Signature
p = 1 - exp(-h * Delta)
| Inputs | Definition | Unit |
|---|---|---|
h | Constant hazard (instantaneous rate) of the event among people still at risk | events per person per unit of time, for example per year |
Delta | Length of the interval, usually the model cycle length, in the same time unit as h | time, for example years |
p | Probability that a person at risk at the start of the interval has the event before its end | 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
Probability from a rate in one cell
Excel returns the interval probability from named cells holding the rate and the interval length in matching units.
=1-EXP(-Rate*CycleLength)
Assumptions
Constant hazard across the interval
The rate h does not change within the interval. When the hazard changes with age or time since an event, the conversion is applied cycle by cycle with the rate for each cycle, or the probability is taken from a fitted survival curve.
One exit from the starting state
The formula describes a single event that removes the person from the starting state. When a state has two or more exits, each exit probability comes from the joint formula for competing hazards on this page, not from this formula applied to each exit separately.
Worked examples
Annual cycle at a rate of 0.10 per year
A constant rate of 0.10 per year gives a one-year probability of about 0.0952, slightly below the rate because people who have the event early in the year are no longer at risk. The figure matches the comparator probability in the Hazard Ratio article.
h = 0.10; Delta = 1; p = 0.0952
Three-month cycle at a total exit rate of 0.20
A constant total exit rate of 0.20 per year over a three-month cycle gives a probability of leaving the state of about 0.04877, the total exit probability in the article's competing-risks example.
h = 0.20; Delta = 0.25; p = 0.04877
Common errors
Using the rate itself as a probability
Entering a rate directly as a per-cycle probability overstates the risk, and can exceed 1 at high rates. At a rate of 1 per year the one-year probability is about 0.632, not 1.
Sources
Rate and probability equations for two-state transitions
Gidwani R, Russell LB. Estimating transition probabilities from published evidence: a tutorial for decision modelers. PharmacoEconomics. 2020;38(11):1153-1164. Section on the probability-rate equations when there are two state transitions (equations 8 to 10).
Conversion between rates and probabilities in economic models
Fleurence RL, Hollenbeak CS. Rates and probabilities in economic modelling: transformation, translation and appropriate application. PharmacoEconomics. 2007;25(1):3-6.
Canonical Identity
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