Prediction For A Specified Covariate Profile

From a Covariate Profile to an Absolute Risk

A fitted regression model does not give an absolute risk merely by supplying its coefficients. To predict risk for a particular person or profile, one must specify the covariate values, apply the relevant coefficients, combine the result with the baseline estimate, and account for the possibility that another cause occurs first. The target here is a conditional cause-specific cumulative incidence: the probability of experiencing a particular cause by a chosen horizon, for a subject with the stated profile.

The earlier material on absolute-risk prediction establishes that a cause-specific cumulative incidence is not the same as a net risk calculated as if competing causes could not occur. The prediction below retains competing events: each cause contributes to the chance of leaving the event-free state, and therefore affects the risk accumulated for every other cause.

Definition: A specified covariate profile, denoted \(\mathbf{x}^{\ast}\), is a stated set of values for all covariates in the prediction model, using the model’s coding and reference categories. The cause-\(k\) absolute risk by time \(t\) for that profile is \(F_k(t\mid\mathbf{x}^{\ast})=P(T\leq t,J=k\mid\mathbf{X}=\mathbf{x}^{\ast})\), where \(T\) is time to the first event and \(J\) identifies its cause.

How Coefficients and Baseline Estimates Enter

For a cause-specific Cox model for cause \(k\), the hazard among subjects who have not yet had any event is modelled as \(\lambda_k(t\mid\mathbf{x})=\lambda_{0k}(t)\exp(\boldsymbol{\beta}_k^{\mathsf T}\mathbf{x})\). Here \(\lambda_{0k}(t)\) is the cause-specific baseline hazard, corresponding to the reference profile \(\mathbf{x}=\mathbf{0}\), and \(\boldsymbol{\beta}_k\) is the coefficient vector for that cause. At the specified profile, the linear predictor is \(\eta_k^\ast=\boldsymbol{\beta}_k^{\mathsf T}\mathbf{x}^\ast\), and the hazard multiplier is \(\exp(\eta_k^\ast)\). The profile-specific hazard is the estimated baseline hazard multiplied by this factor.

The coefficient multiplier alone is not an absolute risk. Its meaning is a relative change in the cause-specific hazard, conditional on still being event-free. To obtain absolute risk, we need a baseline estimate and the hazards for all relevant causes. For continuous-time hazards, define the profile-specific cumulative hazard by \(\Lambda_k(t\mid\mathbf{x}^{\ast})=\exp(\eta_k^\ast)\Lambda_{0k}(t)\). Under the usual continuous-time formulation, event-free probability is \(S(t\mid\mathbf{x}^{\ast})=\exp\{-\sum_j\Lambda_j(t\mid\mathbf{x}^{\ast})\}\), and the cause-\(k\) cumulative incidence is

F_k(t\mid\mathbf{x}^{\ast}) =\int_0^t S(u-\mid\mathbf{x}^{\ast})\,d\Lambda_k(u\mid\mathbf{x}^{\ast}).

The survival factor in the integral matters: cause \(k\) can occur at time \(u\) only among those who have remained free of every event up to just before \(u\). Thus a competing cause reduces the population still available to experience cause \(k\).

A useful exact calculation is available when the profile-specific cause hazards are constant within each time interval. If those hazards on an interval of length \(\Delta\) are \(\lambda_1,\ldots,\lambda_K\), then the probability of any event during that interval, conditional on being event-free at its start, is \(1-\exp(-\Delta\sum_j\lambda_j)\). Cause \(k\) receives the fraction \(\lambda_k/\sum_j\lambda_j\) of that interval’s event probability. This follows because its cumulative incidence increment is the integral of its constant hazard against event-free probability:

\int_0^\Delta e^{-u\sum_j\lambda_j}\lambda_k\,du =\frac{\lambda_k}{\sum_j\lambda_j} \left(1-e^{-\Delta\sum_j\lambda_j}\right).

This calculation assumes the hazards are constant within the interval. With estimated hazard jumps, such as those used in a Cox model, the prediction must instead use a compatible cumulative-incidence calculation for those jumps. Treating a jump as a constant rate over an arbitrary interval changes the model and can change the prediction.

Worked Predictions

Worked Example: Two-Interval Cause-Specific Cox Prediction

Hypothetical data. Consider two event causes and one binary covariate, \(X\), with \(X=0\) as the reference. The profile of interest has \(X=1\). Suppose the cause-specific Cox coefficients are \(\beta_1=\log(2)\) and \(\beta_2=\log(1.25)\). Estimated baseline hazards are constant within each of two one-year intervals: for cause 1 they are \(0.05\) in year 1 and \(0.08\) in year 2; for cause 2 they are \(0.04\) and \(0.016\), respectively. Hazards are in inverse years.

For \(X=1\), the multipliers are \(\exp(\beta_1)=2\) and \(\exp(\beta_2)=1.25\). Therefore the profile-specific cause hazards are \(0.10\) and \(0.05\) per year in year 1, and \(0.16\) and \(0.020\) per year in year 2. We assume these piecewise-constant hazards are appropriate for this calculation, and that the fitted models and baseline estimates are valid for this profile.

In year 1, the total hazard is \(0.15\) per year. The event-free probability at year 1 is \(S(1)=\exp(-0.15)=0.860708\). The total probability of an event during that year is \(1-\exp(-0.15)=0.139292\). Allocating this by the cause-specific hazard shares gives

F_1(1)=\frac{0.10}{0.15}(1-e^{-0.15})=0.0928613,\qquad F_2(1)=\frac{0.05}{0.15}(1-e^{-0.15})=0.0464307.

In year 2, the total hazard is \(0.18\) per year, and the conditional event probability for someone event-free at year 1 is \(1-e^{-0.18}=0.164730\). Multiply each cause’s share of this probability by the probability of having reached year 2 event-free:

\begin{aligned} \Delta F_{1,2} &=e^{-0.15}\frac{0.16}{0.18}(1-e^{-0.18}) \approx 0.12603,\\ \Delta F_{2,2} &=e^{-0.15}\frac{0.02}{0.18}(1-e^{-0.18}) \approx 0.01575. \end{aligned}

Adding the increments, the two-year risks are \(F_1(2)\approx0.0928613+0.12603=0.2189\) and \(F_2(2)\approx0.0464307+0.01575=0.06218\). The event-free probability at year 2 is \(\exp[-0.15-0.18]=0.718924\), so the two cause risks sum to approximately \(1-0.718924=0.281076\), as they should. These are predicted probabilities by two years for the specified profile, not causal effects of changing \(X\).

Worked Example: A Constant-Hazard Profile Over Three Years

Hypothetical data. Suppose a binary profile indicator \(X\) has cause-specific coefficients \(\beta_1=\log(2)\) and \(\beta_2=\log(0.5)\). The baseline hazards are constant over three years: \(0.05\) per year for cause 1 and \(0.04\) per year for cause 2. Predict for \(X=1\). Assume the constant hazards adequately describe the full three-year period.

The profile multipliers are \(2\) and \(0.5\), so the profile-specific hazards are \(\lambda_1=0.05(2)=0.10\) and \(\lambda_2=0.04(0.5)=0.020\) per year. The total hazard is \(0.12\) per year. The probability of any event by year 3 is \(1-e^{-0.12(3)}=1-e^{-0.36}=0.302324\). The cause shares are \(0.10/0.12=5/6\) and \(0.020/0.12=1/6\). Hence

F_1(3\mid X=1)=\frac{0.10}{0.12}(0.302324)=0.251937,\qquad F_2(3\mid X=1)=\frac{0.020}{0.12}(0.302324)=0.0503873.

The predictions are about \(25.19\%\) for cause 1 and \(5.039\%\) for cause 2 by three years. Their sum, \(0.302324\), is the probability of either event by that horizon. Omitting cause 2 from the event-free calculation would overstate the amount of time during which subjects remain at risk for cause 1.

Worked Example: Predicting a Target Cause From a Fine–Gray Model

Hypothetical data. A Fine–Gray model for cause 1 has a binary profile indicator \(X\), coefficient \(\beta=\log(1.5)\), and estimated baseline subdistribution cumulative hazard \(\widetilde{\Lambda}_{01}(2)=0.20\) at two years. For this illustration, take the subdistribution hazard as continuous over the horizon and predict for \(X=1\). Assume the Fine–Gray model and its baseline estimate are suitable for that profile.

The profile multiplier is \(\exp(\beta)=1.5\), giving a profile-specific subdistribution cumulative hazard of \(0.20(1.5)=0.30\). In this continuous-hazard formulation, the predicted cause-1 cumulative incidence is

F_1(2\mid X=1)=1-\exp(-0.30)=0.259182.

Thus the model predicts a cause-1 risk of about \(25.92\%\) by two years for this profile. This calculation predicts the target cause’s cumulative incidence directly; it does not by itself provide a coherent prediction for the risk of every other cause. In applications with baseline hazard jumps, use the corresponding jump-based Fine–Gray prediction formula rather than silently treating the jumps as continuous.

Assumptions, Checks, and Alternatives

A numerical prediction is meaningful only if the model and the profile are well specified. For cause-specific Cox prediction, proportional hazards means that, for a fixed profile, the covariate multiplier for a cause does not vary with time. The model also requires an appropriate functional form for its covariates, a suitable baseline estimate, and censoring that is independent of event times conditional on the modelled information (or an estimation procedure that addresses the censoring structure). Predictions are especially uncertain for profiles poorly represented in the data: a model may technically return a number outside the covariate patterns with substantial support, but that is extrapolation rather than strong empirical evidence.

Before reporting a prediction, check the following:

Profile coding: confirm reference categories, indicator values, and any transformations match the fitted model. A coefficient is applied to the model’s encoded value, not to a prose description of the person.
All causes: when using cause-specific hazards, include each competing cause in the event-free probability. A missing cause can inflate the target-cause risk.
Time scale and units: baseline hazards and interval widths must use the same time units. A rate per year cannot be multiplied by an interval measured in months without conversion.
Probability bounds and totals: each cumulative incidence must be between 0 and 1. For mutually exclusive causes, their sum should not exceed one; together with event-free probability, the total should equal one up to numerical rounding.
Model fit and support: assess proportional-hazards diagnostics and whether the profile is represented in the observed covariate data. A plausible-looking prediction can still be unreliable if the model form is wrong or the profile is far outside the data.

The main alternatives answer somewhat different questions. A cause-specific Cox approach models each cause’s instantaneous hazard among those still event-free, then combines the hazards to obtain competing-risk cumulative incidence. A Fine–Gray model directly models the subdistribution hazard for a target cause and can predict that cause’s cumulative incidence; separate target-cause fits do not automatically guarantee that all predicted causes form a coherent joint distribution. Aalen–Johansen estimation can estimate cumulative incidence directly for a group without imposing a covariate regression model, but it does not by itself provide a model-based prediction for an arbitrary profile. The suitable choice depends on the prediction target, data support, and modelling assumptions.

These methods have limitations. A specified-profile prediction is conditional on the fitted model and does not automatically describe the average risk in a population. Regression coefficients are associations in the model, not necessarily causal effects. Baseline estimation, coefficient estimation, and model misspecification all affect the result; a point prediction alone does not communicate its uncertainty. Uncertainty intervals require additional calculations and should not be inferred from the coefficient multiplier.

Common Mistake

A frequent error is to calculate \(1-\exp[-\Lambda_k(t\mid\mathbf{x}^{\ast})]\) from one cause-specific hazard and call it the absolute risk of that cause. That expression treats the target cause as though it were the only way to leave the event-free state. In competing risks, the correct cause-specific risk accumulates the target hazard only among those still free of all causes:

F_k(t\mid\mathbf{x}^{\ast}) =\int_0^t \exp\left\{-\sum_j\Lambda_j(u-\mid\mathbf{x}^{\ast})\right\} \,d\Lambda_k(u\mid\mathbf{x}^{\ast}).

The correction is to predict each relevant cause-specific hazard at the profile, combine them to obtain event-free probability, and then accumulate the target cause’s events from that event-free risk set.

Key takeaway: A profile-specific absolute-risk prediction needs both parts of the model: the coefficient-based multiplier for the specified profile and a baseline estimate. For cause-specific models, all competing hazards must also enter the calculation of who remains event-free.

Check Your Understanding

Answer each question, then compare with the solutions below.

A binary profile has coefficient \(\log(3)\) in a cause-specific Cox model. What hazard multiplier does the profile receive for that cause?
With constant profile-specific hazards \(0.06\) and \(0.04\) per year for two causes, calculate each cause’s one-year cumulative incidence.
A predicted target-cause risk is computed from one cause-specific hazard alone, without including a competing cause. What is the conceptual error?
A Fine–Gray prediction for one cause is available. Does it, by itself, determine a coherent prediction for all other causes?

Solutions. (1) The multiplier is \(\exp\{\log(3)\}=3\); the profile’s cause-specific hazard is three times the baseline hazard. (2) The total hazard is \(0.10\) per year and the event probability is \(1-e^{-0.10}=0.0951626\). Therefore \(F_1=(0.06/0.10)(0.0951626)=0.0570976\), and \(F_2=(0.04/0.10)(0.0951626)=0.0380650\). The risks sum to the probability of either event during the year. (3) The calculation fails to reduce the event-free risk set for subjects who experience the competing event, so it generally overstates the target cause’s absolute risk. (4) No. A Fine–Gray model for one target cause predicts that cause’s cumulative incidence; separate cause-specific predictions are needed for other causes, and separate fits do not automatically ensure a coherent joint distribution.