Exposure And Time-To-Event Efficacy

From Exposure to Time-to-Event Efficacy

A binary efficacy endpoint records whether an event occurred by a chosen visit. A time-to-event endpoint also uses when it occurred, while retaining information from participants whose event has not been observed by the end of follow-up. This distinction matters when follow-up differs across participants: an event-free participant observed for two months does not provide the same information as one observed for twelve months.

An exposure-dependent hazard model links a participant’s exposure to the instantaneous rate of the efficacy event, conditional on remaining event-free up to that time. The model can then predict event-free survival at a specified follow-up time. Those predictions are conditional on the model and its assumptions; an exposure association alone does not establish that changing exposure would cause the predicted change in efficacy.

For a concrete model, let \(x\) denote a participant’s exposure, measured in a defined unit, and let \(x_{\mathrm{ref}}\) be a reference exposure. Use a Weibull baseline hazard with an exposure effect proportional on the hazard scale:

Here \(\eta>0\) is the Weibull scale parameter, in units of time, \(k>0\) is the shape parameter, and \(\beta\) is the log hazard ratio for a 10-unit increase in exposure. The exposure unit must be stated in context; for example, if \(x\) is measured in mg·h/L, the increment is 10 mg·h/L. At \(x=x_{\mathrm{ref}}\), the exposure multiplier is 1, so the hazard is the baseline hazard. The model implies

The baseline event risk by time \(t\), at the reference exposure, is \(1-S(t\mid x_{\mathrm{ref}})\). This is an absolute probability over a stated period. By contrast, \(\exp(\beta)\) is a hazard ratio: at any time when the model applies, it compares the instantaneous hazards for exposures differing by 10 units. It is not a risk ratio, and it does not by itself specify the difference in event probabilities.

Assumptions and Right-Censoring

The model is appropriate only when its structure and data collection are defensible. The Weibull baseline allows the hazard to increase over time when \(k>1\), decrease when \(k<1\), or remain constant when \(k=1\). The exposure effect is proportional: the hazard ratio for a given exposure contrast does not change with time. Exposure must be defined consistently, and the model treats the specified exposure value as the relevant predictor throughout the modeled risk period. If exposure changes over time, this fixed-exposure model does not represent that change.

The analysis also assumes that event times are defined and observed consistently, participants are independent conditional on included predictors, and right-censoring is noninformative conditional on those predictors. In practical terms, after accounting for modeled information, a participant’s censoring time should not provide additional information about their unobserved event time. Loss to follow-up related to worsening efficacy can violate this assumption. Administrative study end can often be reasonable censoring, but that judgment depends on study conduct and the event process.

For participant \(i\), let \(Y_i\) be the observed time, the smaller of the event time and censoring time, and let \(\delta_i=1\) if the event was observed and \(0\) if follow-up was right-censored. The likelihood contribution is

An observed event contributes its event-time density \(f=hS\); a censored participant contributes the probability of remaining event-free through the observed follow-up, \(S\). This follows from \(f(t)=-dS(t)/dt\) and \(h(t)=f(t)/S(t)\). Under the stated independent-censoring assumption, the censoring distribution does not need to be modeled to estimate the event-time parameters using these contributions. Treating a censored participant as though an event occurred at the censoring time would instead use the wrong contribution.

Working Through Censored Follow-Up

Censoring preserves partial information. A participant censored at month 6 is known to have remained event-free through month 6, even though the later event time is unknown. The Weibull model assigns a survival contribution through that time rather than inventing an event time.

For a data set, multiply the individual likelihood contributions, or equivalently sum their log contributions, and estimate \(\eta\), \(k\), and \(\beta\) by maximizing that likelihood. A reproducible analysis should record the event definition, time origin, exposure summary and units, censoring rule, model parameterization, and estimation settings. The fitted model should be checked before its predictions are used.

Interpreting Time-Based Predictions

The main alternatives answer related but different questions. A Kaplan–Meier estimate describes event-free survival over time without imposing this parametric Weibull shape; it is useful for descriptive comparisons but does not by itself provide the same exposure-adjusted parametric predictions. A Cox proportional-hazards model leaves the baseline hazard unspecified while estimating hazard ratios, so it is less restrictive about baseline shape but does not directly supply a Weibull-based absolute survival curve without additional baseline estimation. The Weibull model gives a compact baseline and direct survival predictions, at the cost of assuming that baseline form. These methods do not remove confounding in observational exposure comparisons.

Diagnostics, Failure Modes, and Limitations

Diagnostics should address both the event-time form and how exposure and censoring enter the analysis. Compare model-predicted survival with descriptive survival patterns where data support that comparison. For a Weibull baseline, a plot of log cumulative hazard against log time should be approximately linear if the Weibull shape is adequate; systematic curvature is a warning, not an automatic proof of failure. Assess whether exposure contrasts appear compatible with a constant hazard ratio over time. Marked time-dependent differences challenge proportional hazards and may require a different model.

Also inspect event counts and follow-up across the exposure range. Sparse events, little exposure overlap, or few participants remaining under observation late in follow-up can make estimates and long-term predictions unstable. Compare censoring patterns by exposure and relevant covariates; concentration of early loss to follow-up in a group may signal a problem with the conditional independent-censoring assumption. A model cannot recover event-time information that was never observed without assumptions.

• Confusing hazard and risk. A hazard ratio of 0.67 is not necessarily a 33% reduction in event probability by a fixed time. Calculate \(1-S(t\mid x)\) to report an absolute risk.
• Calling censored follow-up an event. A participant event-free through month 6 contributes \(S(6)\), not the event density at month 6.
• Ignoring time and units. A prediction at 12 months is not interchangeable with one at 12 weeks. State exposure units, the exposure contrast, and the time origin.
• Over-interpreting exposure association. Differences in exposure may reflect prognosis or other covariates. A fitted association does not by itself show that increasing dose will improve efficacy.

The model concerns time to a first event and right-censoring. It does not automatically handle competing event types, recurrent efficacy events, exposure changing over time, or informative dropout. Its predictions also depend on the population and exposure range represented in the data; extrapolating to unsupported exposure levels or follow-up times can be misleading.