Signature
p_T = 1 - (1 - p_C)^HR
| Inputs | Definition | Unit |
|---|---|---|
p_C | Probability that a comparator person at risk at the start of the cycle has the event within the cycle | probability from 0 to 1 |
HR | Hazard ratio of treatment to comparator, assumed constant within the cycle | ratio, no unit |
p_T | Probability that a treated person at risk at the start of the cycle has the event within the cycle | probability from 0 to 1 |
|---|
Function
Relative hazard function
Maps the hazards of an event in a treatment group and a comparator group at the same time t to their ratio. Under proportional hazards the ratio is constant over time, and it can then be applied to a baseline survival curve or a baseline transition probability to obtain absolute outcomes for the treatment group.
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Implementations
Excel
Treatment probability from named cells
Excel raises the comparator survival probability for the cycle to the power of the hazard ratio and subtracts the result from 1.
=1-(1-ComparatorProb)^HazardRatio
Assumptions
Proportional hazards within the cycle
The hazard ratio applies throughout the cycle. The comparator hazard need not be constant, because the relationship holds for the cumulative hazard over the cycle.
Single exit or cause-specific use
The probability refers to one event from the starting state. When a state has competing exits, the ratio is applied to the cause-specific hazard of the event it describes, and the row is then rebuilt with the competing-hazards formula on the Transition Probability page.
Worked examples
Annual cycle with a comparator hazard of 0.10
A comparator probability of about 0.0952 per annual cycle, from a hazard of 0.10 per year, and a hazard ratio of 0.70 give a treatment probability of about 0.0676, as in the article. Multiplying 0.0952 by 0.70 would give about 0.0666.
p_C = 0.0952; HR = 0.70; p_T = 0.0676
Comparator probability of 0.20 per cycle
With a comparator probability of 0.20 per cycle and a hazard ratio of 0.70, the treatment probability is about 0.1446, compared with 0.14 from direct multiplication, as in the Transition Probability article.
p_C = 0.20; HR = 0.70; p_T = 0.1446
Common errors
Multiplying a probability by the hazard ratio
Direct multiplication treats the hazard ratio as a ratio of probabilities. The error is small at low probabilities, 0.0666 against 0.0676 in the first example, but grows as probabilities rise or cycles lengthen, and a hazard ratio above 1 can then push the product above 1.
Sources
Relative rates applied under proportional hazards
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 (relative rates from randomised trials are incorporated into rate estimates using the assumption of proportional hazards, and the simple rate to probability formula).
Rate and probability conversion used with a hazard ratio
Gidwani R, Russell LB. Estimating transition probabilities from published evidence: a tutorial for decision modelers. PharmacoEconomics. 2020;38(11):1153-1164. Equations 8 and 9.
Canonical Identity
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