Time-to-First Adverse Event Models

Why Model Time to the First Event?

A first-adverse-event analysis asks not only whether an event occurred, but also when it occurred. Two treatment groups can have the same proportion of patients with an event by the end of follow-up while differing in how quickly events arise. Survival methods use the event times of patients who experience an event and the partial follow-up information from those who do not.

The event and time origin must be defined before fitting a model. For example, an analysis might measure time from the first dose to the first occurrence of a specified adverse event. A patient who completes follow-up without that event is right-censored: the event time is known to exceed the observed follow-up time, but its later value is unknown. A patient who stops observation early without a recorded event is also censored at the last time the event status is known. Censoring is not the same as an event.

Definition: Let \(T\) be the time from a specified origin to the first adverse event of interest. The survival function is \(S(t)=P(T>t)\), the probability that the event has not occurred by time \(t\). The hazard \(h(t)\) is the instantaneous event rate at time \(t\), conditional on remaining event-free just before \(t\). The cumulative hazard is \(H(t)=\int_0^t h(u)\,du\). These quantities satisfy \(S(t)=\exp[-H(t)]\).

The hazard has units of inverse time, such as events per patient-year. It is a conditional instantaneous rate, not the probability of an event during the entire study. The survival function, by contrast, is a probability; \(1-S(t)\) is the probability of an event by time \(t\), under the model and endpoint definition.

Cox and Parametric Exposure Models

A Cox proportional-hazards model relates covariates, including exposure, to the hazard while leaving the baseline hazard unspecified:

h(t\mid x,z)=h_0(t)\exp(\beta x+\gamma^\mathsf{T}z).

Here \(h_0(t)\) is the baseline hazard, \(x\) is exposure, and \(z\) denotes other covariates. If \(x\) is a fixed exposure contrast, \(\exp(\beta)\) is the hazard ratio for a one-unit increase in \(x\), holding the included covariates fixed. The proportional-hazards assumption means that this ratio is constant over time. It does not mean that either group’s hazard is constant over time.

The Cox model is often useful when the relative hazard is the main target and a particular baseline-hazard shape is not justified. Its partial likelihood estimates the regression coefficients from the ordering of event times and the covariates of those at risk at each event time. To estimate absolute event probabilities, however, one also needs an estimate of the baseline survival or cumulative hazard.

A parametric model specifies the baseline hazard’s shape as well. For example, a Weibull proportional-hazards model can be written as:

S(t\mid x)=\exp\left[-(t/\lambda)^k\exp(\beta x)\right], \qquad h_0(t)=\frac{k}{\lambda}(t/\lambda)^{k-1}.

Here \(k>0\) is the shape parameter and \(\lambda>0\) is a time-scale parameter. When \(k=1\), the baseline hazard is constant; when \(k\ne1\), it changes with time. Parametric models can provide smooth survival estimates and extrapolation, but those advantages depend on the assumed distribution being adequate over the period of interest.

Key result: Under proportional hazards with a fixed exposure contrast \(x\), \(H(t\mid x)=H_0(t)\exp(\beta x)\) and \(S(t\mid x)=S_0(t)^{\exp(\beta x)}\). Thus an estimated hazard ratio can be converted to an event probability only when the corresponding baseline survival is also available.

Assumptions and a Short Derivation

The survival relationship follows directly from the definition of the hazard. For a continuous event-time distribution, the event density is \(f(t)=-S'(t)\). Since \(h(t)=f(t)/S(t)\), it follows that \(h(t)=-S'(t)/S(t)=-d\log S(t)/dt\). Integrating from time zero, where \(S(0)=1\), gives \(\log S(t)=-H(t)\), and therefore \(S(t)=\exp[-H(t)]\). This also shows why cumulative hazards add across time intervals.

Interpretation requires attention to the design and data, not just the model equation. The time origin and event definition should be clinically coherent, and exposure information must be aligned with time. If exposure changes during follow-up, using a future or post-event exposure summary as though it were known at baseline can introduce bias. A time-updated exposure should use values available up to each risk time, with its timing and measurement process made explicit.

Censoring: Standard likelihood and partial-likelihood interpretations require censoring to be non-informative conditional on the modeled information: among patients with the same modeled history, censoring should not convey additional information about their unobserved event time.
Proportional hazards: A single Cox exposure coefficient assumes a constant exposure hazard ratio over time. This is a modeling assumption to assess, not a consequence of fitting the model.
Exposure contrast and confounding: Exposure coefficients describe conditional associations given included covariates. They do not by themselves establish a causal effect, particularly when exposure is related to patient characteristics or treatment changes.
Endpoint and follow-up: The event must be ascertained consistently, and the chosen time window must support the intended interpretation. Other events that prevent observation of the adverse event may require additional methods; they should not automatically be treated as ordinary independent censoring.

Worked Example: Interpreting a Cox Exposure Estimate

Suppose a hypothetical Cox model relates a baseline exposure measure to time until a first specified adverse event. Exposure is scaled in increments of 5 mg/L. The fitted coefficient is \(\hat{\beta}=0.350\), with standard error \(SE=0.120\). Assume the proportional-hazards diagnostic does not show a material departure, the exposure contrast is defined before follow-up, and censoring is plausibly non-informative conditional on the model covariates.

Hazard ratio. For an increase of 5 mg/L, the estimated hazard ratio is:

\exp(\hat{\beta})=\exp(0.350)=1.419.

In this hypothetical analysis, patients differing by 5 mg/L in baseline exposure have an estimated 1.419-fold instantaneous hazard of the first adverse event at a given time, conditional on still being event-free and holding modeled covariates fixed. This is not a 41.9 percentage-point increase in event probability.

Uncertainty. A large-sample Wald interval for the coefficient is \(\hat{\beta}\pm1.96SE\). Therefore:

0.350\pm1.96(0.120) =0.350\pm0.2352 =(0.1148,\ 0.5852).

Exponentiating the endpoints gives the approximate 95% confidence interval for the hazard ratio:

\left(\exp(0.1148),\exp(0.5852)\right) =(1.122,\ 1.795).

The interval describes values compatible with the data and model under the interval procedure; it is not a 95% probability statement about this particular fixed hazard ratio. The Wald statistic is \(z=0.350/0.120=2.917\), giving an approximate two-sided \(p\)-value of 0.0035 under \(H_0:\beta=0\). This is evidence against a zero exposure association under the model, not proof of causation.

From Hazard Ratios to Event Probabilities

A hazard ratio is not itself an event probability. Under proportional hazards, use the baseline survival at the same time point and covariate reference to obtain a probability. This calculation depends on the proportional-hazards model being appropriate and on having a suitable baseline survival estimate.

Worked Example: Translating a Hazard Ratio Into 12-Month Risk

In hypothetical data, suppose the estimated 12-month survival at a reference exposure is \(S_0(12)=0.900\), and the hazard ratio for a specified higher-exposure contrast is 1.5. Assume proportional hazards over the 12-month period and a consistent endpoint and follow-up definition.

The survival at the higher exposure is:

S(12)=S_0(12)^{1.5} =0.900^{1.5} =0.8538.

Thus the model-implied probability of a first event by 12 months is:

P(T\leq12)=1-S(12)=1-0.8538=0.1462.

The estimated event probability is 14.62% at the higher exposure, compared with \(1-0.900=0.100\), or 10.0%, at the reference exposure. These are model-based probabilities, not a direct conversion of the hazard ratio into a risk difference. They also do not account for uncertainty unless uncertainty in both the hazard ratio and baseline survival is propagated.

When the Exposure Effect Changes Over Time

A changing exposure value and a changing exposure effect are distinct. A time-dependent covariate represents exposure that varies during follow-up; for example, the current measured concentration can be updated at each risk time. A time-varying coefficient represents an exposure association whose hazard ratio itself changes over time. Either or both may be needed, but neither should be added merely to improve a numerical fit without a defensible interpretation.

One way to represent a changing coefficient is \(h(t\mid x)=h_0(t)\exp[\beta(t)x]\). A practical model may use separate coefficients in prespecified time intervals or an interaction between exposure and a function of time. The resulting hazard ratio must be reported for the relevant time or interval, rather than as though one constant ratio applied throughout follow-up.

Worked Example: Calculating Risk With a Changing Exposure Effect

Consider a hypothetical patient whose exposure is \(x=1\) in the model’s chosen units. The baseline cumulative hazard increment is 0.020 in the first interval and 0.030 in the second. The exposure coefficient is 0.60 in the first interval and 0.10 in the second. Suppose the baseline increments and coefficients are appropriate for this two-interval model, censoring is handled under the stated assumptions, and there are no unmodeled competing events in this calculation.

In each interval, the patient’s cumulative-hazard increment is the baseline increment multiplied by \(\exp(\beta x)\). The total cumulative hazard through the second interval is:

\begin{aligned} H(12) &=0.020\exp(0.60)+0.030\exp(0.10)\\ &\approx0.020(1.8221188)+0.030(1.1051702)\\ &\approx0.03644+0.03316\\ &\approx0.0695975. \end{aligned}

The event probability through both intervals is therefore:

P(T\leq12)=1-\exp[-H(12)] =1-\exp(-0.0695975) \approx1-0.93277 =0.06723.

The model-implied probability is about 6.723%. The two interval-specific exposure hazard ratios are \(\exp(0.60)=1.822\) and \(\exp(0.10)=1.105\); it would be misleading to summarize this model with just one constant hazard ratio. The time units here are those used to define the intervals, and the cumulative-hazard increments are dimensionless.

Diagnostics, Alternatives, and Limitations

A Cox model’s proportional-hazards assumption can be assessed using scaled Schoenfeld residuals, plots of residual patterns against time, and comparisons of observed and fitted survival curves. A systematic time trend in residuals for exposure is a warning that a constant coefficient may be inadequate. Such diagnostics are not automatic proof of an assumption: with limited events, they may have little power, and a visually irregular pattern may be random variation. Check whether the apparent departure is clinically meaningful and whether it persists under reasonable modeling choices.

Also examine whether exposure values are measured at the intended times, whether influential observations dominate the estimate, whether event times and censoring are recorded consistently, and whether estimated event counts support the model complexity. For parametric models, compare fitted survival patterns with observed data over the supported follow-up range and inspect whether the assumed hazard shape is plausible. Good fit within observed follow-up does not guarantee reliable extrapolation beyond it.

Approach	Strength	Main caution
Cox proportional hazards	Estimates relative hazards without specifying a baseline hazard shape	Assumes a constant hazard ratio unless the model is extended; absolute risks require baseline survival estimation
Parametric survival model	Provides a specified survival curve and can support smooth prediction	Results depend on the chosen distribution, especially for extrapolation
Binary event analysis at a fixed time	Directly compares event status by a chosen time point	Discards event timing and may not handle unequal follow-up as efficiently as a survival model

The fixed-time binary approach may be suitable when the clinical question is explicitly about status at a fixed, well-observed time, but it does not use the timing information in the same way. Conversely, survival models require careful handling of censoring and time definitions. In either approach, an exposure association can reflect confounding or selection as well as a pharmacologic relationship; design knowledge and appropriate covariate modeling remain essential.

Common Pitfall

Common mistake: “The hazard ratio is 1.5, so the event probability is 50% higher.” A hazard ratio compares instantaneous hazards among patients still event-free at each time; it is not a risk ratio or a percentage-point difference. To estimate event probability at a time point under proportional hazards, combine the hazard ratio with baseline survival using \(S(t\mid x)=S_0(t)^{\exp(\beta x)}\). If proportional hazards is not credible, a single hazard ratio and this conversion may be misleading.

Specify the estimand.
Define the first event, time origin, follow-up window, exposure contrast, and covariates.

Choose the model.
Use Cox when avoiding a baseline-shape assumption is valuable; use a parametric model when its shape is defensible and the survival curve is needed.

Check assumptions and timing.
Assess proportional hazards, censoring, exposure measurement timing, and model fit over observed follow-up.

Report in context.
Give hazard ratios with uncertainty, and report time-specific event probabilities only with a corresponding baseline survival estimate and clear assumptions.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

A patient is followed for 8 months without an event, then withdraws. What information does this contribute to a first-event survival analysis?
A Cox model gives \(\hat{\beta}=0.20\) per 2 mg/L exposure increase. Calculate and interpret the hazard ratio.
At a reference exposure, 6-month survival is 0.80. Under proportional hazards, the hazard ratio for a higher exposure is 2. Calculate the higher-exposure event probability by 6 months.
State one diagnostic signal that could indicate a time-varying exposure effect and one limitation of that diagnostic.
Why is a fitted hazard ratio not by itself an absolute event probability?

Solutions. (1) The observation is right-censored at 8 months: the patient is known to have remained event-free through that time, but the later event time is unknown. (2) \(\exp(0.20)=1.221\); the estimated instantaneous hazard is 1.221-fold per 2 mg/L increase, conditional on the model covariates and being event-free. (3) \(S(6)=0.80^2=0.640\), so \(P(T\leq6)=1-0.640=0.360\), or 36.0%. (4) A systematic trend in scaled Schoenfeld residuals against time can signal non-proportionality; with few events the diagnostic may have low power. (5) A hazard ratio is relative and conditional, whereas absolute probability requires the baseline survival at the specified time as well as the model assumptions.