Dear Allstaters,
Thanks for all those who enlightened me on the
differences between the HR and OR.
Due to some requests, I forward to the list the
responses I received:
***********
The hazard function is an estimate of the probability
that a subject who survived the beginning of a study
period will experience an event during that period.
The hazard rate is plotted along with the standard
error of the estimated hazard rate.
The odds ratio is the proportional change in the
variability of the
independent variable as it is increased by one unit.
******
A hazard ratio is the ratio of two probablities, not
two odds. It arises in survival analysis. Any
substantial book on medical statistics should explain.
*****
The hazard is the probability of an event in a given
time period if you survived, event free, to the
beginning of the interval. Cox regression
produces hazard ratios, which are instantaneous -
which is to say that
the time interval is arbitrarily small. Odds ratios,
autem, are produced by logistic regression; odds are
simply the probability of the event having happened
divided by the probability of its not having
happened(no time period invoked)
*****
The odds tells you something about the probability
of getting the disease, i.e.
P(disease)
odds = ---------------
1 - P(disease)
whereas the hazard rate is additionally concerned with
the time of the onset of disease, it is an
'instantaneous rate'. For a continuous time variable
T for disease onset (i.e. age) the definition of the
hazard rate h(t) is rather technical, i.e.
P( t <= T < t + delta t | T >= t)
h(t) = lim ---------------------------------
delta t -> 0 delta t
The definition for discrete time T is
h(t) = P(T=t | T >= t)
the probability of getting the disease at time t,
given
that one is disease free until time t, see the book of
Kalbfleisch and Prentice (1980).
The odds ratio (hazard rate ratio) is the quotient of
the odds (hazard rates) of the exposed and the
unexposed.
However, although odds and hazards deal with different
aspects( disease yes/no vs. time of disease onset )
I feel that they have a lot in common, but I can not
make this precise.
*******
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