From Exposure Records to a Decision
An exposure-event analysis has to connect several kinds of information without confusing them: concentration over time, time at risk, event or censoring status, and the clinical decision the analysis is intended to inform. A concentration recorded at one visit is not necessarily the exposure relevant to an event that occurred between visits. Likewise, a hazard ratio is not itself an absolute risk or a dose recommendation. This tutorial follows a hypothetical dataset through those steps, then makes the assumptions and limitations visible.
The worked analysis uses a deliberately simple model so that its likelihood and calculations can be checked directly. It is an illustration of an analysis workflow, not evidence about a real drug or a clinically validated dosing rule. Earlier material on integrating time-varying concentration into event risk, censoring, hazard-model diagnostics, and selecting doses by predicted event risk supplies the broader framework.
Build and Check the Exposure and Event Data
The analysis begins by choosing an exposure summary that matches the scientific question and the time scale of the event process. If concentration measurements are available at discrete times, a simple construction between two adjacent measurements is linear interpolation. The area under the interpolated concentration curve over an interval is the trapezoid area. Dividing by interval duration gives the interval-average concentration.
This is an approximation when the true concentration curve is not linear between samples. A population or individual PK model may instead be used to predict \(C_i(t)\) between observations; the choice should reflect the available data and the intended exposure measure. For an instantaneous exposure effect, the relevant predictor may be \(C_i(t)\) at each time. For a cumulative or lagged effect, the predictor must be constructed to represent that mechanism. A single interval average is not automatically appropriate for every exposure-response question.
Worked Example: Constructing an Interval-Average Concentration
Suppose a hypothetical participant has concentrations of 1.0 mg/L at hour 0, 1.6 mg/L at hour 4, and 1.2 mg/L at hour 8. Under linear interpolation between measurements, calculate the average concentration over the 8-hour interval.
The trapezoid areas are \(4(1.0+1.6)/2=5.2\) mg·h/L for hours 0–4 and \(4(1.6+1.2)/2=5.6\) mg·h/L for hours 4–8. Thus the interval area is \(5.2+5.6=10.8\) mg·h/L, and
The arithmetic is correct for the stated interpolation assumption. It does not establish that 1.35 mg/L is the biologically relevant exposure summary; that depends on the exposure-effect mechanism and event timing.
Next, audit the event record. An observed event time is the time from the defined origin to the first qualifying event. A right-censored time is the time from that same origin to the last time the participant is known to be event-free. Censoring must not be coded as an event. For the basic model below, the censoring mechanism is assumed independent of the event process conditional on the modeled information; if participants leave follow-up because their event risk is changing, this assumption may fail. Entry time, competing events, recurrent events, and interval-censored event times also require the corresponding likelihood and risk-set treatment rather than a simple substitution into the example model.
For illustration, imagine 14 hypothetical participants with time-aligned concentration records and complete event/censoring dates. Their follow-up is summarized by exposure level as follows. Person-time is accumulated only while a participant is under observation and at risk; an event contributes an event count and ends first-event follow-up.
| Interval-average concentration | Person-time at risk | Observed events | Event rate |
|---|---|---|---|
| 1.0 mg/L | 20 person-years | 4 | 0.200 events/person-year |
| 3.0 mg/L | 10 person-years | 5 | 0.500 events/person-year |
These totals are meaningful only if the concentration assignments are aligned to the time each participant was actually at risk. For example, a concentration measured after an event must not be used as though it preceded that event. A practical audit checks time origin, units, duplicate or impossible dates, event definitions, exposure timing, and whether each censored participant was event-free at last contact.
Fit a Simple Exposure-Hazard Model
To make the fitting transparent, assume a constant baseline hazard over the analyzed period and a constant interval-average concentration within each exposure stratum. Model the hazard at concentration \(C\) as \(\lambda(C)=\lambda_0\exp\{\beta(C-1)\}\), where concentration is in mg/L, \(\lambda_0\) is the hazard at 1 mg/L, and \(\beta\) is the log hazard ratio per 1 mg/L increase. This model assumes the specified concentration relationship is log-linear, that the hazard is constant over time at a fixed concentration, and that the observed censoring is noninformative conditional on the modeled information. The two-stratum illustration does not assess whether these assumptions are realistic.
With \(d_j\) events and \(T_j\) person-years in stratum \(j\), the exponential event-time likelihood, apart from factors not involving the parameters, is proportional to \(\lambda_j^{d_j}\exp(-\lambda_jT_j)\). Its log likelihood is \(d_j\log(\lambda_j)-\lambda_jT_j\). Differentiating with respect to \(\lambda_j\) gives \(d_j/\lambda_j-T_j=0\), so the maximum-likelihood estimate is \(\hat{\lambda}_j=d_j/T_j\). In this two-concentration example, the estimated log hazard ratio per mg/L is therefore the log rate ratio divided by the concentration difference.
Worked Example: Estimating the Exposure-Hazard Relationship
Use the table’s 4 events in 20 person-years at 1.0 mg/L and 5 events in 10 person-years at 3.0 mg/L. The estimated rates are \(4/20=0.200\) and \(5/10=0.500\) events per person-year. The rate ratio is \(0.500/0.200=2.50\), so
The fitted model is \(\hat{\lambda}(C)=0.200\exp\{0.4581(C-1)\}\) events per year. Under this model, a 1 mg/L increase multiplies the hazard by \(\exp(0.4581)=1.581\), or about 58.1%. That is a relative hazard comparison, not a 58.1 percentage-point increase in event risk.
For a simple Wald calculation, the approximate variance of the log rate ratio is \(1/d_H+1/d_L\), giving \(\operatorname{SE}(\hat{\beta})=\sqrt{1/5+1/4}/2=\sqrt{0.45}/2=0.3354\) per mg/L. Testing \(H_0:\beta=0\) against \(H_a:\beta\ne0\), the Wald statistic is \(z=0.4581/0.3354=1.366\), with a two-sided \(p\)-value of about 0.172. The approximate 95% confidence interval is \(0.4581\pm1.96(0.3354)=[-0.1993,1.116]\) per mg/L. The data do not provide strong evidence against \(H_0\) at the 0.05 level, and the interval includes no exposure association as well as potentially substantial positive association. This calculation uses a large-sample approximation that may be imprecise with only nine events.
The derivation also indicates why exposure time matters: counting events alone would give \(5/4=1.25\) as the high-to-low event-count ratio, which is not the rate ratio because the high-exposure stratum has half as much person-time. The likelihood uses both events and time at risk. In a richer analysis with time-varying concentration and covariates, the hazard and its integrated value must be evaluated along each participant’s observed exposure and risk history; the grouped constant-hazard calculation is not a replacement for that model.
Translate the Hazard into Absolute Risk
For a constant hazard \(\lambda\) over a time horizon \(\tau\), the event-free survival probability is \(S(\tau)=\exp(-\lambda\tau)\), and the probability of an event by \(\tau\) is \(1-S(\tau)=1-\exp(-\lambda\tau)\). The conversion requires a stated time horizon and a hazard model applicable over that horizon. It also assumes the event is modeled as the event of interest without unaccounted competing events.
Worked Example: Predicting One-Year Risk at an Intermediate Exposure
At 1.5 mg/L, the fitted hazard is \(\hat{\lambda}(1.5)=0.200\exp\{0.4581(1.5-1)\}=0.2515\) events per year. The predicted one-year event risk is
The model therefore predicts a 22.2% one-year event risk at 1.5 mg/L, under its assumptions. For comparison, at 1.0 mg/L the predicted one-year risk is \(1-\exp(-0.200)=0.1813\), or 18.1%; at 3.0 mg/L it is \(1-\exp(-0.500)=0.3935\), or 39.3%. These are model-based risks, not observed proportions, and their precision is limited by the small illustrative dataset.
Use Predictions Carefully in a Dose Recommendation
A dose decision requires a clinically specified risk threshold, a relationship between dose and exposure, a defined prediction horizon, and consideration of uncertainty and other treatment objectives. Suppose, only for illustration, that the decision criterion is a predicted one-year risk below 25%, and that average concentration is proportional to dose: 50 mg corresponds to 1.0 mg/L and 150 mg to 3.0 mg/L. This implies \(C=0.0200D\), with \(D\) in mg and \(C\) in mg/L.
Worked Example: Applying a Prespecified Risk Threshold
Solve for the maximum concentration with predicted one-year risk no greater than 0.25. From \(1-\exp\{-\hat{\lambda}(C)\}=0.25\), the threshold hazard is \(-\log(0.75)=0.2877\) per year. Equating this to the fitted hazard gives
Under the assumed proportional dose-exposure relationship, this corresponds to \(D=1.79/0.0200=89.5\) mg. A conservative illustrative choice of 75 mg gives \(C=0.0200(75)=1.50\) mg/L, for which the predicted risk is 22.2%, below the stipulated 25% threshold. This arithmetic does not establish that 75 mg is safe or effective. It ignores uncertainty in the fitted hazard, the dose-exposure relation, individual PK variability, and the clinical consequences of alternative doses. A real recommendation would require an appropriate uncertainty analysis, external or prospective predictive evaluation, and review of the full benefit-risk context.
Diagnostics, Alternatives, and Failure Modes
An end-to-end workflow should treat diagnostics as part of the analysis, not as a final decoration. Check whether event times and risk intervals are correctly represented; compare observed and predicted event counts and risks over time and across exposure groups; inspect residuals and calibration; and examine whether the exposure effect changes over time or departs from the assumed log-linear form. Earlier material on proportional-hazards diagnostics and event-model residuals gives tools for more flexible models. The grouped example cannot diagnose these features from two exposure strata.
The constant-hazard model is attractive here because its rate estimates and risk conversion are transparent. A Cox model is a common alternative when a flexible baseline hazard is desired and proportional hazards is defensible; it estimates relative exposure effects without requiring a particular baseline-hazard shape, but absolute-risk prediction needs a baseline survival estimate. Parametric survival models specify a time-varying hazard and can support extrapolation, but their predictions depend on the chosen distribution. For genuinely time-varying concentration, models that integrate the exposure-linked hazard over time are more appropriate than assigning each participant a single post hoc average. The model should be selected for the estimand and data structure, not because one method is universally preferable.
- Exposure timing error: using concentrations measured after an event to predict that event can reverse temporal ordering and bias the exposure association.
- Informative censoring: if dropout depends on unobserved or changing event risk, treating it as independent can distort both fitted hazards and absolute risk.
- Model form mismatch: a constant baseline hazard or linear log-hazard exposure effect may not describe the data. Systematic residual patterns or poor calibration are warnings.
- Sparse data: few events make coefficient estimates, standard errors, and risk predictions unstable; a nonsignificant result does not demonstrate no effect.
- Overconfident dose choice: a point prediction under one model is not a guarantee that an individual’s risk is below a threshold.
A further limitation is that this example uses grouped event counts and person-time, not a full individual-level analysis. It does not adjust for confounding, represent between-person PK variability, or account for competing events. If such features are present, the corresponding modeling framework and assumptions are needed. A dose recommendation should also be tested against uncertainty in both the exposure-response model and the dose-to-concentration relationship.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A participant has 6 events over 30 person-years in one exposure group and 4 events over 10 person-years in another. What is the event-rate ratio for the second group versus the first?
- Why is a participant whose follow-up ends event-free not counted as an event?
- Under a constant hazard of 0.10 events per year, calculate the three-year event risk.
- In the fitted example, what is the predicted one-year risk at 1.0 mg/L, and why is it not the same as the hazard?
- Name one reason the illustrative 75 mg dose should not be treated as a clinical recommendation.
Solutions. (1) The rates are \(6/30=0.200\) and \(4/10=0.400\) events per person-year, so the rate ratio is \(0.400/0.200=2.00\). (2) The participant is known to be event-free only through the last observed time; censoring contributes time at risk but does not establish that an event occurred. (3) The risk is \(1-\exp(-0.10\times3)=1-\exp(-0.30)=0.2592\), or 25.9%. (4) At 1.0 mg/L, the fitted hazard is 0.200 per year and the one-year risk is \(1-\exp(-0.200)=0.1813\), or 18.1%. Hazard is an instantaneous rate; risk is the probability of an event over a time interval. (5) For example, the dose is based on a small illustrative dataset and omits uncertainty in the exposure-hazard fit and the dose-exposure relationship.