The hazard ratio is the ratio of (chance of an event occurring in the treatment arm)/ (chance of an event occurring in the control arm) (20 ). The HR has also been defined as, the ratio of (risk of outcome in one group)/ (risk of outcome in another group), occurring at a given interval of time ( 21 ). The formula for the hazard rate is C C T T C T CM O E O E H H HR / / = = where O i is the observed number of events (deaths) in group i, E i is the expected number of events (deaths) in group i, and H i is the overall hazard rate for the ith group. The calculation of the E i is explained in Parmar and Machin (1995). If the hazard ratio is not consistent over time, the value that Prism reports for the hazard ratio will not be useful. If two survival curves cross, the hazard ratios are certainly not consistent (unless they cross at late time points, when there are few subjects still being followed so there is a lot of uncertainty in the true position of the survival curves). Hazard ratio is a ratio of two hazard functions HR(t) = 1(t;x 1) 2(t;x 2) (3.1) and we remind the reader that the hazard function is defined as (t;x) = lim +t!0 P(t T<t+ tjT t;X= x) t and that hazard is connected to the survival function via the following formula S(t;x) = e: 1)) = 1 1) = + p) Since the hazard is a function of time, the hazard ratio, say, for exposed versus unexposed, is also a function of time; it may be different at different times of follow up. For example, if the exposure is some surgery (vs. no surgery), the hazard ratio of death may take values as follows: Time since baseline Hazard ratio 1 day 9 2 days 3.5 Survival Distributions, Hazard Functions, Cumulative Hazards 1.1 De nitions: The goals of this unit are to introduce notation, discuss ways of probabilisti- the ratio d=Nestimates the (discrete) hazard function of T =age at death. We will see that H() has nice analytical properties. The hazard ratio was derived as the ratio of the hazard of death for the intervention group to the hazard of death for the placebo group across the study period. To derive the hazard of death for a treatment group, the study period was conceptually divided into very short time intervals. The hazards ratio associated with a predictor variable is given by the exponent of its coefficient; this is given with a confidence interval under the "coefficient details" option in StatsDirect. The hazards ratio may also be thought of as the relative death rate, see Armitage and Berry (1994) . This is the math equation used in this hazard ratio calculator. The formula for the mean hazard ratio is the same, but instead of observed and expected at time t, we sum the observations and expected observations across all time slices. From this we can see why the hazard ratio is also called the relative failure rate or relative event rate. 2.Compute L = (O1 - E1) / V, where O1 is the total observed number of events in group1, and E1 is the total expected number of events in group1. You'd get the same value of L if you used the other group. 3.Note that L is the natural logarithm of the hazard ratio. So the hazard ratio equals exp (L).
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