Conditional Event-Risk Prediction From Intermediate Data

Why Condition Event Risk on an Intermediate Response?

At an interim look, an intermediate outcome may be available before the final time-to-event endpoint is mature. A response recorded by a fixed landmark can provide information about the risk of a later event, but the response rate alone is not an event-risk prediction. To make that prediction, we need the event risk associated with each response category and the proportion of patients in each category.

This tutorial develops that calculation for a binary response and a fixed event horizon. The same logic extends to more response categories: calculate the later-event risk within each category, then average those risks using the category proportions in the arm. The calculation describes predicted risk under an explicit model; by itself, it does not establish that treatment caused a response or that changing the response would change event risk.

Define the Landmark Estimand

Let \(A=a\) denote assignment to arm \(a\), and let \(L\) be the time by which the intermediate response is assessed. Fix a later horizon \(\tau>L\). Define \(E=1\) if the event occurs by \(L\). Among participants with \(E=0\), let \(R\) denote the response measured at \(L\), with categories such as response and no response. Let \(Z=1\) indicate an event in the interval \((L,\tau]\). Thus, \(Z\) is defined for the landmark risk set: participants who have not had the event by \(L\).

The landmark definition matters. A response assessed at \(L\) cannot be used to classify someone who had already experienced the event before \(L\). If the target is event risk by \(\tau\) from the original arm population, early events must also be included. Write \(e_a=P(E=1\mid A=a)\). The total predicted risk by \(\tau\) is then \(e_a+(1-e_a)\sum_r\pi_{ar}q_{ar}\). This separates observed early risk from predicted later risk in the event-free group.

Deriving the Arm-Level Prediction

Among participants event-free at \(L\), the response categories partition the risk set. The law of total probability therefore gives the later-event risk as a weighted average of the category-specific risks. For a binary response, denoted \(R=1\) for response and \(R=0\) for no response, the formula is:

The weights are response proportions in the arm’s landmark risk set, not proportions in the original randomized group if some participants experienced an event before \(L\). Each product is the fraction of the landmark risk set expected to have a later event through that response category. Summing the products gives the predicted later-event proportion.

For risk by \(\tau\) in the original arm population, partition first by early event status and then, for those event-free at \(L\), by response category:

This expression assumes the response is defined and ascertained in the event-free group at \(L\), and that the \(q_{ar}\) values refer to the same later interval and event definition as the target prediction. If the target is only the conditional risk among patients event-free at \(L\), the early-event term is not part of that estimand.

Using Interim Arm Evidence

At an interim look, the response distribution \(\pi_{ar}\) may be observed in the current arm while the later-event risks \(q_{ar}\) come from mature participants in the same trial, earlier data, or a prespecified model. The calculation is useful only if the source of each component is clear. Substituting an observed response distribution into a conditional-risk model produces a plug-in prediction, not a direct observation of the arm’s final event risk.

If early events have also been observed in the original arm, the same predicted later risk can be combined with the early-event proportion. In the next example, the response-specific risks are assumed applicable in both arms. That assumption lets the calculation isolate how the different response distributions contribute to the predicted risks.

Estimating the Conditional Risks and Checking the Calculation

A simple estimate of \(q_{ar}\) is the number of later events divided by the number of participants in response category \(r\) who have adequate follow-up through \(\tau\). For example, if 12 of 80 responders experience the event by \(\tau\), the crude estimate is \(12/80=0.15\). This ratio is appropriate only when the relevant outcomes are observed for everyone in the denominator or when censoring is handled by a method justified for the data. Treating participants censored before \(\tau\) as event-free generally understates risk.

A basic derivation shows why category-specific risks must be combined using their arm-specific proportions. In hypothetical mature data from one landmark risk set, suppose 12 of 80 responders and 24 of 120 nonresponders experience an event by \(\tau\). Then \(q_{a1}=12/80=0.15\), \(q_{a0}=24/120=0.20\), and \(\pi_{a1}=80/200=0.40\). The weighted prediction is:

There are 36 events among 200 participants, so the directly observed risk is \(36/200=0.18\), matching the weighted calculation. In general, if every response category is represented and its risk is estimated in the same complete dataset, the weighted average reproduces the overall risk. When category-specific risks are transported from another source, however, the weighted result is a prediction and need not equal the current arm’s observed risk.

Useful checks include comparing predicted with observed risks in mature data, separately by arm and response category; checking that every category used in the prediction has data; and verifying that the landmark, response definition, event definition, and follow-up horizon match across sources. Large differences in calibration across arms or time periods warn that a shared conditional-risk mapping may not transport. Very few participants or events in a category make its estimated risk unstable, even when the overall event count appears adequate.

Assumptions, Alternatives, and Limitations

The prediction relies on several substantive and data-quality assumptions. First, response and event status must be defined consistently, and response must be measured by the stated landmark for the people included in the landmark risk set. Second, the conditional risks must apply to the current arm and time period. If conditional risks are pooled across arms, equality across arms is an extra assumption, not a consequence of observing a response. If risks come from earlier data, changes in population, care, or follow-up can undermine transportability.

Third, censoring before \(\tau\) must not be treated as absence of an event. Complete follow-up through \(\tau\) is one simple setting; otherwise, risk estimation needs a method that accounts for censoring under appropriate assumptions. Fourth, each response category used in the calculation needs enough data to estimate its risk. A response category absent from the interim arm cannot support a data-based within-category estimate for that arm without additional modeling assumptions.

A common mistake is to argue, “The response rate is higher, so the final event risk must be lower.” That conclusion does not follow from response rates alone. It requires information about later-event risk by response category and a defensible way to apply those risks to the arm. Even when a response is prognostic, its association with later events does not show that treatment’s effect on response explains treatment’s effect on the final endpoint. Predictive use is not proof of surrogacy or a causal mechanism.

This calculation is transparent and easy to audit, but it reduces the intermediate information to response categories and uses specified conditional risks. Direct analysis of sufficiently mature final outcomes is preferable when those outcomes answer the target question with adequate precision. More flexible models can incorporate covariates or time-to-event information, but they bring additional modeling assumptions and diagnostics. The introductory material on delayed intermediate outcomes distinguishes availability from maturity; here, even an observed response does not eliminate the need to assess whether later-event data are mature enough to estimate the conditional risks. The calculation also does not quantify the uncertainty in the plug-in prediction: uncertainty in response proportions, conditional risks, and their dependence should be considered before treating small arm differences as persuasive interim evidence.