does not estimate treatment efficacy.

Deriving the Expected Resolved Count

The expected-count calculation follows directly from indicator variables. For the \(j\)th participant assigned to arm \(a\), let \(R_j=1\) if the outcome is resolved by the update and \(R_j=0\) otherwise. Given follow-up age \(u_j\), \(E(R_j)=P(D_j\leq u_j\mid A_j=a)=F_a(u_j)\). Since \(M_a=\sum_jR_j\), linearity of expectation gives

This derivation does not require the resolution indicators to be independent. Independence is needed for the simple variance expression, not for the expected count. If the follow-up ages are equal to \(t\), the result becomes \(E(M_a)=n_aF_a(t)\). This makes clear why arm-specific distributions are needed whenever the probability of resolution differs by arm.