Estimating mortality using Cox model with baseline I have been reading online notes and papers on how to build survival models using CPH, and I think I've a good idea how things work. However, there are two questions I still have in mind:
1) let's say I have some mortality data (the probability of survival) by age (e.g., for 55 years old, the probability of survival next year is 0.855, and so on). How can I incorporate this piece of info in my model (with different covariate: age, sex, income, etc.) . Is it something like this:
Let $S(t) = p$ be the survival probability.
Then, by the definition of CPH, we have
$S_i(t)=S_0(t)e^{(x′_i \bf{\beta})}$ 
Let's say I fit the model and obtained the parameters estimate, and now I need to perform prediction for 55 y.o. person as follows
$\hat{S}(t)=(0.855)e^{(age \times\hat{\beta}_{age} + \; sex\times\hat{\beta}_{sex} +\;income\times\hat{\beta}_{income})}$  
Is that correct?
2) How can I include an intercept in my model (i.e., $\beta_0$).
 A: The Cox model is not evaluating the survival function $S(t)$ but the Hazard function $h(t)$, which gives the instantaneous probabilities of the event, given that the individual has survived up to time $t$.
It is called proportional hazard (PH) model because the hazard at any time $t$ is the product of two quantities: the baseline hazard, $h_0(t)$, and the exponential expression to the linear sum of $\beta_iX_i$:
$h(t, X) = h_0(t)e^{\sum\limits_{i=1}^p \beta_iX_i}$
This construction implies that the baseline hazard $h_o(t)$ is a function of $t$ but does not involve $X$'s; we do not need to assume a specific shape for the baseline hazard function. 
For this reason the model is semiparametric and has no intercept. The intercept would correspond to the hazard when all covariates are equal to 0 - that is - the baseline hazard.
You can use the Cox model when your interest is in evaluating hazard ratios between different groups of individuals. If the baseline hazard is a quantity of interest in your study you may consider fully parametric approaches to survival data.
