Absolute Risk With Continuous Covariates

Why Predict Across a Continuous Covariate?

A prediction for one covariate profile answers a specific question: what is the estimated probability of each event type by a chosen horizon for a person with these recorded characteristics? A continuous covariate makes it possible to ask a broader question: how does that predicted probability change as one measurement changes, when the other predictors are held fixed? Examples include age, a laboratory value, or a continuous measure of disease severity.

The result is a model-based curve, not a collection of unrelated predictions. Its shape reflects both the fitted relationship between the covariate and each cause-specific hazard and the way competing events alter who remains event-free. The prediction horizon and the values held fixed must therefore be stated along with the curve.

Definition: For a continuous covariate \(x\), a fixed vector \(w\) of the other predictors, and horizon \(\tau\), an absolute-risk prediction curve for cause \(k\) is the function \(x\mapsto \widehat{F}_k(\tau\mid x,w)\). It gives the model-estimated probability of experiencing cause \(k\) by \(\tau\), before any event, conditional on the covariate profile \((x,w)\). Each point on the curve is a prediction for that profile; it is not a population-average risk.

From a Continuous Predictor to Absolute Risk

One common approach models each cause-specific hazard. With \(K\) mutually exclusive causes, write the hazard for cause \(k\) as

\lambda_k(t\mid x,w)=\lambda_{k0}(t)\exp\{\beta_{kx}x+\gamma_k^\mathsf{T}w\}.

Here, \(\lambda_{k0}(t)\) is the cause-specific baseline hazard, \(\beta_{kx}\) is the coefficient for \(x\) for cause \(k\), and \(\gamma_k\) contains the coefficients for the other predictors. The model allows the association with \(x\) to differ across causes. For each value of \(x\) on a chosen grid, the fixed vector \(w\) is entered unchanged.

The cause-specific hazard alone is not the probability of that cause by \(\tau\). The probability also depends on remaining free of every event up to each time. For a profile \((x,w)\), let \(\Lambda_k(t\mid x,w)\) be the cause-specific cumulative hazard and let \(S(t\mid x,w)\) be the probability of no event by \(t\). In continuous time,

S(t\mid x,w)=\exp\left\{-\sum_{j=1}^{K}\Lambda_j(t\mid x,w)\right\}, \qquad F_k(\tau\mid x,w)=\int_0^\tau S(u\mid x,w)\,d\Lambda_k(u\mid x,w).

The integral weights the cause-\(k\) hazard at each time by the probability of still being event-free just before that time. Thus a competing cause reduces the probability of remaining at risk for the cause of interest. With fitted cause-specific Cox models, one obtains each profile-specific cumulative hazard from the fitted baseline cumulative hazard and coefficient estimates, then combines the causes to obtain the cumulative incidence.

Assumptions and conditions: These predictions rely on the fitted models being suitable for the intended prediction population and horizon. In a cause-specific proportional-hazards model, each cause-specific hazard must have the specified covariate form, including any proportional-hazards assumption and the chosen functional form for \(x\). The causes must be defined as mutually exclusive first events, and the models must account for all relevant competing causes. Estimation also requires an appropriate censoring assumption, commonly independent censoring conditional on included predictors. A curve is most credible over covariate values supported by the data. For this tutorial’s fixed-profile predictions, predictors are treated as constant over follow-up.

Calculating and Reading a Risk Curve

In practice, choose a meaningful set of values of \(x\), keep \(w\) fixed, and calculate \(\widehat{F}_k(\tau\mid x,w)\) at each value. A grid can be a short table or a smooth plotted curve. The units and scale of \(x\) matter: if \(x\) is age in years, a coefficient is per year; if it is entered per 10 years, the coefficient is per 10-year increase. Centering \(x\) changes the reference value, not the fitted predictions, when the model is transformed consistently.

The effect on absolute risk is generally not a straight line, even when \(x\) enters linearly in the log hazard. The exponential link changes the hazards, and the cumulative incidence combines them through survival from all causes over time. The cause-specific risk can rise less steeply, flatten, or even change direction relative to another cause as \(x\) changes.

Worked Example: A Linear Predictor and Two Competing Causes

Suppose a hypothetical model has two constant cause-specific hazards over three years:

\lambda_1(x)=0.04\exp(0.5x),\qquad \lambda_2(x)=0.02\exp(-0.25x).

Here \(x\) is a continuous measurement recorded in units of 10 points, and all other predictors are fixed at zero. The hazards are per year. Assume the constant hazards are appropriate over this three-year horizon. With constant hazards, the total hazard is \(\lambda_\cdot=\lambda_1+\lambda_2\), and

F_k(3\mid x)=\frac{\lambda_k(x)}{\lambda_\cdot(x)} \left[1-\exp\{-3\lambda_\cdot(x)\}\right].

This follows because event-free probability at time \(t\) is \(\exp(-t\lambda_\cdot)\), so integrating \(\exp(-t\lambda_\cdot)\lambda_k\) from zero to three gives the displayed expression.

At \(x=0\), the hazards are \(0.04000\) and \(0.02000\) per year, so \(\lambda_\cdot=0.06000\). The total failure risk by year 3 is \(1-\exp(-3(0.06000))=0.1647\). The cause-specific risks are \(F_1=(0.04000/0.06000)(0.1647)=0.1098\) and \(F_2=(0.02000/0.06000)(0.1647)=0.05491\).

At \(x=1\), \(\lambda_1=0.04\exp(0.5)=0.06595\) and \(\lambda_2=0.02\exp(-0.25)=0.01558\) per year. Their sum is \(0.08152\). The total risk is \(1-\exp(-3(0.0815249))=0.2170\), giving \(F_1=(0.0659489/0.0815249)(0.216963)=0.1755\) and \(F_2=0.04146\).

At \(x=2\), the hazards are \(0.1087\) and \(0.01213\) per year, for a total of \(0.1209\). The total risk is \(1-\exp(-3(0.1208619))=0.3041\). Using the unrounded hazards, the risks are \(F_1\approx0.2736\) and \(F_2\approx0.03054\). The risks for the causes sum to the total failure risk at each covariate value, up to rounding. Across these three values, cause 1’s predicted risk rises while cause 2’s falls; neither curve should be interpreted as a causal effect.

Piecewise Hazards and Changing Cause Shares

A fitted model need not have constant hazards over time. To see how a curve is constructed when the relative contributions of causes change across intervals, suppose a hypothetical piecewise-constant model gives the following baseline integrated hazards in two successive two-year intervals. Let \(x\) be continuous, let the other predictors be fixed at zero, and let the cause-specific coefficients be \(0.4\) and \(-0.2\). Within each interval, assume constant hazards and use \(x=1\).

Worked Example: Combining Hazards Over Two Intervals

For the first interval, the baseline integrated hazards are \(0.03\) for cause 1 and \(0.02\) for cause 2. At \(x=1\), they become \(a_1=0.03\exp(0.4)=0.04475\) and \(a_2=0.02\exp(-0.2)=0.01637\). The interval’s total integrated hazard is \(a_\cdot=0.06113\). Its event probability, conditional on being event-free at the start, is \(q_1=1-\exp(-0.06113)=0.05930\). The cause shares are \(a_1/a_\cdot=0.7321\) and \(a_2/a_\cdot=0.2679\). Since everyone is event-free at time zero, the first-interval contributions to the cumulative incidences are \(0.05930(0.7321)=0.04341\) and \(0.05930(0.2679)=0.01589\).

For the second interval, the baseline integrated hazards are \(0.01\) and \(0.04\). At \(x=1\), they are \(0.01\exp(0.4)=0.01492\) and \(0.04\exp(-0.2)=0.03275\), with total \(0.04767\). The conditional event probability in this interval is \(q_2=1-\exp(-0.04767)=0.04655\). The probability of reaching the interval event-free is \(\exp(-0.06113)=0.9407\), so the event probability contributed over the interval is \(0.9407(0.04655)=0.04378\). Its cause 1 share is \(0.01492/0.04767=0.3130\), giving increments of approximately \(0.01370\) for cause 1 and \(0.03008\) for cause 2.

Adding the increments gives \(F_1(4\mid x=1)\approx0.05711\) and \(F_2(4\mid x=1)\approx0.04597\), for total event risk \(0.1031\). Event-free probability at year 4 is \(\exp\{-(0.06113+0.04767)\}=0.8969\); it complements the total event risk, up to rounding. The cause shares differ by interval, so assigning a single cause share to the four-year total would not give the same result.

Functional Form Matters

A continuous covariate need not enter the model as a single linear term. A quadratic term or a spline can represent curvature in the log hazard. The same prediction steps apply: evaluate the fitted covariate terms at each grid value, calculate the hazards for all causes, and combine them to obtain cumulative incidence. A flexible fit can improve representation of a curved relationship, but it does not remove the need to inspect how much data support the curve at its extremes.

Worked Example: A Curved Hazard Relationship

Suppose a hypothetical two-cause model over five years uses \(\lambda_1(x)=0.02\exp(0.3x-0.1x^2)\) and \(\lambda_2(x)=0.01\exp(-0.1x)\), with \(x\) measured in the original scale and other predictors fixed. Assume constant hazards over the horizon. At \(x=-1\), the hazards are \(0.02\exp(-0.4)=0.01341\) and \(0.01\exp(0.1)=0.01105\) per year. The total hazard is \(0.02446\), the total risk is \(1-\exp(-5(0.02446))=0.1150\), and the cause 1 risk is \((0.01341/0.02446)(0.1150)=0.0630\); cause 2 risk is about \(0.0520\).

At \(x=0\), the hazards are \(0.02000\) and \(0.01000\). The total risk is \(1-\exp(-0.15)=0.1393\), split into \(F_1=0.09286\) and \(F_2=0.04643\). At \(x=1\), the hazards are \(0.02\exp(0.2)=0.02443\) and \(0.01\exp(-0.1)=0.009048\) per year. The total risk is \(1-\exp(-5(0.03348))=0.1541\), with \(F_1\approx0.1125\) and \(F_2\approx0.04163\). These calculated points illustrate a rising cause 1 risk and a falling cause 2 risk over this range. A plotted curve would require evaluating the same fitted equations at additional values; it should not be inferred that the risk changes linearly between the displayed points.

Diagnostics, Alternatives, and Common Pitfalls

Before interpreting a curve, check its construction and its domain. Confirm the covariate’s units, centering, and fixed values for the other predictors; verify that the chosen horizon is the same at every grid point; and check that all competing causes are included in the calculation. At each point, cause-specific cumulative incidences should be nonnegative, should not decrease with time, and should sum to the total probability of failure by the horizon. Their sum plus event-free probability should equal one, subject to the prediction procedure and rounding.

A smooth-looking curve is not evidence that the functional form is adequate. Sparse observations at one end of the covariate range can make predictions there depend strongly on model assumptions. Residual or model-fit checks for the relevant hazard models, inspection of the observed covariate distribution, and comparison of plausible functional forms can reveal problems. A curve that behaves implausibly near the observed range may indicate a poor functional form, influential observations, or an implementation error. Predictions far beyond the observed range are extrapolations, not data-supported conclusions.

Cause-specific Cox models and Fine–Gray models provide different routes to absolute-risk prediction. Cause-specific models describe the instantaneous rate of a particular event among people who have not yet had any event, and predictions combine the cause-specific hazards. Fine–Gray models directly model a subdistribution hazard associated with the cumulative incidence. Either route requires correct model specification and appropriate estimation; their coefficients do not have the same interpretation. For a given profile, the main practical check is whether the predicted risks are computed for the intended cause, horizon, and competing-event structure.

Common mistake: “The cause 1 hazard ratio is constant per unit increase in \(x\), so the absolute risk must increase by a constant amount per unit.” This is wrong. A hazard ratio describes a relative change in a cause-specific hazard, not an additive change in cumulative incidence. Absolute risk depends on the baseline hazards, time horizon, other causes, and the functional form. Calculate the cumulative incidence at each covariate value instead of translating a hazard ratio into an absolute-risk slope.

A final limitation is that a fixed-profile curve does not describe the average risk in a population with varying values of the other predictors. Nor does it show how prediction uncertainty changes across \(x\); point estimates alone should not be mistaken for equally precise estimates everywhere. Those are separate prediction questions. Here, the curve answers a conditional question: how do the fitted risks vary with \(x\), for the stated fixed values of \(w\) and the chosen horizon?

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

For a prediction curve indexed by age, what does “holding other predictors fixed” mean?
Why can a competing cause change the absolute-risk curve for the cause of interest even if its own hazard does not depend on \(x\)?
In the two-cause constant-hazard example, what is the total three-year risk at \(x=0\), and how does it relate to the two cause-specific risks?
What is one reason not to interpret the far end of a continuous-covariate curve when few observations occur there?
Does a cause-specific hazard ratio give the additive change in absolute risk per unit of the predictor? Explain briefly.

Solutions. (1) Use the same specified values of every predictor other than age at each age-grid value; this defines a conditional profile, not a population average. (2) The competing cause contributes to the total hazard and lowers event-free probability, which weights the cause-of-interest hazard in the cumulative-incidence calculation. (3) The total risk is \(1-\exp(-3(0.06))=0.1647\); the cause-specific risks are approximately \(0.1098\) and \(0.05491\), which sum to \(0.1647\). (4) The prediction may be extrapolating beyond well-supported data and can depend heavily on the assumed functional form. (5) No. A hazard ratio is a relative hazard association; absolute risk also depends on time, baseline hazards, and competing causes.