Signature
p_1 = h_1 / (h_1 + h_2) * (1 - exp(-(h_1 + h_2) * Delta)); p_2 = h_2 / (h_1 + h_2) * (1 - exp(-(h_1 + h_2) * Delta)); p_stay = exp(-(h_1 + h_2) * Delta)
| Inputs | Definition | Unit |
|---|---|---|
h_1 | Constant cause-specific hazard of moving to destination 1 | events per person per unit of time |
h_2 | Constant cause-specific hazard of moving to destination 2 | events per person per unit of time |
Delta | Length of the model cycle, in the same time unit as the hazards | time, for example years |
p_1 | Probability of leaving the starting state for destination 1 within the interval | probability from 0 to 1 |
|---|---|---|
p_2 | Probability of leaving the starting state for destination 2 within the interval | probability from 0 to 1 |
p_stay | Probability of still being in the starting state at the end of the interval | probability from 0 to 1 |
Function
Rate and transition probability conversion function
Maps an event rate and a time interval to the probability that the event occurs within that interval, and back again. Rates are converted rather than probabilities divided, because a probability is tied to the length of the interval over which it was observed. The converted probability then enters a transition matrix used in the cohort update s_(t+1) = s_t P described on the Markov Model page.
Implementations
Excel
Competing exit probabilities in three cells
With named cells Hazard1, Hazard2 and CycleLength, the three formulas return p_1, p_2 and p_stay respectively.
=Hazard1/(Hazard1+Hazard2)*(1-EXP(-(Hazard1+Hazard2)*CycleLength)); =Hazard2/(Hazard1+Hazard2)*(1-EXP(-(Hazard1+Hazard2)*CycleLength)); =EXP(-(Hazard1+Hazard2)*CycleLength)
Assumptions
Constant cause-specific hazards
Both hazards are constant within the interval and their sum is greater than zero.
Absorbing destinations within the cycle
A person who reaches destination 1 or 2 does not move again within the same cycle. If a further transition is possible from a destination, for example from ill to dead within one cycle, the formulas are no longer valid and the full matrix is derived from the transition rate matrix.
Worked examples
Two exits over a three-month cycle
With hazards of 0.12 and 0.08 per year and a cycle of 0.25 years, the destination probabilities are about 0.02926 and 0.01951 and the probability of remaining is about 0.95123. The unrounded values sum to 1 and match the article's table.
h_1 = 0.12; h_2 = 0.08; Delta = 0.25; p_1 = 0.02926; p_2 = 0.01951; p_stay = 0.95123
Common errors
Converting each competing rate with the single-event formula
Using p equal to 1 minus exp of minus h_k times Delta for each exit ignores that people who leave by one route are no longer at risk of the other. With both hazards equal to 1 per cycle, each exit would receive about 0.632 and the probability of remaining would be about minus 0.264, which is impossible. The joint formula gives about 0.432 for each exit and about 0.135 for remaining.
Sources
Competing-risk rate to probability conversion
Jones E, Epstein D, García-Mochón L. A procedure for deriving formulas to convert transition rates to probabilities for multistate Markov models. Medical Decision Making. 2017;37(7):779-789. Introduction, which shows that the simple formula is always wrong with competing risks, and the special case with two competing absorbing exits.
Canonical Identity
Stable URI · Machine-readable · Resolvable · CC BY 4.0