GIGAMIND

Folder:
113 Math
File:
113.020.020.50.10 Proportional Hazards - Mathematical desc of Cox PH

PH - mathematical description of Cox PH

Cox PH allows the hazard to change over time (as opposed to 113.020.020.40 Survival analysis - Kaplan-Meier Model or 113.020.010.10.20 Regression models - Exponential Distribution). Can increase/decrease as time goes by.

BUT, Hazards between groups are proportional, and HR is constant.

e.g.
$$HR = \frac{HAZ(males)}{HAZ(females)} = 2$$
- At any time a male is twice as likely to die as a female.
- This is the definition of "proportional hazards". The proportion of top-to-bottom value is always the same. If the top if .5, the bottom will be .25. If the top if 10, the bottom will be 5.

$$ln(HAZ) = ln(h_0(t)) + b_1 X_1 + b_2 X_2 + ... + b_k X_k$$
Same as
$$HAZ = h_0(t) * e^{b_1 X_1 + b_2 X_2 + ... + b_k X_k}$$

The big thing about Cox is that he came up with a way to estimate the b_1, b_2, coefficients without having to specify what the baseline hazard is.

Cox PH Assumptions
1. Non-informative censoring - if someone is censored it is ignored, doesn't mean they are more or less likely to die (all models assume this)
2. Survival times (t) are independent (all models assume this)
3. Hazards are proportional - the HR is constant over time (also assumed for exponential and weibull)
- If hazards are not the same, we can stratify on the variable making it change
- Time dependent coefficients
4. ln(HAZ) is a linear function of the X's (also assumed for exponential and weibull)
5. VALUES of X DON'T change over time
6. Baseline hazard (h_0(t)) is unspecified


Source:
  • Me