Signature
p_new = 1 - (1 - p_old)^(Delta_new / Delta_old)
| Inputs | Definition | Unit |
|---|---|---|
p_old | Probability of the event over the interval in which it was reported | probability from 0 to 1 |
Delta_new | Length of the model cycle | time, in the same unit as Delta_old |
Delta_old | Length of the interval over which p_old was reported | time, in the same unit as Delta_new |
p_new | Probability of the event within one model cycle of length Delta_new | 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.
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Implementations
Excel
Rescaled probability in one cell
Excel raises the complement of the source probability to the ratio of the interval lengths, using named cells.
=1-(1-OldProbability)^(NewLength/OldLength)
Assumptions
Constant hazard across both intervals
The hazard is the same throughout the source interval and the model cycle, so compounding the new probability over Delta_old divided by Delta_new cycles recovers p_old.
Two-state transition only
The rescaling is exact only when the starting state has a single exit. With three or more possible transitions from a state, the rows of the transition matrix are converted jointly, for example from the competing hazards or from a matrix root.
Worked examples
Annual probability rescaled to one month
An annual probability of 0.20 becomes a monthly probability of about 0.01842, as in the article. Dividing 0.20 by 12 would give about 0.01667 instead.
p_old = 0.20; Delta_old = 12; Delta_new = 1; p_new = 0.01842
Five-year probability rescaled to one year
A five-year probability of 0.30 becomes an annual probability of about 0.0689, larger than 0.30 divided by 5 because fewer people remain at risk in later years.
p_old = 0.30; Delta_old = 5; Delta_new = 1; p_new = 0.0689
Common errors
Dividing an annual probability by 12
Dividing 0.20 by 12 gives a monthly probability of about 0.01667. Applied for 12 months it implies an annual probability of about 0.1826 rather than 0.20, so the model understates the event. The error grows with the size of the probability.
Rescaling each exit of a multi-exit state separately
In a state with more than one exit, rescaling each destination probability separately with this formula gives a row that no longer reproduces the source data. In the article's example with annual exit probabilities of about 0.10876 and 0.07251, separate rescaling to three months keeps about 0.8248 of the cohort in the starting state after four cycles rather than 0.8187.
Sources
Warning against dividing probabilities when changing cycle length
Chhatwal J, Jayasuriya S, Elbasha EH. Changing cycle lengths in state-transition models: challenges and solutions. Medical Decision Making. 2016;36(8):952-964. Introduction (dividing an annual transition probability by 12 is not appropriate) and the section on the traditional approach, which gives the rescaling formula and shows that it applies only to two-state models.
Equivalent single-step rescaling formula
Gidwani R, Russell LB. Estimating transition probabilities from published evidence: a tutorial for decision modelers. PharmacoEconomics. 2020;38(11):1153-1164. Equation 10 and the section on changing cycle length when there are three or more state transitions.
Canonical Identity
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