Joint Models of Response and Resolution Time

Why Model Response and Resolution Together?

In response-adaptive randomization, an outcome that has not yet resolved cannot be treated as a known response. Its resolution time can nevertheless carry information: if response and resolution time are associated, knowing that the outcome is still pending changes the probabilities of the possible responses. A model that ignores this association can therefore give a misleading prediction for a pending participant.

Here, \(Y\) denotes a binary final response and \(T\) the elapsed time from randomization to resolution and availability of that response. Thus \(Y=1\) might represent a defined clinical response and \(Y=0\) its absence; \(T\) is measured in days, for example. The model is conditional on treatment arm \(a\). It describes a participant's final response and the time at which that response becomes known, not merely the time until an interim data update.

The factorization does not itself require a particular parametric family. Its purpose is to separate two questions that must be specified: how often each response occurs, and how quickly outcomes resolve among participants with each response. For positive resolution times, write \(f_{ay}(t)\) for the conditional density of \(T\) given \(Y=y\) and arm \(a\), and \(S_{ay}(t)=P(T>t\mid Y=y,a)\) for its survival function. Then

The response probabilities are \(P(Y=1\mid a)=\pi_a\) and \(P(Y=0\mid a)=1-\pi_a\). The model assumes, at minimum, that the response definition and time origin are consistent, that \(T\) is recorded for resolved participants, and that observed pending status means \(T>c\) at the participant's current follow-up time \(c\). A likelihood based on administrative censoring also assumes that censoring is independent of the unobserved response and resolution time, conditional on modeled variables. If withdrawal or loss to follow-up is related to outcome or delay, that process needs separate treatment rather than being silently counted as ordinary administrative censoring.

What Does Pending Status Tell Us?

Suppose a participant in arm \(a\) has been followed for \(c\) time units and remains unresolved. By Bayes' rule, the probability that the eventual response is positive is

The denominator is the probability of still being unresolved at time \(c\), averaging over the two possible responses. The numerator is the probability of both a positive response and no resolution by \(c\). If positive responses tend to resolve more slowly, pending status raises the probability of a positive response; if they resolve more quickly, pending status lowers it.

This probability is conditional on the model parameters. With estimated parameters, a plug-in prediction substitutes estimates for \(\pi_a\) and the delay distributions. A Bayesian predictive probability instead averages the displayed conditional probability over the parameter distribution. Neither calculation turns an unresolved outcome into an observed response.

Likelihood Contributions for Resolved and Pending Outcomes

A joint model uses both kinds of records. For a participant whose response \(y\) is observed at resolution time \(t\), the likelihood contribution in arm \(a\) is the joint density

For a participant still pending at follow-up time \(c\), the response is unknown. The likelihood contribution is the probability of remaining unresolved, summed over both possible responses:

The pending contribution is a probability, not a density, because it records an event of the form \(T>c\). For independent participants conditional on their treatment assignments and model parameters, the full likelihood is the product of their individual contributions. This conditional independence assumption is commonly used for a basic model; clustering or other dependence between participants requires an extended likelihood.

Derivation and a Second Model-Based Prediction

The pending-response formula follows directly from the definition of conditional probability. The event \(Y=1,T>c\) has probability \(\pi_aS_{a1}(c)\). The event \(T>c\) can occur with either response, and those disjoint possibilities have probabilities \(\pi_aS_{a1}(c)\) and \((1-\pi_a)S_{a0}(c)\). Dividing the first event probability by their sum gives the conditional probability:

This derivation also shows why the relevant delay quantities are the response-specific survival probabilities at the participant's elapsed follow-up time, not just the marginal delay distribution. A single pending participant at day 1 and one at day 10 generally have different predictions, even in the same arm.

Assumptions, Diagnostics, and Failure Modes

A joint model is only as credible as its specifications and observation process. In the examples, exponential times are convenient, but their constant-hazard assumption may be implausible. For instance, resolution may be concentrated around scheduled visits, or the hazard may rise or fall with time. More flexible parametric distributions, piecewise hazards, or other survival models can replace the exponential distribution while retaining the joint response-delay structure.

Useful checks include comparing predicted and observed resolution patterns over time, separately by response among participants whose responses are known; checking whether the model reproduces the number still pending at relevant follow-up times; and examining calibration of predicted response fractions as pending outcomes later resolve. Such checks should respect the fact that early resolved participants are not necessarily representative of all participants. Where data are sparse, response-specific delay estimates can be unstable, and the apparent association may be driven by a small number of resolved outcomes.

A key limitation is identification. At an interim look, pending responses are unobserved, so the observed data may not determine how response and delay are associated without adequate follow-up, assumptions about the delay distributions, or external information. A model can produce a precise numerical prediction while its dependence assumptions remain weakly supported. Sensitivity analysis across plausible response-specific delay patterns is therefore important, especially when predictions will inform allocation decisions.

The approach also presumes that response is ultimately defined for each participant and that the resolution process is being modeled consistently. If some participants withdraw, become unobservable, or never have an outcome recorded, their status is not automatically equivalent to \(T>c\). A model for loss to follow-up or missing outcomes may be needed. Likewise, treatment arms may have different response probabilities and different response-specific delay distributions; pooling them without justification can obscure real differences.

How the Joint Model Differs From Alternatives

The independence model is a special case: \(S_{a1}(c)=S_{a0}(c)\) for all \(c\). Substitution into the conditional probability gives \(P(Y=1\mid T>c,a)=\pi_a\), so pending status changes nothing. A treatment-specific marginal delay model is not enough to avoid that restriction unless it also allows delay to vary with response. The choice is therefore substantive: use a joint model when the response-delay association is plausible and supported well enough to estimate, and make the uncertainty in that choice visible.