Why Fit Both Models to the Same Data?
Cause-specific Cox regression and Fine-Gray regression can use the same people, event times, event types, and covariates, yet return different coefficients. The difference is not a computational inconsistency: the models describe different hazards and use different risk sets. A useful way to make that distinction concrete is to fit both models to one small event history and trace the calculations event by event.
The cause-specific model describes the instantaneous rate of a particular cause among people who have not yet experienced any event. The Fine-Gray model describes the subdistribution hazard for that cause, using a risk set that retains people after a competing event. Neither coefficient is a direct risk ratio, and coefficients from the two models should not be treated as estimates of the same quantity.
Assumptions and the Event History
The calculations below use a binary, fixed covariate \(X\), coded 1 for exposed and 0 for unexposed. For each model, the covariate effect is assumed proportional over time: the relevant hazard ratio is constant over the modeled follow-up. Observations are independent, event causes are classified correctly, and censoring is independent of the event process conditional on the modeled covariate. A fitted coefficient describes the specified model; it does not establish that its assumptions hold.
The example has no censoring before the final cause-1 event. Thus, for the Fine-Gray calculations shown, censoring weights are 1. In data with censoring before or during follow-up, the Fine-Gray weighted partial likelihood must account for censoring; a competing-event subject’s contribution is not automatically an unweighted contribution of 1 at every later time. The hypothetical data are deliberately small enough to calculate by hand, not large enough to support reliable diagnostics or substantive conclusions.
Worked Example: Fit Both Models to One Six-Person Cohort
Consider six hypothetical participants followed in time units. The covariate is exposure \(X\). Cause 1 is the event of interest, and cause 2 is the competing event.
| Participant | Exposure \(X\) | Observed time | Outcome |
|---|---|---|---|
| A | 1 | 1 | Cause 1 |
| B | 1 | 2 | Cause 2 |
| C | 0 | 3 | Cause 1 |
| D | 0 | 4 | Cause 2 |
| E | 1 | 5 | Cause 1 |
| F | 0 | 6 | Censored |
There are three cause-1 failures, at times 1, 3, and 5, with covariate values 1, 0, and 1. The cause-specific risk sets at those times contain people who have not yet had any event. Their exposed and unexposed counts are, respectively, \((3,3)\), \((1,3)\), and \((1,1)\). For example, at time 3, A has had cause 1 and B has had cause 2, so neither remains in the cause-specific risk set.
For Fine-Gray regression, previously observed cause-2 participants remain in the risk set for cause 1. With no censoring before these cause-1 event times, the exposed and unexposed counts in the corresponding Fine-Gray risk sets are \((3,3)\), \((2,3)\), and \((2,2)\). At time 3, participant B remains alongside C, D, E, and F. At time 5, B and D remain after their cause-2 events, while A and C, who have already had cause 1, do not.
Let \(r=\exp(\beta)\), the hazard ratio for a given model. With no tied cause-1 event times, the cause-specific partial log likelihood, up to constants, is
Its score is the sum of the observed event covariates minus the model-based mean covariate in each risk set. Here the observed values sum to 2, so setting the score to zero gives
The first and third terms are both \(r/(r+1)\). Rearranging,
The positive solution is \(r=3\), so \( \hat{\beta}_{\mathrm{CS}}=\log(3)\approx1.099 \). In this fitted cause-specific model, the estimated cause-1 hazard ratio for exposed versus unexposed participants is 3.
For Fine-Gray regression, the same cause-1 event covariates contribute to the score, but the risk-set denominators differ. The score equation is
The first and last terms simplify to \(r/(r+1)\). Therefore,
The positive root is \(r=(1+\sqrt{13})/2\approx2.303\), giving \( \hat{\beta}_{\mathrm{FG}}=\log(2.303)\approx0.8341 \). The fitted Fine-Gray subdistribution hazard ratio is about 2.303. This is not a second estimate of the same cause-specific hazard ratio: the two fitted coefficients refer to different hazards and use different risk sets.
Why the Score Equations Have Different Roots
At each failure time, the partial-likelihood score adds the covariate value of the person who failed and subtracts the model-weighted mean covariate in that model’s risk set. For a binary covariate and risk-set counts \(n_1\) exposed and \(n_0\) unexposed, that mean is
The cause-specific and Fine-Gray models therefore differ in the expected-covariate terms whenever their risk-set compositions differ. In the example, the difference first appears at time 3: the cause-specific risk set has one exposed participant, whereas the Fine-Gray risk set has two because participant B had a competing event and remains in the latter risk set. At time 5, D’s competing event also makes the Fine-Gray risk set differ.
The score equation is not merely an arithmetic recipe. Each model’s partial log likelihood is concave in the coefficient. Differentiating a risk-set contribution twice gives the negative variance of \(X\) under the model’s risk-set weights. For the simple binary, unweighted calculations here, that variance is \(p_j(1-p_j)\), where \(p_j=n_1r/(n_1r+n_0)\). Thus,
When at least one event-time risk set contains both covariate values, the sum is positive and the score decreases strictly with \(\beta\). A score root is then unique if it exists. In the example both roots exist and are unique. This does not guarantee that a useful estimate exists in every data set: if failures are completely separated by covariate level, an estimate can diverge rather than settle at a finite value.
Worked Example: Audit the Time-3 Fine-Gray Contribution
At time 3, participant C fails from cause 1 with \(X=0\). The Fine-Gray risk set includes exposed B and E and unexposed C, D, and F. With \(r=3\), the model-weighted mean covariate is
The event’s score contribution is its observed covariate minus that mean:
If B were incorrectly removed after cause 2, the time-3 denominator would instead use one exposed and three unexposed participants, giving a mean of \(3/(3+3)=1/2\) at \(r=3\). That is the cause-specific risk-set mean, not the Fine-Gray one. This single event-time calculation shows why using the wrong risk set changes the fitted score.
Worked Example: Check the Fine-Gray Score at the Cause-Specific Estimate
The cause-specific fit gave \(r=3\). To see why that value does not also solve the Fine-Gray equation, substitute \(r=3\) into all three Fine-Gray expected-covariate terms:
The observed cause-1 covariate values sum to 2, so the Fine-Gray score at \(r=3\) is \(2-2.1667=-0.1667\), not zero. Since the score decreases with \(\beta\), its root lies below \(\log(3)\); the exact Fine-Gray calculation gives \(r\approx2.303\). This is a direct numerical check that the risk-set difference changes the fit, rather than a rounding discrepancy.
Interpreting the Comparison and Checking the Fit
In this hypothetical data set, the cause-specific estimate of 3 compares instantaneous cause-1 rates among participants still free of both event types. The Fine-Gray estimate of about 2.303 compares cause-1 subdistribution hazards, whose risk sets retain participants after cause 2. The smaller numerical estimate in this example does not imply that Fine-Gray regression will always produce a smaller coefficient, nor does either estimate alone give a probability or risk difference by a specified time.
A reliable comparison begins by auditing the event coding, time ordering, and risk sets. At each cause-1 failure time, verify that the failing participant is included in the denominator and that previously observed cause-1 failures are excluded. For the cause-specific fit, remove people after any event. For Fine-Gray, retain people after competing events, with the appropriate censoring weights when required. Then check that the reported coefficient and hazard ratio correspond to the intended model and covariate coding.
The example also illustrates what cannot be checked from a tiny hand calculation. Three target events cannot establish proportional hazards or provide meaningful evidence that either model fits well. In applied analyses, inspect the event counts and covariate overlap, consider whether the proportional-effect specification is plausible, and use appropriate model diagnostics. Those checks do not make the estimands interchangeable; they assess the adequacy of each model for its own target.
A common mistake is to say, “Competing events are removed from both models, so the two coefficients ought to agree.” That is false for Fine-Gray regression: competing-event participants leave the cause-specific risk set but remain in the Fine-Gray risk set. The correction is to write down the risk sets separately before interpreting or comparing coefficients. Another mistake is to call a subdistribution hazard ratio a risk ratio. The coefficient is a relative effect on the subdistribution hazard; translating it to absolute cumulative incidence at a time requires additional prediction work.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- In the six-person example, which participants are in the Fine-Gray risk set at time 5, and why is participant C not included?
- At \(r=2\), calculate the cause-specific expected-covariate contribution at time 3.
- Why can the Fine-Gray and cause-specific coefficients differ even when both models use the same observed cause-1 failures?
- Can the Fine-Gray hazard ratio of about 2.303 be interpreted as a 2.303-fold cause-1 cumulative incidence? Explain.
Solutions. (1) The time-5 Fine-Gray risk set is B, D, E, and F. B and D remain after cause 2, E is the current cause-1 failure, and F is still followed; C is excluded because C already experienced cause 1. (2) At time 3, the cause-specific counts are one exposed and three unexposed, so the expected covariate is \(2/(2+3)=0.4000\). (3) The score subtracts a model-weighted mean covariate at each event time, and that mean depends on the risk-set membership. Competing-event participants remain in Fine-Gray risk sets but not cause-specific risk sets. (4) No. It is a relative effect on the subdistribution hazard, not a direct ratio of cumulative incidences; absolute incidence comparisons require predictions at a specified time.