Combining Models When Causes Use Different Predictors

Why Different Predictor Sets Can Be Combined

A competing-risk prediction often uses one model for each event cause. The model for one cause may use age and a biomarker, while the model for another uses age and a comorbidity score. This is not inherently a problem: each model can provide its cause-specific hazard for the same prediction target. The essential requirement is to combine all the cause-specific hazards in the event-free survival calculation before calculating any cause-specific cumulative incidence.

The procedure does not require every cause model to have the same predictors. It does require their predictions to be compatible: they must refer to the same time origin, prediction horizon, event-free state, and target population or covariate profile. In particular, models cannot be combined coherently if one hazard prediction is conditional on a full covariate profile while another is a marginal prediction for a different population, unless those predictions have been brought to a common target.

Definition: Let \(K\) be the number of mutually exclusive causes of a first event. For cause \(k\), let \(\lambda_k(t\mid z_k)\) be its cause-specific hazard at time \(t\), evaluated using the predictors \(z_k\) included in that cause's model. For a shared prediction target, the event-free survival is \(S(t)=\exp\{-\sum_{k=1}^{K}\Lambda_k(t)\}\), where \(\Lambda_k(t)=\int_0^t\lambda_k(u\mid z_k)\,du\). The cumulative incidence for cause \(k\) is \(F_k(t)=\int_0^t S(u)\lambda_k(u\mid z_k)\,du\).

The definition uses \(z_k\) to emphasize that the predictor sets can differ by cause. A covariate needed by one model need not appear in every other model. For a particular target profile, each model is evaluated using the values of the predictors that it uses; the resulting hazards are then combined. If a predictor is omitted from a cause model, its omission must be appropriate for the intended prediction. For example, the model must not need that predictor to capture the cause-specific hazard for the target being predicted.

These formulas apply to continuous time with nonnegative, integrable cause-specific hazards. The causes must be defined consistently as the mutually exclusive types of first event, and the cause-specific hazards must describe the same event-free process. When fitting the models, censoring must be handled appropriately; a standard sufficient condition is independent censoring conditional on the predictors in the relevant cause-specific model. The formulas do not require independent latent failure times for the causes.

Why the Hazards Must Share Event-Free Survival

At time \(t\), a person can experience a first event from any of the modeled causes. Thus the instantaneous rate of leaving the event-free state is the sum of the cause-specific hazards. The survival calculation must account for that combined rate, even though the separate cause models use different predictors.

\frac{dS(t)}{dt} =-S(t)\sum_{j=1}^{K}\lambda_j(t\mid z_j), \qquad S(0)=1.

Dividing by \(S(t)\), integrating from 0 to \(t\), and using \(S(0)=1\) gives the expression for event-free survival. To see how this leads to cumulative incidence, over a short interval of length \(dt\), the probability of still being event-free just before \(t\) is approximately \(S(t)\), and the conditional probability of a cause-\(k\) event in that interval is approximately \(\lambda_k(t\mid z_k)dt\). Consequently, the cause-\(k\) event probability accumulated by time \(t\) is the integral of their product.

\begin{aligned} \frac{d}{dt}\log S(t) &=-\sum_{j=1}^{K}\lambda_j(t\mid z_j),\\ \log S(t) &=-\sum_{j=1}^{K}\int_0^t\lambda_j(u\mid z_j)\,du,\\ S(t) &=\exp\left\{-\sum_{j=1}^{K}\Lambda_j(t)\right\},\\ F_k(t) &=\int_0^t S(u)\lambda_k(u\mid z_k)\,du. \end{aligned}

This also verifies that all cause-specific risks and event-free probability form a coherent partition. Since \(F_k'(t)=S(t)\lambda_k(t\mid z_k)\), summing across causes gives \(\sum_k F_k'(t)=S(t)\sum_k\lambda_k(t\mid z_k)=-S'(t)\). Both sides start at zero after integration, so \(\sum_k F_k(t)=1-S(t)\). Every cause contributes to the reduction in event-free survival, whether or not its model uses the same predictors as the others.

Combining Continuous-Time Predictions

The following example uses exponential cause-specific models. An exponential model has a constant cause-specific hazard over time. The model rates below are hypothetical and are evaluated for the same target profile: cause 1 uses a biomarker indicator, while cause 2 uses a comorbidity indicator. The different predictors produce different linear predictors, but both models refer to the same event-free state and time origin.

Worked Example: Two Exponential Models With Different Predictors

Suppose the cause-1 model gives baseline rate \(0.05\) per year and linear predictor \(1.10\) for the target's biomarker value. The cause-2 model gives baseline rate \(0.03\) per year and linear predictor \(0.80\) for the target's comorbidity value. Find each cause's cumulative incidence by 3 years.

The cause-specific rates are the baseline rates multiplied by the exponentiated linear predictors:

\begin{aligned} \lambda_1&=0.05\exp(1.10)=0.1502083\text{ per year},\\ \lambda_2&=0.03\exp(0.80)=0.0667662\text{ per year},\\ \lambda_1+\lambda_2&=0.2169745\text{ per year}. \end{aligned}

Both rates enter the event-free survival. At 3 years, \(S(3)=\exp\{-(0.2169745)(3)\}=\exp(-0.6509235)\approx0.5216\). Thus the probability of any first event by 3 years is \(1-S(3)\approx0.4784\).

For constant rates, the cause-specific cumulative incidence is the cause's share of the total rate multiplied by the probability of any event:

\begin{aligned} F_1(3) &=\frac{0.1502083}{0.2169745} \{1-\exp[-(0.2169745)(3)]\} \approx 0.3312,\\ F_2(3) &=\frac{0.0667662}{0.2169745} \{1-\exp[-(0.2169745)(3)]\} \approx 0.1472. \end{aligned}

The predicted 3-year risks are therefore about 33.12% for cause 1 and 14.72% for cause 2. They sum to 47.84%, the probability of any first event; adding the event-free probability, 52.16%, gives 100%. Each model used only its own predictor, but both predicted rates contributed to the common survival calculation.

With time-varying hazards, the integral defining \(F_k(t)\) may require numerical integration. In a discrete-time model, the same logic appears as a recursion: at each interval, calculate each cause's event probability among those still event-free, use all causes to update event-free probability, and add the cause-specific event probability weighted by the event-free probability at the start of that interval.

Worked Example: Discrete-Time Models With Separate Predictors

Suppose two cause-specific models are evaluated for the same target. Cause 1 uses a biomarker predictor, and cause 2 uses a comorbidity predictor. They return conditional interval probabilities of \(0.10\) and \(0.05\) in interval 1, and \(0.20\) and \(0.10\) in interval 2. These probabilities are conditional on being event-free at the start of the interval. Calculate the cause-specific risks through the end of interval 2.

At the start of interval 1, event-free probability is 1. The interval-1 contributions are \(F_1=1(0.10)=0.10\) and \(F_2=1(0.05)=0.05\). The probability of remaining event-free is \(1-0.10-0.05=0.85\).

At the start of interval 2, the remaining event-free probability is 0.85. The interval-2 contributions are \(0.85(0.20)=0.17\) for cause 1 and \(0.85(0.10)=0.085\) for cause 2. The updated event-free probability is \(0.85(1-0.20-0.10)=0.595\).

Adding the contributions across intervals gives \(F_1=0.10+0.17=0.27\) and \(F_2=0.05+0.085=0.135\). Their sum, 0.405, equals \(1-0.595\). The conditional interval probabilities do not need the same predictors, but they must refer to the same intervals and the same event-free population.

Assumptions, Checks, and Common Failure Modes

Before combining model outputs, check that each cause-specific model supplies a hazard or interval probability that can be interpreted for the same prediction target. A cause model's covariates can differ from another's, but the prediction targets cannot silently differ. For example, a hazard conditional on a detailed covariate profile is not automatically interchangeable with a hazard averaged over a population with a different covariate distribution. Omitted covariates can also matter: if a predictor absent from one model is associated with that cause's hazard and the target requires conditioning on it, treating the model output as the hazard for the full profile can be misleading.

Further practical checks include:

Common time scale. Align time zero, units, and prediction horizon. A rate per month cannot be combined directly with a rate per year without conversion.
Cause definitions. Causes must classify the same kind of first event. An event that occurs after another cause has already happened is not a competing first event in this calculation.
Valid hazards or probabilities. Hazards must be nonnegative. In a discrete interval, the sum of conditional probabilities for all mutually exclusive causes must be at most 1.
Model adequacy. Check whether the chosen functional forms, time dependence, and predictor effects are plausible for each cause. A well-computed combination cannot repair a badly specified component model.
Support for prediction. Predictions for covariate values or combinations poorly represented in the data can be extrapolations. Different cause models may have different support, so check each one.

Cause-specific hazard models are one approach to absolute-risk prediction. A nonparametric Aalen–Johansen estimate describes observed transition probabilities without modeling covariate effects in the same way. Fine–Gray models, covered in the earlier material on Fine–Gray fits, model a subdistribution hazard as a route to cause-specific cumulative incidence. These approaches have different modeling targets. In particular, a Fine–Gray prediction is not a cause-specific hazard to be inserted into the total-hazard survival formula above.

A common mistake is to calculate a separate “net risk” from each cause model and add those risks. That ignores the fact that a competing event prevents the other first event from occurring. The correction is to combine the cause-specific hazards in \(S(t)\), then integrate each cause's hazard against that shared event-free survival.

Worked Example: Why Separate Net Risks Cannot Be Added

Suppose two constant cause-specific hazards are \(0.20\) and \(0.10\) per year. Compare the correct 2-year cumulative incidences with the separate risks \(1-\exp(-\lambda_k t)\), which would treat each cause as if the other could not occur.

The combined hazard is \(0.20+0.10=0.30\) per year. Event-free survival at 2 years is \(S(2)=\exp[-(0.30)(2)]=\exp(-0.60)\approx0.5488\), so total first-event risk is \(1-0.5488=0.4512\). The cause-specific risks are:

\begin{aligned} F_1(2) &=\frac{0.20}{0.30}(1-\exp(-0.60)) \approx 0.3008,\\ F_2(2) &=\frac{0.10}{0.30}(1-\exp(-0.60)) \approx 0.1504. \end{aligned}

They sum to 0.4512, as required. By contrast, the separate net risks are \(1-\exp[-(0.20)(2)]\approx0.3297\) and \(1-\exp[-(0.10)(2)]\approx0.1813\), which sum to 0.5109. That sum exceeds the actual probability of any first event because it counts each cause as if the competing cause could not preclude it.

Key takeaway: Different predictor sets are compatible with a cause-specific hazard combination only when every predicted hazard refers to a common event-free prediction target. Use all cause-specific hazards to calculate event-free survival, then use that shared survival to calculate each cause's cumulative incidence.

Check Your Understanding

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

Two cause-specific models use different covariates. Name two features of their predictions that must still be aligned before combining them.
In a discrete interval, cause-specific conditional event probabilities are 0.12 and 0.08, and event-free probability at the interval's start is 0.70. What are the cause-specific risk contributions, and what is event-free probability at the interval's end?
Why is it generally incorrect to add two cause-specific net risks to obtain the probability of any first event?
For constant cause-specific hazards, write the expression for \(F_k(t)\) using the total hazard.

Solutions. (1) They must refer to a common time origin and horizon, and to a compatible event-free state and prediction target; the covariate sets themselves may differ. (2) The contributions are \(0.70(0.12)=0.084\) and \(0.70(0.08)=0.056\). Event-free probability becomes \(0.70(1-0.12-0.08)=0.56\); the contributions plus 0.56 equal the starting probability 0.70. (3) Each net risk treats its cause as if the other cause could not prevent it, so their sum can exceed the probability of a first event. Use the shared event-free survival instead. (4) If the constant cause-specific hazards are \(\lambda_1,\ldots,\lambda_K\), then \(F_k(t)=\frac{\lambda_k}{\sum_j\lambda_j}\{1-\exp[-t\sum_j\lambda_j]\}\), provided the total hazard is positive.

Combining Models When Causes Use Different Predictors

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