Why a Smaller Criterion May Not Mean a Better Model
A cause-specific Cox model and a Fine-Gray model can use the same cohort, event of interest, and covariates. Their fitted coefficients may even look similar. But the models do not describe the same hazard: the cause-specific model concerns the instantaneous rate of the event among people still free of every event, whereas Fine-Gray regression concerns the subdistribution hazard for a particular cause. Their risk sets, and therefore their partial-likelihood contributions, differ.
This distinction matters when software reports partial log-likelihoods, likelihood-ratio statistics, or AIC-like criteria. Such numbers are useful only in comparisons justified by the underlying model and estimation procedure. A smaller number from one model family does not, by itself, show that it fits better than a number from the other family.
Different Hazards, Different Risk Sets
Let \(T\) be the time to the first event, \(J\) its cause, and \(k\) the event cause of interest. For covariates \(Z\), the cause-specific hazard for cause \(k\) is the instantaneous event rate among individuals who have experienced no event just before time \(t\):
A cause-specific Cox model specifies \(\lambda_k(t\mid Z)=\lambda_{k0}(t)\exp(\beta_k^\top Z)\). Its risk set at \(t\) contains people still event-free and under observation just before \(t\). A person who has already had a competing event is no longer at risk for a first event of cause \(k\), so that person leaves the cause-specific risk set.
The Fine-Gray model instead concerns the subdistribution hazard for cause \(k\). Write \(F_k(t\mid Z)=P(T\leq t,J=k\mid Z)\) for the cumulative incidence function and \(f_k(t\mid Z)\) for its event-time density. Where the relevant quantities are defined, the subdistribution hazard is
Its denominator reflects people who have not yet had the event of interest, including people who have already had a competing event. In the Fine-Gray construction, those competing-event individuals remain in the subdistribution risk set. Censoring requires an appropriate adjustment, commonly using inverse-probability-of-censoring weights; the exact estimating criterion depends on the implementation and tie handling.
Both model types require assumptions for their usual interpretation and estimation. Each needs an appropriate proportional-hazards specification if a constant regression coefficient over time is assumed. Cause-specific Cox estimation also relies on suitable censoring assumptions for the cause-specific risk set. Fine-Gray estimation requires suitable censoring assumptions for its weighting procedure, including adequate probability of remaining uncensored over the times used. These conditions do not make the two hazards or their likelihood contributions the same.
Worked Example: The Same Event Time, Two Denominators
Consider four hypothetical individuals with complete follow-up until their listed outcomes. A and C have the event of interest at times 2 and 3, respectively; B has a competing event at time 1; D is censored at time 4. There is no censoring before either event of interest, so the Fine-Gray censoring weights at times 2 and 3 are 1. Let their covariate values be \(Z_A=1\), \(Z_B=2\), \(Z_C=0\), and \(Z_D=-1\). Evaluate the event contributions at \(\beta=0\), with distinct event times.
| Individual | Covariate | Observed outcome |
|---|---|---|
| A | 1 | Event of interest at time 2 |
| B | 2 | Competing event at time 1 |
| C | 0 | Event of interest at time 3 |
| D | -1 | Censored at time 4 |
At a distinct event time \(t\), the cause-specific Cox partial-likelihood contribution from an event of interest is \(\exp(\beta Z_i)/\sum_{j\in R_k(t)}\exp(\beta Z_j)\), where \(R_k(t)\) is the cause-specific risk set. The Fine-Gray contribution uses its modified subdistribution risk set, with censoring adjustment where needed. Here all weights are 1 before the events of interest.
At time 2, the cause-specific risk set is \(\{A,C,D\}\). B has already experienced a competing event and is removed. At \(\beta=0\), each exponential equals 1, so A’s cause-specific contribution is
For Fine-Gray regression, B remains in the subdistribution risk set at time 2, giving \(\{A,B,C,D\}\). A’s contribution is therefore \(1/4\). At time 3, the cause-specific risk set is \(\{C,D\}\), giving C a contribution of \(1/2\). The Fine-Gray risk set is \(\{B,C,D\}\), giving C a contribution of \(1/3\). Thus, at this fixed coefficient value, the products of the two event contributions are
These products illustrate different denominators, not a contest in which \(1/6\) is evidence that the cause-specific model fits better. They are contributions under different risk-set constructions, and the values at a fixed \(\beta\) are not the maximized criteria.
Why the Partial-Likelihood Comparison Fails
The difference follows directly from the conditional event probabilities used to form the Cox partial likelihood. Under a proportional-hazards model, with one event at a time and a specified risk set \(R(t)\), the probability that individual \(i\) is the one who fails at time \(t\), conditional on one failure in that risk set, is
A partial-likelihood contribution is this probability for the observed failing individual. Its denominator depends on the definition of \(R(t)\). The cause-specific Cox and Fine-Gray procedures use different risk sets; consequently they condition on different risk-set comparisons. Fine-Gray weighting for censoring further affects its estimating criterion. The resulting quantities are not two versions of the same observed-data likelihood evaluated under competing candidate models.
This explains both why a raw partial-likelihood difference is not a general cross-model fit measure and why a likelihood-ratio test across these model types is not justified. A likelihood-ratio statistic has a reference distribution only under appropriate nesting and regularity conditions for the likelihoods being compared. A cause-specific Cox model and a Fine-Gray model do not become nested models merely because they share covariates and an event code.
The familiar AIC formula, \(\mathrm{AIC}=-2\ell+2p\), is defined using a maximized likelihood \(\ell\) for a specified data model, with \(p\) parameters. Partial-likelihood and pseudo-likelihood versions of criteria may be used within particular modelling frameworks, subject to their conventions and assumptions. They are not automatically commensurate across frameworks. In particular, equal parameter counts do not repair the difference in risk sets, targets, or censoring adjustments.
Worked Example: An AIC-Like Ranking That Is Not Supported
Suppose a hypothetical software report gives a cause-specific Cox partial-likelihood criterion of \(-2\ell_{\mathrm{CS}}=18\) and a Fine-Gray pseudo-partial-likelihood criterion of \(-2\ell_{\mathrm{FG}}=21\). Each model has three regression parameters. A user applies the formula \(-2\ell+2p\) mechanically:
The arithmetic is correct, but the conclusion “24 is lower, so the cause-specific model is preferred” does not follow. The two reported quantities arise from different criteria and risk sets, so their numerical difference of 3 is not a calibrated measure of relative fit. Nor may it be treated as a likelihood-ratio statistic with a chi-squared reference distribution. Each number may still be useful for a comparison that is valid within its own modelling procedure; the cross-model ranking is the unsupported step.
What Comparisons Are Appropriate?
Within a cause-specific Cox analysis, models using the same cause-specific risk-set definition can be compared using procedures appropriate to that analysis, such as a partial-likelihood ratio test for nested models under its regularity conditions. Corresponding comparisons can be made within Fine-Gray analyses when the same outcome definition, censoring-weight method, and estimation conventions are maintained. Changes to the weights or risk-set construction should be treated as changes to the procedure, not ignored as cosmetic details.
If the goal is to compare predictions from the two model types, define a common prediction target first. For example, one can evaluate predictions of the same cause-specific cumulative incidence on the same held-out observations using a suitable scoring rule that accounts for censoring. The scoring rule, prediction horizon, and censoring adjustment must be specified consistently. This compares performance for a common target; it does not make the fitted hazards or their partial likelihoods identical.
A model’s purpose also matters. Cause-specific coefficients and Fine-Gray coefficients describe different hazard relationships, so a preferred model cannot be chosen just by asking which coefficient is larger or which criterion is numerically smaller. State whether the analysis is aimed at a cause-specific hazard or at modelling the subdistribution hazard, and assess the assumptions and diagnostics relevant to that aim.
Specify whether the analysis concerns the cause-specific hazard or the subdistribution hazard for a named event cause.
Verify how competing events are handled, how censored observations are treated, and whether the required censoring and positivity assumptions are plausible.
Use a model-specific procedure for comparisons within one framework, or define a common prediction target and evaluation measure for comparing predictions across frameworks.
Diagnostics, Pitfalls, and Limits
Warning signs include a report that labels a Fine-Gray pseudo-likelihood as an ordinary full likelihood without qualification, a comparison that ignores different censoring weights, or a model-selection claim based only on which cross-model AIC-like number is smaller. Check that the cause code, event times, censoring indicators, risk sets, and estimation options match the intended analysis. Where available, inspect how the software defines its partial or pseudo-partial likelihood and any information criterion it reports.
A second pitfall is treating the event of interest as though the competing event were merely ordinary censoring in every analysis. That removal is part of the cause-specific risk-set construction. Fine-Gray regression instead retains competing-event individuals in its subdistribution risk set. Neither construction is a data-cleaning choice that can be changed while assuming the likelihood remains comparable.
The distinction does not imply that one method is always preferable, or that their predictions must disagree in every dataset. It means that the models answer different hazard questions and that the numerical criteria have to be interpreted in that context. Cross-validated prediction comparisons also have limits: their conclusions depend on the target, time horizon, scoring rule, censoring adjustment, and validation sample. A common score provides a defined basis for prediction comparison, not a universal ranking for every scientific purpose.
Correction: Same data do not imply the same likelihood construction. The risk sets and hazards differ, and Fine-Gray estimation may use censoring weights. Compare criteria only when the comparison is supported by the same model framework, or compare predictions using a common, explicitly defined target and scoring procedure.
Check Your Understanding
Answer each question, then check the concise solutions below.
- At time \(t\), why is a person with an earlier competing event absent from a cause-specific risk set but retained in the Fine-Gray subdistribution risk set?
- In the worked risk-set example, what are the cause-specific and Fine-Gray contributions for the event of interest at time 2 when \(\beta=0\)?
- Two models have the same number of regression parameters. Does that make their AIC-like values directly comparable if one is a cause-specific Cox criterion and the other a Fine-Gray pseudo-likelihood criterion?
- Name one defensible way to compare predictions from the two model types.
Solutions. (1) The cause-specific hazard is defined among people still free of every event; the Fine-Gray subdistribution risk set retains those without the event of interest, including people who have had a competing event. (2) The cause-specific risk set is \(\{A,C,D\}\), so the contribution is \(1/3\); the Fine-Gray risk set is \(\{A,B,C,D\}\), so the contribution is \(1/4\). (3) No. Equal parameter counts do not align their different risk sets, targets, or estimation criteria. (4) Evaluate both models’ predictions of the same cumulative incidence on the same validation observations using a suitable censoring-adjusted scoring rule.