Cause-Specific Cox or Fine-Gray: Two Hazards, Two Questions
On this page
อ่านฉบับภาษาไทย (Thai version)
Abstract
When patients can die before the event of interest, a treatment has two hazard ratios that answer different questions. The cause-specific hazard ratio compares the event rate among patients still alive and event-free, an aetiological question. The subdistribution hazard ratio, from the Fine-Gray model, keeps patients who already died in the group counted as at risk (the risk set), with a weight. It therefore tracks the cumulative incidence of the event, a prognostic question. A hand example with twelve patients builds both risk sets at one time point. A simulated heart-failure trial of 1,500 patients then fits both models for a first admission. In one version the drug leaves the admission rate among surviving patients unchanged but raises mortality. There the cause-specific hazard ratio is close to 1, but the subdistribution hazard ratio falls below 1 because more treated patients die first. This article concludes that both hazard ratios belong in a report, beside cumulative incidence curves for every cause and absolute differences at stated times.
A reviewer's question about fewer admissions
A trial team is answering reviewers on the main paper of a randomised heart-failure trial. The paper reports that a new drug lowered the cumulative incidence of a first admission. That is the proportion of patients admitted by a given time, where a patient who dies first is never counted as admitted. The drug is fictional and the data are simulated.
One reviewer asks whether the drug protects the heart, or simply lets fewer patients live long enough to be admitted. Both stories lower the cumulative incidence of admission, and telling them apart takes two different hazard ratios. Each is a ratio of hazards, where a hazard is an event rate measured among the patients counted as at risk. The two hazard ratios count different patients as at risk.
Death is a competing event here: once a patient dies, an admission can no longer happen. The cause-specific hazard ratio (csHR) compares the admission rate between arms among patients still at risk. The subdistribution hazard ratio (sHR), estimated with the Fine-Gray model, follows the cumulative incidence itself, whichever story produced it.
This article builds both risk sets by hand. It then fits both models to a simulated trial in which, in one version, the drug lowers admissions only by raising deaths.
Two ways to count who is still at risk
Write $T$ for the time from randomisation to a patient's first event and $D$ for its cause. $D = 1$ marks the event of interest (here a first admission) and $D = 2$ the competing event (death before any admission). $D = 0$ marks censoring: follow-up stopped before any event was seen, for example at the trial's closing date or after loss to follow-up. $A$ is the arm, 1 for the drug and 0 for control.
A hazard is an instantaneous event rate. Among patients still at risk at time $t$, it is the chance of the event in the next short interval, divided by its length. The patients counted as at risk form the risk set, the hazard's denominator. A competing event allows two reasonable risk sets, each with its own hazard.
The cause-specific risk set holds the patients still free of any event and under observation, so a patient who dies leaves it at death. The subdistribution risk set also keeps patients who already had the competing event, as if they could still be admitted. In reality they never can be [1]. Write $\lambda_k(t)$ for the hazard of cause $k$ over the first set and $h_k(t)$ for its hazard over the second.
Keeping the dead raises a problem under censoring: nobody knows how long a dead patient would have stayed under observation. The Fine-Gray method answers it with censoring weights [2]. Each patient who died stays in with a weight: the estimated chance of remaining under observation from the death to time $t$. That chance comes from a Kaplan-Meier curve of the censoring times.
Estimating either hazard assumes that censoring, for example at a closing date or on loss to follow-up, is unrelated to a patient's chance of either event. Because the weights come from one censoring curve for all patients, they also assume that censoring does not differ by arm.
Hand example: two risk sets at month 6
Twelve invented patients are followed for recurrence after cancer surgery, and death before recurrence is the competing event. Their follow-up, in months after surgery:
- deaths at 2, 5 and 9;
- recurrences at 3, 6, 7 and 10;
- censored at 4, 8, 11, 12 and 12.
A recurrence happens at month 6, so build both risk sets at that time.
-
Cause-specific risk set
\[ n_{\mathrm{cs}}(6) = 8 \]
Eight patients are still free of any event and under observation: those followed to month 6 or later.
-
Subdistribution risk set, unweighted
\[ n_{\mathrm{sd}}(6) = 8 + 2 = 10 \]
The deaths at months 2 and 5 stay in. The patient censored at month 4 is in neither set, and the recurrence at month 3 has left both.
-
Chance of staying under observation
\[ G(6^{-}) = 1 - \frac{1}{10} = 0.9 \]
$G(t)$ is the Kaplan-Meier chance of still being uncensored at $t$, and $6^{-}$ means just before month 6. The only earlier censoring is at month 4, when 10 patients are still under observation.
-
Weight of the death at month 2
\[ w = \frac{G(6^{-})}{G(2^{-})} = \frac{0.9}{1} = 0.9 \]
This patient died before the censoring at month 4, so the patient counts as 0.9 of a patient.
-
Weight of the death at month 5
\[ w = \frac{G(6^{-})}{G(5^{-})} = \frac{0.9}{0.9} = 1 \]
Nobody was censored between months 5 and 6, so this patient counts in full.
-
Weighted subdistribution risk set
\[ 8 + 0.9 + 1 = 9.9 \]
Patients still under observation count with weight 1.
-
Cause-specific hazard at month 6
\[ \hat\lambda_1(6) = \frac{1}{8} = 0.125 \]
One recurrence over the 8 patients of the cause-specific set; a hat marks an estimate from data.
-
Subdistribution hazard at month 6
\[ \hat h_1(6) = \frac{1}{9.9} = 0.101 \]
The same recurrence over the weighted set. Without the weights it would be 1/10, or 0.10.
Result: One recurrence, two denominators. A death removes a patient from the cause-specific set but not from the subdistribution set. More deaths therefore lower the subdistribution hazard, even when the recurrence rate among the living is unchanged.
The cause-specific hazard ratio: the aetiological question
The cause-specific hazard of cause $k$ is the hazard over the cause-specific risk set:
$$\lambda_k(t) = \lim_{\Delta t \to 0} \frac{P(t \le T < t + \Delta t,\ D = k \mid T \ge t)}{\Delta t}$$Here $\Delta t$ is the length of a short interval after $t$. The condition $T \ge t$ keeps only patients with no event of any kind before time $t$.
A Cox proportional hazards model is a regression for a hazard that leaves its shape over time unspecified. It assumes a constant hazard ratio between arms, the proportional hazards assumption. Fitted to this hazard, with deaths treated as censored when they happen, its exponentiated arm coefficient is the csHR. The csHR compares admission rates between arms among patients still alive and admission-free [3].
That is the aetiological question: does the treatment change the event rate among patients still at risk? Here, that means how fast admissions happen in patients who can still be admitted. Treating deaths as censored is right for this question, because the target is a rate among the living, not a probability of being admitted.
The csHR cannot give the cumulative incidence on its own. The cumulative incidence of cause 1 accumulates, over time, the chance of reaching each moment free of every event multiplied by the cause-1 hazard at that moment [4]:
$$\mathrm{CIF}_1(t) = \int_0^t S(u^{-})\,\lambda_1(u)\,du$$where the chance of staying free of every event depends on both hazards:
$$S(u) = \exp\left\{-\int_0^u \left[\lambda_1(v) + \lambda_2(v)\right] dv\right\}$$Here $S(u^{-})$ is the probability of being free of every event just before time $u$. It falls faster when either hazard is higher. A drug that leaves $\lambda_1$ alone but raises the death hazard $\lambda_2$ lowers $S$, and with it the cumulative incidence of admission. With competing events, the one-to-one link between a hazard and a risk is lost [5].
The subdistribution hazard ratio: the prognostic question
The subdistribution hazard of cause $k$ is the hazard over the subdistribution risk set, which keeps patients whose earlier event had another cause:
$$h_k(t) = \lim_{\Delta t \to 0} \frac{P(t \le T < t + \Delta t,\ D = k \mid T \ge t \text{ or } \{T < t,\ D \ne k\})}{\Delta t}$$In this definition and the cause-specific one, $T$ and $D$ are the time and cause of the first event as complete follow-up would show them, so $D$ is 1 or 2. Censoring is handled only when the hazards are estimated, here through the censoring weights. The condition now also admits patients who had a competing event before time $t$. Nobody leaves this risk set except through cause $k$ or censoring, so the subdistribution hazard maps one-to-one onto the cumulative incidence of its own cause [2]:
$$\mathrm{CIF}_k(t) = 1 - \exp\left\{-\int_0^t h_k(u)\,du\right\}$$The integral in this equation is the cumulative subdistribution hazard, equal to $-\log\{1 - \mathrm{CIF}_k(t)\}$. The larger the subdistribution hazard accumulated up to time $t$, the higher the cumulative incidence of that cause at $t$, and the reverse. A subdistribution hazard that is higher at every time up to $t$ therefore gives a higher cumulative incidence at $t$.
The Fine-Gray model is a proportional hazards regression for $h_k(t)$, fitted with the censoring weights of the hand example. Its exponentiated arm coefficient is the sHR. When the subdistribution hazards are proportional, the two arms' curves are linked at every time:
$$1 - \mathrm{CIF}_1(t \mid A = 1) = \left\{1 - \mathrm{CIF}_1(t \mid A = 0)\right\}^{\mathrm{sHR}}$$The equation shows that an sHR below 1 lowers the treated arm's cumulative incidence at every time. The amount depends on the control arm's level and on the time. This is the prognostic question: how likely is a patient in each arm to have been admitted by a given time, when some patients die first? A review of published Fine-Gray analyses found many unclear or incorrect readings of the sHR [3].
A subdistribution hazard ratio of 0.70 means a lower subdistribution hazard in the treated arm, which, when the subdistribution hazards are proportional, implies a lower cumulative incidence of the event at every time. It does not mean a 30% lower risk. Read the absolute difference in risk at a stated time from the cumulative incidence curves.
The sHR is also not a rate that any living patient experiences, because its risk set holds people who can no longer have the event [1]. It describes where the cumulative incidence goes, not why it goes there. When the subdistribution hazards are not proportional, a single sHR is a summary over follow-up that also depends on how long patients were followed. That is the case in both versions of the simulated trial below.
Cause-specific Cox versus Fine-Gray at a glance
| Feature | Cause-specific Cox model | Fine-Gray model |
|---|---|---|
| Hazard modelled | Cause-specific hazard, $\lambda_1(t)$ | Subdistribution hazard, $h_1(t)$ |
| A patient who dies first | Leaves the risk set at death | Stays in the risk set with a censoring weight |
| Effect measure | csHR | sHR |
| Question | Aetiological: the event rate among patients still event-free | Prognostic: the cumulative incidence of the event |
| Link to the cumulative incidence | Indirect: needs the hazards of every cause | Direct: one-to-one for its own cause |
| Fitted in Stata | stcox after stset | stcrreg with compete() |
| Fitted in R | survival::coxph | cmprsk::crr |
A simulated heart-failure trial in two versions
The dataset example is a simulated randomised trial of 1,500 adults enrolled after a heart-failure admission, 750 per arm. Follow-up lasts 24 to 36 months, ending at a common closing date, and 80 patients (5.3%) were lost to follow-up. By design, both kinds of censoring are unrelated to arm and to either event. The outcome is the time to a first heart-failure admission, and death before any admission is the competing event.
In the main version, the drug lowers the admission rate (rate ratio 0.70) and the cardiovascular death hazard (hazard ratio 0.70). It leaves non-cardiovascular death unchanged (hazard ratio 1.00). Patients also differ in their underlying tendency to be admitted, a patient-level multiplier on the admission rate called frailty.
In the harms-through-death version, the control arm is the same. The treated arm has an admission rate ratio of 1.00 and both death hazard ratios at 1.80. That drug leaves the admission rate alone and raises deaths.
Simulation code generated both versions from the settings above and then fitted the Cox and Fine-Gray models reported below. Only excerpts of that code, with their output, appear later in this article; the full code and the simulated trial files are not published. Where a number comes from the model that generated the data rather than from the tables or the code output, the text says so.
First events by arm, main version
| First event | Control (patients) | Treated (patients) | Total (patients) |
|---|---|---|---|
| First heart-failure admission | 364 | 319 | 683 |
| Death before any admission | 258 | 256 | 514 |
| Censored with neither event | 128 | 175 | 303 |
| All patients | 750 | 750 | 1,500 |
Hazard ratios for a first admission, treated versus control
| Version | csHR (95% CI) | sHR (95% CI) | Large-sample csHR | Large-sample sHR |
|---|---|---|---|---|
| Main | 0.77 (0.66 to 0.89) | 0.83 (0.71 to 0.96) | 0.75 | 0.84 |
| Harms through death | 1.03 (0.88 to 1.20) | 0.78 (0.67 to 0.91) | 1.01 | 0.81 |
Cumulative incidence by arm at 12 and 24 months
| Version and event | Control, 12 months (proportion) | Treated, 12 months (proportion) | Control, 24 months (proportion) | Treated, 24 months (proportion) |
|---|---|---|---|---|
| Main: first admission | 0.353 | 0.270 | 0.461 | 0.412 |
| Main: death before any admission | 0.229 | 0.209 | 0.333 | 0.310 |
| Harms through death: first admission | 0.353 | 0.306 | 0.461 | 0.388 |
| Harms through death: death before any admission | 0.229 | 0.377 | 0.333 | 0.524 |
Main version: two hazard ratios, one consistent story
In the main version the csHR for a first admission is 0.77 (95% CI 0.66 to 0.89). Among patients still alive and admission-free, the treated arm is admitted at 0.77 times the control arm's rate, summarised over follow-up.
The drug multiplies every patient's admission rate by 0.70 throughout, yet the large-sample csHR, a model value in the table above, is 0.75. Frailty explains the gap from 0.70, and the rest of the gap to 0.77 is chance. Admission-prone patients leave the first-admission risk set early, and faster in the control arm. So in later months, the control patients still at risk are less admission-prone than the treated patients still at risk, and the two arms' admission rates move closer together.
As a result, in the model that generated the data, the true csHR for a first admission drifts from 0.70 at the start of follow-up to 0.80 at 24 months. The drug does not act differently; the patients still at risk change. Randomisation therefore no longer balances the patients still at risk. The csHR may then be read as a comparison of admission rates among those patients, rather than as the drug's effect on any one patient's rate.
The sHR is 0.83 (0.71 to 0.96), closer to 1 than the csHR. It is likewise a summary over follow-up. In the model that generated the data, the ratio of the two arms' cumulative subdistribution hazards is 0.76 by month 12 and 0.82 by month 24. These figures, like the drift of the true csHR above, are computed from the model that generated the simulated data; they are not in the tables above and the full code is not published, so treat them as the author's stated model values.
Part of the gap between the two hazard ratios comes from the drug also lowering cardiovascular death in the simulation (hazard ratio 0.70). More treated patients therefore stay alive and can still be admitted. The trial shows this as a lower cumulative incidence of death before any admission in the treated arm, 0.310 against 0.333 at 24 months.
Part is built into the subdistribution hazard itself, even with no effect on death. The control arm loses more patients to admission, so patients who have already died make up a larger share of its subdistribution risk set. That pulls its subdistribution hazard down more.
At 24 months the cumulative incidence of admission is 0.461 in the control arm and 0.412 in the treated arm. The absolute difference is 0.049, a point estimate; intervals for the differences in this article are not shown.
A drug that lowers admissions by letting more patients die first
In the harms-through-death version the data were simulated with no effect on the admission rate. The csHR agrees: 1.03 (0.88 to 1.20), against a true value of 1.00 at every time. That interval is consistent with no effect on the admission rate among patients still at risk, although it does not rule out a modest one.
A Cox model for death from any cause, including deaths after an admission, gives a hazard ratio of 1.89, close to the 1.80 the data were simulated with. The Stata excerpt at the end of this section shows that fit and its output. The 1.89 is not the cause-specific hazard ratio for death before any admission, which this article does not show. The cumulative incidence of death before any admission in the table above stands in for it.
Yet the sHR for admission is 0.78 (0.67 to 0.91), below 1 with an interval that excludes 1. Nobody has died at the start of follow-up, so the two arms begin with the same subdistribution hazard. The gap opens only as deaths build up faster in the treated arm.
The subdistribution hazards are therefore not proportional. In the model that generated the data, the ratio of cumulative subdistribution hazards is 0.86 by month 12 and 0.78 by month 24. These values are also computed from the model that generated the simulated data; they are not in the tables above and the full code is not published, so treat them as the author's stated model values. The sHR of 0.78 summarises follow-up, which is why the absolute differences at stated times carry the message.
At 24 months the cumulative incidence of admission is 0.388 in the treated arm against 0.461 in the control arm, 0.073 lower. The cumulative incidence of death before any admission, the competing event itself, is 0.524 against 0.333, 0.191 higher.
Each hazard ratio is right for its own question: treated patients really are admitted less often by 24 months, because more of them have died. A report with only the sHR, or only the admission curve, would read as a benefit. The csHR near 1 and the death curve point instead to a drug whose only effect in this version is on death.
Stata: a Cox model for death from any cause (harms-through-death version)
stset fu_months, failure(death == 1)
stcox arm. stset fu_months, failure(death == 1)
Survival-time data settings
Failure event: death==1
Observed time interval: (0, fu_months]
Exit on or before: failure
--------------------------------------------------------------------------
1,500 total observations
0 exclusions
--------------------------------------------------------------------------
1,500 observations remaining, representing
1,056 failures in single-record/single-failure data
24,475.123 total analysis time at risk and under observation
At risk from t = 0
Earliest observed entry t = 0
Last observed exit t = 35.9327
. stcox arm
Failure _d: death==1
Analysis time _t: fu_months
Iteration 0: Log likelihood = -7101.905
Iteration 1: Log likelihood = -7050.0919
Iteration 2: Log likelihood = -7050.0815
Refining estimates:
Iteration 0: Log likelihood = -7050.0815
Cox regression with no ties
No. of subjects = 1,500 Number of obs = 1,500
No. of failures = 1,056
Time at risk = 24,475.1228
LR chi2(1) = 103.65
Log likelihood = -7050.0815 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
_t | Haz. ratio Std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
arm | 1.885638 .1184866 10.09 0.000 1.667139 2.132773
------------------------------------------------------------------------------stset fu_months, failure(death == 1) sets the follow-up time in months with a death at any time, before or after an admission, as the event, and the arm row of the stcox arm output is the hazard ratio of 1.89 for death from any cause, with its 95% CI.Fitting both models in Stata and R
Both models need only the time to the first event, t_rec, and its cause, rec_status: 0 censored, 1 admission, 2 death before any admission. Each script fits both models in both versions. The excerpts below show the main version, with the Fine-Gray fit in Stata and the cause-specific fit in R.
Stata: the Fine-Gray model for a first admission (main version)
* Fine-Gray: subdistribution hazard, people who died stay in the risk set (weighted)
stcrreg arm, compete(rec_status == 2). * Fine-Gray: subdistribution hazard, people who died stay in the risk set (weighted)
. stcrreg arm, compete(rec_status == 2)
Failure _d: rec_status==1
Analysis time _t: t_rec
Iteration 0: Log pseudolikelihood = -4769.8195
Iteration 1: Log pseudolikelihood = -4769.3374
Iteration 2: Log pseudolikelihood = -4769.3374
Competing-risks regression No. of obs = 1,500
No. of subjects = 1,500
Failure event: rec_status == 1 No. failed = 683
Competing event: rec_status == 2 No. competing = 514
No. censored = 303
Wald chi2(1) = 6.27
Log pseudolikelihood = -4769.3374 Prob > chi2 = 0.0123
------------------------------------------------------------------------------
| Robust
_t | SHR std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
arm | .8254225 .0632281 -2.50 0.012 .7103519 .9591336
------------------------------------------------------------------------------stset t_rec, failure(rec_status == 1), which declares the time to the first event and a first admission as the event of interest; just before the excerpt, the script fits the cause-specific model with stcox arm. In stcrreg arm, compete(rec_status == 2), compete() names death before any admission as the competing event. The header counts 683 admissions, 514 competing deaths and 303 censored patients, and the SHR row is the sHR with its 95% CI.R: the cause-specific Cox model for a first admission (main version)
# cause-specific Cox: hazard of admission among patients still alive and admission-free
cs <- coxph(Surv(t_rec, rec_status == 1) ~ arm, data = d, ties = "breslow")
print(summary(cs))Call:
coxph(formula = Surv(t_rec, rec_status == 1) ~ arm, data = d,
ties = "breslow")
n= 1500, number of events= 683
coef exp(coef) se(coef) z Pr(>|z|)
arm -0.2673 0.7655 0.0768 -3.48 0.000502 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
exp(coef) exp(-coef) lower .95 upper .95
arm 0.7655 1.306 0.6585 0.8898
Concordance= 0.533 (se = 0.01 )
Likelihood ratio test= 12.14 on 1 df, p=5e-04
Wald test = 12.11 on 1 df, p=5e-04
Score (logrank) test = 12.18 on 1 df, p=5e-04d, here the main version. The cause-specific fit coxph(Surv(t_rec, rec_status == 1) ~ arm, data = d, ties = "breslow") treats deaths as censored, and exp(coef) is the csHR with its 95% CI. ties = "breslow" is a standard way to handle tied event times, patients whose events fall at the same recorded time, and is the method Stata's stcox uses by default. The same block of the script also fits the Fine-Gray model with crr() from the cmprsk package (failcode = 1, cencode = 0); survival::finegray() followed by a weighted coxph() is an alternative.Readings that sound right and are not
-
"An sHR of 0.70 means a 30% lower risk."
The sHR is a ratio of subdistribution hazards, not of risks. How far the cumulative incidence moves depends on the control arm's level and on time.
Fix: A subdistribution hazard ratio of 0.70 means a lower subdistribution hazard in the treated arm, which, when the subdistribution hazards are proportional, implies a lower cumulative incidence of the event at every time. It does not mean a 30% lower risk. Read the absolute difference in risk at a stated time from the cumulative incidence curves.
-
"The csHR shows how much the cumulative incidence fell."
The cumulative incidence of admission also depends on the death hazard. In the harms-through-death version the csHR is 1.03, yet the cumulative incidence of admission at 24 months is 0.073 lower in the treated arm.
Fix: Read the csHR as an effect on the event rate among patients still event-free, and read risk from the cumulative incidence curves of every cause.
-
"The sHR measures how the drug acts on the event."
Its risk set keeps patients who have died, so the sHR blends the drug's effect on the event with its effect on death. The harms-through-death sHR is 0.78 for a drug with no effect on the admission rate.
Fix: For a question about whether the drug changes the rate of each event among patients still at risk, fit cause-specific Cox models for the event and for each competing event.
-
"Separate Fine-Gray models for admission and for death fit together."
Each model assumes proportional subdistribution hazards for its own cause, and nothing makes the two assumptions consistent. Their predicted cumulative incidences can add up to more than 1 for the same patient.
Fix: Use a Fine-Gray model for one named cause. To describe every cause at once, report the Aalen-Johansen curves by arm, or predict them from cause-specific models for all causes.
What to do in your own analysis
A report that answers both of the reviewer's questions usually needs the following [6].
- Name the question before the model: whether the drug changes the event rate among patients still at risk points to the csHR; how many patients have the event by a stated time points to the cumulative incidence and, if a regression is needed, the sHR [1].
- Fit cause-specific Cox models for the event and for each competing event, so a reader can see where a change in cumulative incidence comes from.
- Show Aalen-Johansen cumulative incidence curves by arm for every cause, not one minus Kaplan-Meier.
- Report absolute differences in cumulative incidence at stated times, for example 12 and 24 months, with confidence intervals.
- Compare censoring by arm, for example the proportion lost to follow-up, before trusting the censoring weights; if it differs, weights from one pooled censoring curve may mislead.
- Check proportional hazards for each model; when it fails, a single csHR or sHR is only a summary over follow-up, so give the curves more weight.
Glossary
- cause-specific hazard
- The instantaneous rate of one cause among patients still free of any event and under observation; a patient who has a competing event leaves its risk set.
- cause-specific hazard ratio (อัตราส่วนฮาซาร์ดเฉพาะสาเหตุ)
- The ratio between arms of the cause-specific hazard, the instantaneous rate of one cause among patients still free of any event.
- subdistribution hazard
- The instantaneous rate of one cause over a risk set that also keeps patients who already had a competing event; it maps one-to-one onto the cumulative incidence of that cause.
- subdistribution hazard ratio (อัตราส่วนฮาซาร์ดของ subdistribution)
- The ratio between arms of the subdistribution hazard, whose risk set keeps patients who had a competing event; it is tied to the cumulative incidence.
- risk set
- The patients counted as at risk at a given time, the denominator of a hazard.
- Fine-Gray model (แบบจำลอง Fine-Gray)
- A proportional hazards regression for the subdistribution hazard of one cause, fitted with censoring weights.
- aetiological versus prognostic question
- An aetiological question asks whether a treatment changes the event rate among patients still at risk; a prognostic question asks how likely the event is by a given time.
- censoring weight
- The estimated chance of staying under observation from a patient's competing event to the current time, used to keep that patient in the subdistribution risk set.
References
- Lau B, Cole SR, Gange SJ. Competing risk regression models for epidemiologic data. Am J Epidemiol. 2009;170(2):244-256. doi:10.1093/aje/kwp107 https://doi.org/10.1093/aje/kwp107
- Fine JP, Gray RJ. A proportional hazards model for the subdistribution of a competing risk. J Am Stat Assoc. 1999;94(446):496-509. doi:10.1080/01621459.1999.10474144 https://doi.org/10.1080/01621459.1999.10474144
- Austin PC, Fine JP. Practical recommendations for reporting Fine-Gray model analyses for competing risk data. Stat Med. 2017;36(27):4391-4400. doi:10.1002/sim.7501 https://doi.org/10.1002/sim.7501
- Putter H, Fiocco M, Geskus RB. Tutorial in biostatistics: competing risks and multi-state models. Stat Med. 2007;26(11):2389-2430. doi:10.1002/sim.2712 https://doi.org/10.1002/sim.2712
- Andersen PK, Geskus RB, de Witte T, Putter H. Competing risks in epidemiology: possibilities and pitfalls. Int J Epidemiol. 2012;41(3):861-870. doi:10.1093/ije/dyr213 https://doi.org/10.1093/ije/dyr213
- Latouche A, Allignol A, Beyersmann J, Labopin M, Fine JP. A competing risks analysis should report results on all cause-specific hazards and cumulative incidence functions. J Clin Epidemiol. 2013;66(6):648-653. doi:10.1016/j.jclinepi.2012.09.017 https://doi.org/10.1016/j.jclinepi.2012.09.017
Key takeaways
- A cause-specific hazard ratio answers the aetiological question of whether a treatment changes the event rate among patients still alive and event-free, a group that randomisation may not keep balanced as follow-up goes on.
- A subdistribution hazard ratio keeps patients who had the competing event in a weighted risk set, so it follows the cumulative incidence and answers the prognostic question.
- A treatment can lower the cumulative incidence of an event, and its subdistribution hazard ratio, purely by raising the hazard of the competing event.
- Separate Fine-Gray models for each cause are not built to fit together, and their predicted cumulative incidences can add up to more than 1.
- Report both hazard ratios for the event and for the competing event, cumulative incidence curves by arm for every cause, and absolute differences at stated times.
Related in the wiki: [[cumulative-incidence-competing-risks-kaplan-meier]] [[time-to-event-survival-analysis]]