Time-Horizon Risk From Survival Models

From Relative Genetic Effects to Risk by a Time

A hazard ratio describes how quickly an event occurs among people who are still event-free at a given time. It does not, by itself, tell us the probability that a person will experience the event over a specified period. That probability also depends on the underlying event rate and the time horizon. A survival model combines these pieces, allowing a polygenic score and other predictors to be translated into an estimated risk by a chosen future time.

The central calculation is straightforward once its inputs are clear: adjust the model’s baseline survival using the person’s relative hazard, then subtract the resulting survival probability from one. The calculation relies on a proportional-hazards model and on baseline survival that is appropriate to the population, time origin, and predictor reference values in the model.

Definition: Let \(T\) be time to the event of interest, and let \(S(t\mid z,x)=P(T>t\mid z,x)\) denote the probability of remaining event-free beyond time \(t\), conditional on polygenic score \(z\) and other predictors \(x\). The event risk by time \(t\) is \(P(T\leq t\mid z,x)=1-S(t\mid z,x)\). In a proportional-hazards model, \(S(t\mid z,x)=S_0(t)^{\exp\{\eta(z,x)\}}\), where \(S_0(t)\) is baseline survival and \(\exp\{\eta(z,x)\}\) is the relative hazard compared with the model’s reference predictors.

The Proportional-Hazards Calculation

Write the model’s linear predictor as \(\eta(z,x)=\beta z+\gamma^\mathsf{T}x\), where \(z\) is the score on the scale used to fit the model, \(x\) contains other predictors, and \(\beta\) and \(\gamma\) are their fitted coefficients. A coefficient is on a log-hazard scale. For a one-unit increase in \(z\), holding the other predictors fixed, the hazard ratio is \(\exp(\beta)\). For a particular person, \(\exp\{\eta(z,x)\}\) is the hazard ratio relative to the baseline predictor values, conventionally those for which \(\eta=0\).

The proportional-hazards assumption means this relative hazard does not change over time. The baseline hazard itself can change over time; the model does not require the event rate to be constant. If the score was standardized, a one-unit increase usually means one standard deviation on the specified standardization scale. The model’s units and reference values must be used as fitted, not inferred from a score’s raw numerical value.

\lambda(t\mid z,x)=\lambda_0(t)\exp\{\eta(z,x)\}, \qquad S(t\mid z,x)=S_0(t)^{\exp\{\eta(z,x)\}}.

Here \(\lambda(t\mid z,x)\) is the event hazard at time \(t\) among people with predictors \(z,x\) who have remained event-free until \(t\). The baseline hazard \(\lambda_0(t)\) corresponds to the reference predictors, and \(S_0(t)\) is the corresponding baseline survival probability. Baseline survival is not the same as baseline risk: the baseline risk by time \(t\) is \(1-S_0(t)\).

Derivation: The cumulative hazard is the integral of the hazard over time. Since \(\exp\{\eta(z,x)\}\) is constant over time under proportional hazards, \(H(t\mid z,x)=\exp\{\eta(z,x)\}H_0(t)\). Survival equals the exponential of negative cumulative hazard, so \(S(t\mid z,x)=\exp[-H(t\mid z,x)]=\exp[-H_0(t)\exp\{\eta(z,x)\}]\). Because \(S_0(t)=\exp[-H_0(t)]\), this is \(S_0(t)^{\exp\{\eta(z,x)\}}\).

The final subtraction from one is essential: survival is the probability of no event by the horizon, while risk is the probability of an event by then. A hazard ratio greater than one generally lowers survival and raises risk, but it does not multiply the risk by that same factor. The relationship between hazard ratio and risk depends on baseline survival and the chosen time horizon.

Worked Example: Risk by 10 Years for a Score Value

Suppose a hypothetical model reports baseline survival of \(S_0(10)=0.960\). The score is measured in standard-deviation units, and the fitted hazard ratio per one-unit increase is \(1.50\). For a person with score \(z=1.20\) and reference values for other predictors, calculate the 10-year risk.

The score-specific relative hazard is \(1.50^{1.20}\). Thus the person’s 10-year survival is baseline survival raised to this relative hazard:

\begin{aligned} \exp\{\eta(z)\} &=1.50^{1.20}\approx 1.626707,\\ S(10\mid z=1.20) &=0.960^{1.50^{1.20}}\\ &=\exp\!\left(1.50^{1.20}\ln(0.960)\right)\\ &\approx \exp(-0.0664054)\approx 0.9357514,\\ P(T\leq10\mid z=1.20) &=1-0.9357514\approx0.0642486. \end{aligned}

The estimated 10-year event risk is about \(0.0642\), or \(6.42\%\), and the estimated probability of remaining event-free beyond 10 years is about \(0.9358\). These are model-based probabilities for the specified score and reference predictors, not consequences of the hazard ratio alone.

Combining the Score With Clinical Predictors

When other predictors are included, their contributions belong in the same linear predictor. Equivalently, if the model supplies separate hazard ratios for specified predictor changes, multiply those hazard ratios to obtain the person’s relative hazard, provided they come from the same fitted model and the model has no unaccounted interactions for those values. Then apply that relative hazard to the baseline survival. The baseline must correspond to the reference values for all predictors, not merely to a reference score.

Worked Example: A Score and a Binary Clinical Predictor

Consider a hypothetical model with baseline survival \(S_0(8)=0.920\). A one-unit increase in the standardized score has hazard ratio \(1.40\). A binary clinical predictor has hazard ratio \(1.20\) for the value present in this person, relative to its reference value. Assume the model contains these effects additively on the log-hazard scale and has no interaction between them. For score \(z=0.80\), calculate the 8-year risk.

The score contributes a relative hazard of \(1.40^{0.80}\), while the clinical predictor contributes \(1.20\). The combined relative hazard is their product. Carrying the full value through the survival calculation gives:

\begin{aligned} \exp\{\eta(z,x)\} &=1.40^{0.80}\times1.20 \approx1.570666,\\ S(8\mid z,x) &=0.920^{1.40^{0.80}\times1.20}\\ &=\exp\!\left[(1.40^{0.80}\times1.20)\ln(0.920)\right]\\ &\approx\exp(-0.1309646)\approx0.8772489,\\ P(T\leq8\mid z,x) &=1-0.8772489\approx0.1227511. \end{aligned}

The estimated 8-year event risk is about \(0.1228\), or \(12.28\%\), and estimated survival beyond 8 years is about \(0.8772\). This interpretation assumes that the model’s baseline survival and predictor effects apply to the person’s target population and time scale.

Baseline Survival and Right Censoring

In follow-up data, some people have not experienced the event when observation ends. Their event times are right-censored: we know that they remained event-free until at least their censoring time, but do not know their eventual event time. Censoring is not itself an event, and a censored person should not simply be counted as event-free through the end of the study.

Survival methods use the time each person was observed and whether an event occurred or follow-up ended without an event. For example, the Kaplan–Meier estimator multiplies conditional survival proportions at observed event times. Its usual interpretation requires censoring to be independent of event time within the group being analyzed. In a covariate-adjusted survival model, the relevant assumption is typically that censoring is independent of the event time conditional on the modeled predictors. If people at higher unobserved risk are more likely to leave follow-up, this assumption can fail, and estimated survival may be biased.

A baseline survival curve may be estimated from survival data or supplied by a fitted model. It is specific to a time origin and predictor reference values. For example, a risk measured from study enrollment is not automatically a risk from birth or from a person’s current age. For a prediction beginning at a particular age among people event-free at that age, the model and its baseline survival must represent that conditional starting point and follow-up horizon.

Worked Example: Censoring in a Small Reference Group

Suppose a hypothetical reference group has 10 people at risk at time 1, with one event. At time 2, one of the nine remaining people is censored, and no event occurs. At time 3, two of the eight remaining people experience the event. Use the Kaplan–Meier product to calculate estimated survival just after time 3, then use a hypothetical score hazard ratio of \(1.30\) to calculate risk by that time for a person one score unit above reference.

At time 1, the conditional survival proportion is \(9/10\). The censoring at time 2 does not create an event-time factor in the product; it reduces the number at risk at time 3 from nine to eight. At time 3, the conditional survival proportion is \(6/8\), because two of the eight at risk experience the event. Therefore:

S_0(3)=\frac{9}{10}\times\frac{6}{8}=0.675.

For a person with relative hazard \(1.30\), the model-based survival and risk are:

\begin{aligned} S(3\mid z=1) &=0.675^{1.30}\\ &=\exp\!\left(1.30\ln(0.675)\right)\\ &\approx\exp(-0.5109554)\approx0.5999,\\ P(T\leq3\mid z=1) &=1-0.5999\approx0.4001. \end{aligned}

The estimated risk by time 3 is about \(40.01\%\), conditional on the model assumptions and the hypothetical baseline estimate. This small calculation illustrates how censoring affects the number at risk; it is not enough data to support a reliable clinical prediction.

Assumptions and Interpretation in Practice

A computed number is meaningful only if the model and its inputs match the question. Before interpreting a time-horizon risk, check the following:

Proportional hazards: the relative hazard associated with the score and other predictors is assumed constant over the modeled follow-up time.
Appropriate baseline survival: \(S_0(t)\) must refer to the same time origin, outcome definition, population, and reference predictor values used for the prediction.
Correct predictor scale: the score value and hazard ratio must use the same units and standardization. A hazard ratio per standard deviation cannot be applied as though it were per raw score point.
Suitable censoring: estimation assumes censoring is independent of event time, unconditionally for a simple group estimate or conditional on modeled predictors as appropriate.
Single event framework: the formula gives survival for the modeled event process. If another event can prevent the event of interest, treating that event as ordinary censoring does not generally yield the real-world cumulative probability of the event of interest. A competing-risks analysis is needed for that target.

The resulting risk is conditional on the predictors and modeling framework. It is not a guarantee for an individual, nor does it establish that a genetic score causes the event. It is also not automatically calibrated simply because the relative effects were estimated precisely; baseline survival must be suitable for the population and horizon where the prediction will be used.

Common Mistakes

Calling the hazard ratio a risk ratio. A hazard ratio compares instantaneous event rates among those still event-free; risk is a probability accumulated through a specified time. Use \(1-S_0(t)^{\exp\{\eta\}}\) for the model-based risk.
Multiplying baseline risk by the hazard ratio. The proportional-hazards model transforms baseline survival, not baseline risk. For instance, multiplying \(1-S_0(t)\) by a hazard ratio can produce the wrong answer or even a value above one.
Ignoring the reference predictors. Baseline survival applies at the model’s reference values. It cannot be treated as an unconditional population probability unless those are indeed the intended baseline conditions.
Treating censoring as an event or as complete event-free follow-up. A censored observation contributes follow-up only until its censoring time; it does not reveal what happened afterward.
Rounding the relative hazard early. Keep full precision through exponentiation and round the final survival and risk. Small intermediate rounding differences can affect the displayed last digit.

Key takeaway: Under a proportional-hazards model, calculate the person’s relative hazard, raise baseline survival at the chosen horizon to that value, and subtract from one. Interpret the result only with the model’s time origin, reference predictors, population, and censoring assumptions in view.

Check Your Understanding

Use the model assumptions and definitions above to answer each question.

If \(S_0(t)=0.90\) and a person’s relative hazard is \(2\), write the formula for that person’s survival and risk by time \(t\).
Why is a hazard ratio of \(1.5\) not generally the same as a risk ratio of \(1.5\) by a chosen horizon?
In a model with score and clinical predictors, what predictor values does the baseline survival represent?
What does right censoring tell us about a participant’s event time, and what does it not tell us?
Why must the time origin and target population be checked before using a baseline survival value for an individual prediction?