Comparing Predictions From Different Model Strategies

Why Compare Prediction Strategies?

A competing-risk prediction asks for the probability of experiencing a particular event by a stated time, before a different event prevents it from occurring. Two common strategies start from different hazard models: cause-specific hazard models describe each event cause’s instantaneous rate among people still free of every event, while a Fine-Gray model describes the subdistribution hazard associated with a particular cause. Both can be used to predict absolute risk, but they do not generally give identical predictions.

A fair comparison holds the prediction target, covariate profile, time horizon, and target population fixed. The comparison is about the predictive implications of two modeling strategies—not evidence that one model or hazard is a cause of the outcome. A difference in predicted risk can arise from the models’ structures, fitted coefficients, baseline estimates, or assumptions.

How Each Strategy Produces Absolute Risk

A cause-specific strategy models the cause-specific hazard for every relevant cause. Let \(\Lambda_j(t\mid\mathbf{x})=\int_0^t\lambda_j(u\mid\mathbf{x})\,du\) be the cumulative cause-specific hazard for cause \(j\). Under continuous event-time distributions, the probability of remaining free of every event through time \(t\) is

The cause-\(k\) CIF is then obtained by accumulating cause-\(k\) hazard only while a person remains event-free:

This formula shows why a cause-\(k\) hazard model alone is not enough to obtain the cause-\(k\) absolute risk: the hazards of the competing causes also affect \(S(u\mid\mathbf{x})\), and hence the number who remain at risk of cause \(k\).

A Fine-Gray strategy models the subdistribution hazard for the target cause. In its proportional-hazards form, \(\alpha_k(t\mid\mathbf{x})=\alpha_{0k}(t)\exp(\boldsymbol{\gamma}_k^\mathsf{T}\mathbf{x})\). If \(A_k(t\mid\mathbf{x})=\int_0^t\alpha_k(u\mid\mathbf{x})\,du\), the corresponding risk prediction is

The subdistribution hazard is tied directly to the target cause’s CIF. It is not interchangeable with a cause-specific hazard, and a Fine-Gray model for one cause does not by itself provide a complete model for every other cause. In particular, independently fitted Fine-Gray models for multiple causes need not produce risks that sum to at most one; check probability coherence rather than assuming it.

Assumptions Before Comparing Predictions

A model comparison is interpretable only if both predictions refer to the same well-defined quantity. Specify the event cause, time origin, covariate values, prediction horizon, and population of interest. For example, “the risk of cause 1 by five years for a person with \(\mathbf{x}=(1,0)\)” is more precise than “the risk for the treated group.” If the goal is an average over a population rather than risk at a single profile, calculate each strategy’s predictions for the same individuals and average them using the same target-population weights.

Both strategies also rely on sound outcome and follow-up information. Event causes must be classified correctly, and censoring must be independent of the event process conditional on the variables used in the analysis, or otherwise handled by an appropriate method. Prediction at a horizon requires adequate follow-up support there; extending either model past the observed support is extrapolation, not a direct comparison of observed-range prediction performance.

A proportional cause-specific hazards model assumes that each modeled cause-specific hazard has the stated covariate effect over time, unless time-varying effects are included. A proportional Fine-Gray model makes the corresponding proportionality assumption for the subdistribution hazard. Each also requires suitable baseline hazard estimation and an adequate covariate functional form. These assumptions are distinct: satisfying one does not imply that the other holds.

Worked Comparisons

Why the Predictions Can Differ

The cause-specific approach builds the target CIF from the joint pattern of all cause-specific hazards. The Fine-Gray approach models the target CIF through its subdistribution hazard. Even when both are fitted to the same data, their regression structures and assumptions differ. Different covariate effects, baseline estimates, or departures from proportionality can therefore lead to different risks for the same profile and horizon. Neither strategy is guaranteed to be numerically closer to the truth merely because it models a particular type of hazard.

One useful mathematical link makes the distinction clear. For a continuous subdistribution hazard, its definition gives \(dF_k(t\mid\mathbf{x})=\{1-F_k(t\mid\mathbf{x})\}\alpha_k(t\mid\mathbf{x})\,dt\). Dividing by \(1-F_k(t\mid\mathbf{x})\) and integrating from 0 to \(t\), using \(F_k(0\mid\mathbf{x})=0\), yields

Rearranging gives \(F_k(t\mid\mathbf{x})=1-\exp\{-A_k(t\mid\mathbf{x})\}\). This identity explains how a Fine-Gray model’s cumulative subdistribution hazard becomes an absolute-risk prediction. It does not make that hazard the same as a cause-specific hazard.

Diagnostics, Alternatives, and Limitations

Before interpreting a gap between predictions, check that it is not caused by a mismatched target. Confirm the cause label, time origin, horizon, covariate values, and population weights. Then inspect whether each model’s assumptions are credible and whether predictions behave sensibly over time. For example, a CIF should not decrease as the horizon increases, and a predicted probability must lie between zero and one. Check multiple causes jointly when predictions are meant to describe a coherent set of competing outcomes.

Where data permit, evaluate each strategy’s calibration and predictive error at the same horizon in the same target population, using appropriate methods for censoring. A comparison of apparent fit in the data used to fit the models is not, by itself, evidence of better prediction in new individuals. Uncertainty in the difference matters too: predictions from models fitted to the same sample are dependent, so a paired resampling procedure that refits both strategies can preserve that dependence when estimating uncertainty.

A nonparametric competing-risk estimate can be a useful reference for an unadjusted population or group, whereas regression strategies provide covariate-specific predictions under their model assumptions. It does not automatically answer an adjusted prediction question. Conversely, regression predictions depend on correct model specification and adequate data support. No single strategy is uniformly best; the appropriate comparison depends on the prediction target, assumptions, and validation evidence.

A common mistake is to say, “The Fine-Gray model gives a higher risk, so the covariate causes more events under that model,” or to infer that one hazard ratio is the causal effect because its risk prediction is larger. This reasoning is wrong: model choice is not a causal intervention, and hazard parameters from the two strategies describe different quantities. The correction is to report the two absolute-risk predictions at the matched profile and horizon, quantify their difference and uncertainty, then assess predictive accuracy and assumptions without assigning a causal interpretation to the model contrast.