11.1 Competing risks and the cumulative incidence function
Suppose a subject fails at time \(T\) from cause \(D \in \{1, \dots , K\}\). The cause-specific hazard of cause \(k\) is the rate of failing from that cause among those still event-free, \[ h_k(t) = \lim _{\Delta t \to 0}\frac {P(t \le T < t + \Delta t,\ D = k \mid T \ge t)}{\Delta t} , \] and the hazards add, \(h(t) = \sum _k h_k(t)\), so the overall survivor function is \(S(t) = \exp \{-\sum _k H_k(t)\}\). The quantity usually wanted is the probability of having failed from cause \(k\) by time \(t\), the cumulative incidence function \[ F_k(t) = P(T \le t,\ D = k) = \int _0^t h_k(u)S(u^-)\,\mathrm du , \] the rate of cause-\(k\) failure at \(u\) among those still at risk, times the probability of still being at risk. It depends on all the cause-specific hazards through \(S\), not on \(h_k\) alone: a treatment that lowers the risk of a competing cause leaves more subjects exposed to cause \(k\) and so can raise \(F_k\).
A tempting estimate of \(F_k\) is one minus the Kaplan–Meier curve computed with failures from other causes treated as censored. This is wrong. Censoring assumes that the censored subjects could still fail later; a subject who has died of another cause cannot. The curve \(1 - \hat S_{\text {KM}}\) estimates instead the probability of cause-\(k\) failure in a hypothetical world with the other causes removed, which is not identifiable from the data without assuming the causes act independently, and it is always at least as large as the cumulative incidence. The correct estimate replaces \(h_k(u)\,\mathrm du\) and \(S(u^-)\) by their empirical versions. With \(d_{kj}\) failures from cause \(k\) among \(n_j\) at risk at time \(t_{(j)}\) and \(\hat S\) the Kaplan–Meier estimate for failure from any cause, it is the Aalen–Johansen estimator \[ \hat F_k(t) = \sum _{t_{(j)} \le t}\hat S(t_{(j-1)})\frac {d_{kj}}{n_j} . \]
Proof. At \(t_{(j)}\) the Kaplan–Meier estimate drops by \(\hat S(t_{(j-1)}) - \hat S(t_{(j)}) = \hat S(t_{(j-1)})\,d_j/n_j\), where \(d_j = \sum _kd_{kj}\); this is exactly the total of the increases in the \(\hat F_k\) at that time. Summing over event times up to \(t\), the fall of \(\hat S\) from \(1\) equals \(\sum _k\hat F_k(t)\). □
The estimated probabilities of each way the story can end, and of its not having ended yet, therefore add to one, which the sum of the \(1 - \hat S_{\text {KM}}\) curves for the separate causes does not.
Example 11.1.2. Ten subjects have times \(2, 3, 3^+, 5, 6, 8^+, 9, 10^+, 12, 14^+\), with failures at \(2\), \(5\) and \(9\) from cause \(1\) and at \(3\), \(6\) and \(12\) from cause \(2\) (\(+\) for censored). The Kaplan–Meier estimate for any failure is \(0.9\) after time \(2\), \(0.8\) after \(3\), \(0.6857\) after \(5\), \(0.5714\) after \(6\), \(0.4286\) after \(9\) and \(0.2143\) after \(12\). Then \[ \hat F_1(14) = 1 \times \tfrac 1{10} + 0.8 \times \tfrac 17 + 0.5714 \times \tfrac 14 = 0.1 + 0.1143 + 0.1429 = 0.3571 , \] \[ \hat F_2(14) = 0.9 \times \tfrac 19 + 0.6857 \times \tfrac 16 + 0.4286 \times \tfrac 12 = 0.1 + 0.1143 + 0.2143 = 0.4286 , \] and \(0.3571 + 0.4286 + 0.2143 = 1\), as Proposition 11.1.1 says. Treating cause-\(2\) failures as censored instead gives \(1 - \frac 9{10}\cdot \frac 67\cdot \frac 34 = 0.4214\) for cause \(1\), an overestimate by \(0.064\).
Example 11.1.3. The mgus2 data follow \(1384\) patients with monoclonal gammopathy of undetermined significance (MGUS), a blood abnormality that can progress to a plasma cell malignancy (PCM) such as myeloma; median age at diagnosis is \(72\). There were \(115\) progressions and \(860\) deaths without progression, so death is a strong competing risk. The Aalen–Johansen estimate of the cumulative incidence of progression is \(6.4\%\) at \(10\) years and \(10.0\%\) at \(20\) years, while one minus the Kaplan–Meier curve with deaths censored gives \(9.5\%\) and \(21.0\%\), about twice the true risk at \(20\) years and more than three times at \(30\) (\(42\%\) against \(13\%\)) (Software 11.5.1, Figure 11). A patient told the second figure would be misled: most patients with MGUS die of other causes before it progresses.
Questions on this section
Stuck on something here? Ask below and it stays attached to this topic.