Prognostic Association With the Final Endpoint

Why Prognostic Association Needs a Clear Target

In a multi-arm, multi-stage trial, an intermediate outcome may be available before the final time-to-event endpoint is mature. It is useful to know whether that intermediate outcome is associated with the later event: for example, whether patients with a poor early response tend to experience the event sooner or more often over a specified follow-up period. That is a prognostic question about patients’ subsequent outcomes.

It is not, by itself, a question about whether treatment effects on the intermediate outcome reflect treatment effects on the final endpoint. A marker can identify patients at higher risk while failing to capture differences between treatments. The association described here is therefore a patient-level relationship, defined for a specific population, treatment context, and time horizon.

This definition makes the target population explicit: patients who are event-free at \(\tau\) and have an observed intermediate outcome. The association need not describe patients who experienced the event before the landmark. It also depends on the chosen horizon \(u\); a marker may be associated with near-term risk but less informative about risk over a longer period.

For a binary marker, useful descriptive measures include the risk difference, risk ratio, and odds ratio. For instance, the risk difference compares the event probabilities in the two marker groups, while the risk ratio divides one probability by the other. These are association measures, not effects of intervening to change the marker.

Assumptions and Interpretation

Before calculating an association, specify the landmark, marker measurement, event definition, and follow-up horizon. The marker must be measured by the landmark, and the subsequent event window must begin after it. Use a consistent event definition and ascertainment process across marker groups and treatment arms.

A simple comparison of observed risks is most directly interpretable when subsequent event status is known for all eligible patients through the horizon. If some patients are censored before \(\tau+u\), treating them as event-free is generally incorrect. A valid analysis then needs a method that accounts for censoring and assumptions about the censoring process; the risk calculations below assume complete ascertainment through the specified horizon.

Describing an association does not require the marker to be randomized, nor does it require causal exchangeability between marker groups. It does require care about what population is being described. In particular, conditioning on being event-free at \(\tau\) defines a landmark-survivor population; it may differ across arms if treatment affects events before \(\tau\). If the marker is affected by treatment, its distribution may also differ across arms. Neither issue is resolved by merely pooling participants.

For an adjusted model, the functional form and included covariates must be stated. A model that assumes one common marker association across arms should be checked for departures from that assumption. Adequate observations across marker values and covariate patterns are also needed; sparse or empty groups can make estimated associations unstable or impossible to identify.

Risk Contrasts and a Simple Derivation

For a binary marker, let \(p_1\) be the subsequent-event risk for \(M=1\) and \(p_0\) the risk for \(M=0\), in a specified arm and landmark population. The risk difference is \(p_1-p_0\), the risk ratio is \(p_1/p_0\) when \(p_0>0\), and the odds ratio compares the odds \(p/(1-p)\) between groups.

The odds ratio has a direct connection to logistic regression. Suppose that, within a treatment arm and at fixed values of other covariates, the model is \(\operatorname{logit}\{P(Z=1\mid M,A,X,T>\tau)\}=\beta_0+\beta_M M+\beta_A A+\beta_X X\), with no marker-by-arm interaction. The logit is the logarithm of the odds. Increasing \(M\) from 0 to 1 changes the model’s log odds by \(\beta_M\), so exponentiating gives the conditional odds ratio:

This result follows from the stated model, including its assumption that the marker coefficient does not vary with arm or \(X\). The odds ratio is not a risk ratio, and the same odds ratio can correspond to different risk differences at different baseline risks. If marker associations plausibly vary by arm, an interaction or arm-specific estimates are needed rather than interpreting one common coefficient.

Why Treatment Arm Must Be Considered

A pooled marker–event association can reflect differences in treatment-arm composition rather than the association within an arm. This matters in multi-arm trials because arms can have different event risks, and the intermediate outcome may have different distributions across arms. A useful first analysis is therefore to examine risks separately by arm. If a single summary is needed, it should use an explicit adjustment or standardization strategy and state its target population.

For example, a standardized risk can average the arm-specific risks using the same arm weights for each marker group. With two arms and equal weights, the standardized risk for marker value \(m\) is the average of the two risks for that marker value. This removes differences in arm weights from the descriptive comparison, but it does not turn the marker association into a causal effect.

Regression as an Adjusted Description

When the marker and baseline covariates are continuous or numerous, a regression model can summarize the association while adjusting for measured covariates. For a binary event indicator at the stated horizon, logistic regression models the log odds of that event. For a binary marker, its coefficient gives a conditional odds ratio under the specified model. For a continuous marker, the coefficient describes the change in log odds per unit increase, so the unit must be reported.

Adjustment changes the question from an unadjusted comparison to a conditional association at fixed values of included covariates. It does not guarantee that all relevant differences are controlled, nor does it justify a causal interpretation. The model should include treatment arm, and an interaction should be considered when there is reason to expect the marker association to differ by arm. The analysis should also assess whether the chosen functional form is adequate and whether predicted risks agree reasonably with observed risks.

Diagnostics, Alternatives, and Limits

Useful diagnostics are substantive as well as statistical. Check that marker measurement precedes the subsequent event window, follow-up is comparable, and the event counts and denominators are correct. Display arm-specific risks before a pooled estimate. Look for sparse marker-by-arm groups, implausible fitted probabilities, poor calibration, and evidence that the marker relationship is nonlinear or differs by arm. If censoring is substantial, an analysis that assumes everyone without a recorded event remained event-free is not appropriate.

The main alternatives answer related but different descriptive questions. Crude risks are transparent but can be distorted by arm composition or covariate imbalance. Arm-specific risks expose variation across arms but may be imprecise in small groups. Regression can adjust for measured covariates and accommodate continuous markers, but depends on model specification. Standardization gives a comparison for a stated common distribution of arms or covariates; changing that distribution can change the summary.

A strong prognostic association can be useful for identifying patients with higher subsequent event risk, but it does not establish that treatment effects on the intermediate outcome predict treatment effects on the final endpoint. That is a separate surrogacy question. In particular, an association estimated among patients who remain event-free at the landmark is not, without further argument, a measure of how well the marker supports treatment selection across all randomized patients.