Signature
h = -log(1 - p) / Delta
| Inputs | Definition | Unit |
|---|---|---|
p | Probability of the event over the interval in which it was observed | probability from 0 to 1, excluding 1 |
Delta | Length of the interval over which p was observed | time, for example years |
h | Constant hazard of the event implied by the observed probability | events per person per unit of time |
|---|
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
Rate from a probability in one cell
Excel uses LN, the natural logarithm, on named cells holding the probability and the length of its observation interval.
=-LN(1-Probability)/Interval
Assumptions
Constant hazard over the observation period
The implied rate is an average constant hazard. If the hazard rose or fell during the observation period, the rate describes neither the start nor the end of the period.
Worked examples
One-year probability of 0.20
A one-year probability of 0.20 implies a constant rate of about 0.2231 per year, as in the article's rate conversion example.
p = 0.20; Delta = 1; h = 0.2231
Five-year probability of 0.10
A study reports that 10% of people had the event over five years. The implied constant rate is about 0.0211 per year, slightly above 0.10 divided by 5.
p = 0.10; Delta = 5; h = 0.0211
Common errors
Using Excel LOG instead of LN
Excel LOG uses base 10 by default. The formula with LOG gives about 0.0969 per year for a one-year probability of 0.20 instead of 0.2231, understating the rate by a factor of about 2.3.
Sources
Probability to rate conversion in a transition probability tutorial
Gidwani R, Russell LB. Estimating transition probabilities from published evidence: a tutorial for decision modelers. PharmacoEconomics. 2020;38(11):1153-1164. Equation 8 and the worked 12-month to 3-month example.
Canonical Identity
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