The ordinary equation of linear regression has the form:
y=B+b1x1+b2x2+e
Where
B is intercept
b1,b2-beta-coef
x1,x2-real values of obs
I want write the equation for a Cox regression. I get the follow result table
Beta t-value Wald p
AGE 0,105520 3,373554 11,38087 0,000743
ANTIGEN 0,583551 1,366189 1,86647 0,171889
How do I use these beta coefficients to write the equation for Cox regression?
The Cox regression models the log-hazard of the outcome as a linear combination of risk factors and a baseline hazard function. It is written as:
$$\log(\lambda(t) | X) = \log (\lambda_0(t)) + \beta_1 X_1 + \ldots \beta_p X_p$$
where $p$ is the number of "predictors" in the model. The notation may vary from text-to-text. The term "baseline hazard function", $\lambda_0(t)$ does not actually refer to time 0, but having all predictors $X_1, \ldots, X_p$ equal to 0.
Using your output it would be written as:
$$\log(\lambda(t) | X) = \log (\lambda_0(t)) + 0.11 \text{Age} + 0.58 \text{Antigen}$$
The term $\lambda_0(t)$ is not actually estimated. The reason for this is that Cox regression maximizes a partial likelihood. People at risk for the event are only compared to others at times in which events are observed. These groups at event times are called risk sets.
Theoretically, between any two failure times, any amount of time may pass. Whether 1 minute or 400 years, gaps between failure times do not change model coefficient estimates. The only term affected in this case would be the baseline hazard function (which would be very low if 400 years elapse, and very high if only 1 minute elapses until the next failure).
