Selecting Doses by Predicted Event Risk

From Exposure Effects to a Dose Decision

A dose regimen is not chosen because its hazard ratio is small or large in isolation. The clinical meaning of a modeled exposure effect depends on the baseline event risk, the time horizon, and the uncertainty in the exposure-response relationship. A hazard ratio of 1.5 can correspond to a modest absolute risk increase in a low-risk population and a much larger increase in a high-risk population. Dose selection therefore requires predictions on an absolute-risk scale for a stated population and time horizon.

This tutorial considers a single event type and a fixed prediction horizon $t^*$. The model is assumed to describe the event process adequately for the target population, and the predictions are conditional on the covariates and exposure information used by the model. If event hazards vary over time with concentration, the cumulative hazard must account for that time variation; the simple constant hazard-ratio conversion below applies when the dose-specific hazard is a constant multiplier of the reference hazard over the horizon.

Definition: The predicted event risk under regimen $d$ by horizon $t^*$ is the probability that the event has occurred by $t^*$ under that regimen, conditional on the specified baseline covariates and model. In a single-event proportional-hazards model, if the reference cumulative hazard is $H_0(t^*\mid z)$ and regimen $d$ has log hazard ratio $\eta_d$, then $$ R_d(t^*\mid z)=1-\exp\left\{-H_0(t^*\mid z)\exp(\eta_d)\right\}. $$ Here $z$ denotes the covariate profile. A positive $\eta_d$ increases the hazard relative to reference; it does not by itself give the absolute event risk.

Deriving the Risk Conversion

The hazard is an instantaneous event rate among individuals who have not yet had the event. The cumulative hazard through $t^*$ is the integral of that hazard over time. For a single event type, survival through the horizon is the exponential of the negative cumulative hazard. If $\lambda_d(t\mid z)=\lambda_0(t\mid z)\exp(\eta_d)$, with $\eta_d$ constant over the horizon, integration gives $H_d(t^*\mid z)=H_0(t^*\mid z)\exp(\eta_d)$. Thus the regimen-specific risk follows from one minus survival.

\begin{aligned} H_d(t^*\mid z)&=\int_0^{t^*}\lambda_d(t\mid z)\,dt =\exp(\eta_d)H_0(t^*\mid z),\\ R_d(t^*\mid z)&=1-\exp\{-H_d(t^*\mid z)\}. \end{aligned}

Often the model output or clinical benchmark is a reference risk $R_0(t^*\mid z)$, rather than a cumulative hazard. Since $R_0=1-\exp(-H_0)$, it follows that $H_0=-\ln(1-R_0)$. Substituting this identity yields the risk conversion. Its assumptions matter: the reference risk must apply to the same horizon and relevant covariate profile; the hazard ratio must be appropriate over that horizon; and the event definition and competing-event handling must match the prediction target.

R_d(t^*\mid z) =1-(1-R_0(t^*\mid z))^{\exp(\eta_d)}.

Worked Example: Converting a Hazard Effect to Absolute Risk

Suppose the hypothetical reference risk of an event by 12 months is 8.00%. Two regimens have modeled log hazard ratios relative to reference of 0.24 and 0.60. Calculate their 12-month risks and absolute increases over reference.

For the first regimen, the hazard ratio is $\exp(0.24)=1.271249$. Using the reference survival of $1-0.0800=0.9200$,

R_1=1-0.9200^{\exp(0.24)} =1-0.9200^{1.271249} \approx 1-0.89943 =0.10057.

The predicted risk is about 10.06% by 12 months. Its absolute increase over the 8.00% reference risk is $10.057\%-8.000\%=2.057$ percentage points, or approximately 2.06 percentage points.

For the second regimen, $\exp(0.60)=1.822119$. Retaining precision through the calculation,

R_2=1-0.9200^{\exp(0.60)} =1-0.9200^{1.822119} \approx 1-0.859047 =0.140953.

The predicted risk is approximately 14.10%, an absolute increase of $14.0953\%-8.000\%=6.0953$ percentage points, or 6.10 percentage points. These are model-based predictions for the stated hypothetical reference risk and horizon, not observed event rates.

Baseline Risk and the Target Population

A relative hazard effect is not a fixed absolute effect. At the same horizon, the same hazard ratio produces different predicted risks when the reference risks differ. This is why dose comparisons should use a baseline risk appropriate to the patients who will receive the regimen, rather than a convenient but mismatched study average.

Worked Example: Why Patient Mix Changes the Risk Comparison

Consider a hypothetical population split equally between two covariate groups. At 12 months, their reference risks are 5.00% and 20.0%. A regimen has a hazard ratio of 1.50 in both groups. Calculate the regimen risk in each group and the equally weighted population-average risk.

For the first group, the reference survival is 0.9500:

R_{\mathrm{low}} =1-0.9500^{1.50} \approx 1-0.925945 =0.074055.

Thus the risk is about 7.41%, an increase of approximately 2.41 percentage points from 5.00%. For the second group,

R_{\mathrm{high}} =1-0.8000^{1.50} \approx 1-0.715542 =0.284458.

The risk is about 28.45%, an increase of approximately 8.45 percentage points from 20.0%. With equal group weights, the population-average regimen risk is $(0.074055+0.284458)/2=0.179257$, or about 17.93%.

The average reference risk is 12.5%. Applying the hazard ratio directly to that average gives $1-0.8750^{1.50}\approx0.181513$, or 18.15%, not 17.93%. The difference arises because risk is a nonlinear function of baseline survival. For a target population with heterogeneous baseline risks, calculate group-specific risks and average them using the target population’s group proportions.

The same principle applies when patient characteristics and exposure vary continuously. Predictions may be calculated for each target individual or representative covariate profile and then averaged with appropriate population weights. Applying the model only to mean covariates or mean exposure can give a different answer because, in general, the risk function evaluated at an average is not the average of the risks.

Account for Exposure-Response Uncertainty

A fitted exposure-response coefficient is estimated with uncertainty. Treating its point estimate as exact can make two dose options appear more decisively different than the evidence supports. A decision-focused comparison should propagate uncertainty in the exposure-response parameters, the baseline risk, and, where relevant, individual exposure predictions. One practical approach is to calculate predicted risks for each draw from a joint parameter distribution, then summarize the resulting risk distribution.

The resulting distribution describes uncertainty in model-based predictions under the specified assumptions. Its mean is an average predicted risk over the parameter draws, while the fraction of draws exceeding a chosen threshold estimates how often the prediction exceeds that threshold under the uncertainty representation. Neither quantity alone determines the clinical decision: the threshold, benefits, harms, and consequences of being wrong must be specified in context.

Worked Example: Comparing Regimens With a Risk Threshold

In a hypothetical decision analysis, the target population has an 8.00% reference risk by 12 months. Four equally weighted draws from the exposure-response uncertainty distribution produce the following hazard ratios for a lower-dose regimen: 1.10, 1.20, 1.30, and 1.40. For a higher-dose regimen, the corresponding draws are 1.20, 1.50, 1.80, and 2.10. For illustration, suppose the decision rule is to flag a regimen if its 12-month event risk exceeds 12.0%. Calculate the risks and compare the options.

For each draw, use $R=1-0.9200^{HR}$. For the lower-dose regimen, the four risks are approximately 8.76%, 9.52%, 10.27%, and 11.02%. Their equally weighted mean is $(0.08764+0.09522+0.10273+0.11018)/4=0.09894$, or 9.89%. None of the four draws exceeds 12.0%.

For the higher-dose regimen, the risks are approximately 9.52%, 11.76%, 13.94%, and 16.06%. Their mean is $(0.09522+0.11757+0.13937+0.16063)/4=0.12820$, or 12.82%. Two of the four draws exceed 12.0%, so the fraction above the threshold is $2/4=0.50$.

Under this simplified uncertainty representation and the stated rule, the higher-dose regimen has a greater average predicted risk and a 50% fraction of draws above the threshold, while no lower-dose draw exceeds it. This supports closer scrutiny of the higher-dose option; it does not establish that the lower dose is clinically preferable. The calculation omits any efficacy benefit of the higher dose and assumes the 8.00% reference risk applies to the target population. In a real decision, baseline-risk uncertainty and the joint distribution of model parameters should also be propagated.

Making the Comparison Decision-Relevant

Before comparing doses, define the event, prediction horizon, target population, and reference regimen. Then state the decision quantity: for example, absolute risk difference, probability of exceeding a safety threshold, or expected utility that combines benefit and harm. The following sequence helps keep the modeling and the decision aligned.

Specify the target.
Name the event, horizon, population, and covariate distribution. Use the same definitions for each regimen.

Generate regimen-specific predictions.
Use the fitted exposure-response relationship and regimen-specific exposure predictions. Convert cumulative hazard to risk at the horizon.

Represent uncertainty and heterogeneity.
Recalculate predictions over parameter and exposure uncertainty, and average individual or subgroup risks using target-population weights.

Apply a prespecified decision criterion.
Compare absolute risks and risk differences with a clinically justified threshold or an explicit benefit-harm rule.

A simple risk threshold may be suitable for illustrating a safety constraint, but dose choice usually involves multiple outcomes. If a higher dose improves disease control, the decision must weigh that benefit against the event risk; a risk-only comparison cannot resolve the trade-off. A hazard ratio or risk difference is a model summary, not a complete decision rule.

Diagnostics, Alternatives, and Limitations

Predicted-risk comparisons inherit the assumptions and weaknesses of the event model. Check whether the model reproduces observed event frequencies over the relevant horizon and whether predictions are calibrated in important patient subgroups. Examine whether exposure and covariate values for the proposed regimens are supported by the data; extrapolation beyond observed exposure can make an apparently precise prediction unreliable. Review uncertainty in the baseline hazard and exposure predictions, not only uncertainty in the exposure-response coefficient.

A constant hazard-ratio conversion is inappropriate if the exposure effect varies over time in a way the model does not capture. In that situation, use the modeled time-varying hazard and integrate it over the horizon before converting to event probability. If competing events affect whether the event of interest can occur, a single-event survival conversion may not equal the probability of that event; use a prediction method that represents the competing-event process. The target prediction must match the event definition and estimand.

The main alternative summaries answer different questions. A hazard ratio compares instantaneous hazards among individuals still event-free, while an absolute risk difference compares probabilities by a specified time. Relative risk compares event probabilities as a ratio. For decisions, absolute risks and differences are often easier to relate to a threshold, but they depend on the target baseline risk and horizon. An expected-utility approach is more complete when benefits and harms can be assigned meaningful consequences, although it requires explicit values and assumptions.

Common mistake: applying a hazard ratio directly to a probability. Multiplying an 8% risk by 1.5 gives 12%, but that is not the proportional-hazards risk conversion. The correct calculation is $1-(1-0.08)^{1.5}$, which is about 11.76%.
Common mistake: assuming one absolute effect applies to every patient. A common hazard ratio can yield very different risk increases across baseline-risk groups. Calculate risks for relevant groups before averaging.
Common mistake: reporting a point prediction as certain. A single predicted risk omits parameter and exposure uncertainty. Show uncertainty in the decision quantity and explain what distribution it represents.
Common mistake: choosing a dose from event risk alone. If doses differ in efficacy, a risk-only result is incomplete. State the benefit-harm criterion and the consequences of threshold exceedance.

Key takeaway: Dose selection by predicted event risk requires an explicit horizon and target population. Convert modeled hazards to absolute risks using a suitable baseline risk, account for heterogeneity and uncertainty, and interpret the comparison through a stated clinical decision rule.

Check Your Understanding

Answer each question using the stated horizon and assumptions. Short solutions follow the questions.

At a specified horizon, the reference risk is 10% and the regimen hazard ratio is 2. What is the regimen risk under a constant proportional-hazards effect?
Why can applying a hazard ratio to the average baseline risk differ from averaging subgroup-specific risks?
What does it mean if 30% of uncertainty draws for a regimen exceed a prespecified risk threshold?
Name two reasons a proportional-hazards risk conversion may be unsuitable for a proposed dose comparison.

Solutions. 1. $R=1-0.90^2=0.19$, so the predicted risk is 19.0%. 2. The conversion $1-(1-R_0)^{HR}$ is nonlinear in $R_0$, so its value at the average baseline risk need not equal the average of its subgroup-specific values. 3. Under the uncertainty distribution used, 30% of draws produce predicted risks above the threshold; this is not the probability that the model or null hypothesis is true. 4. Examples include a time-varying hazard ratio not represented by the constant multiplier, a mismatched baseline risk or horizon, unsupported exposure extrapolation, and competing events not represented by the single-event calculation.