Q <- rbind(c(0, 1, 0, 0),
c(0, 0, 1, 0),
c(0, 0, 0, 1),
c(0, 0, 0, 0))Competing risks
Links
https://www.emilyzabor.com/tutorials/survival_analysis_in_r_tutorial.html
https://www.danieldsjoberg.com/ggsurvfit/
jmpost: combines survival analysis, mixed effect model https://genentech.github.io/jmpost/main/
Subdistribution hazards: modifies the traditional hazard function to account for probability of NOT experiencing the event of interest due to competing risks
Cumulative incidence function: used to estimate the probability of experiencing each type of event over time
Aalen-Johansen estimators \(\hat{P}(T<=t, X_T = j), j = 1, 2\) add up to 1 minus Kaplan-Meier estimator \(\hat{P}(T>t)\)
Independence assumption: assuming competing events are mutually exclusive
Nonparametric estimation
Cause-specific hazards \(\alpha_{0j}(t), j = 1,2\) are key quantities of the competing risk model, its specification suffice to generate competing risk data (except censoring).
Nelson-Aalen estimator of the cumulative cause-specific hazards \(A_{0j}(t) = \int_0^t \alpha_{0j}(u) du, j = 1,2\)
Multi-state modeling
Multi-state modeling is not only for time-to-event data. Competing risk is a special case for MS.
Models the transition between states: healthy to diseased, diseased to death, healthy to death etc
Transition probabilities
Transition hazards
State occupancy probabilities
Use package msm, mstate, etc
Use msm
Time of transition is unknown. The time in the dataset is observed state
Transition structure: defined by a matrix with \(r, s\) entry. For example, for transitions allowed below
- 1 to 2
- 2 to 3
- 3 to 4
the transition matrix would be
Transitions should only be allowed in adjacent states in continuous time. For example, even if we observe 1 -> 3 without seeing the state 2, we know that patients have been through 1 -> 2 -> 3. The transition matrix should NOT be selected based on what we observe in the data!
Mean sojourn time (waiting time): \(-1/q_{rr}\) is the expected time to stay in the same state (hence the \(rr\)).