8. The Cox Model - also known as the
proportional hazards regression analysis
proportional: the ratio between the
treatment group and control group
hazard: instantaneous risk of an event
regression: an iterative process whereby
the effects of selected explanatory factors
are adjusted to create a model (equation)
which fits the empiric survival curve
10. Mathematically, the hazard function
is estimated by:
h(t) = h0(t)exp(ẞ1x1 +…..+ ẞkxk)
h(t) = h0(t)exp(ẞX)
h(t) = h0(t)exp(ẞ1x1 +…..+
ẞkxk)
where X = 0 for control group
X = 1 for intervention group
h0 = risk of event for
control group at time t
11. Which when we take the natural log
of both sides we get:
= ẞX
Now just try different values of
ẞ until the function best
approximates the actual
survival data.
12. Now we just plug in our
value of ẞ with X = 0 then
X = 1 to solve for the ratio
between the hazard of
dying for the intervention
group divided by the
hazard of dying for the
control group
13. Nota Bene!!
There is no time-dependent
term on the right-hand side
of the equation, thus the
ratio is independent of time,
hence constant throughout
the length of the survival
curve.
14. Remember what this was all about?
Recall our study and the survival
curve:
15. The median survival time
was lower in the intensive-control
group than in
the conventional-control group
(hazard ratio, 1.11;
95% CI, 1.01 to 1.23; P = 0.03)
Having an equation for the hazard
ratio we can thus derive P values for
the results!
