# 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., $\beta_0$).

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• Can you be more specific about what is given in part 1? What is the sex, income, etc distribution of the population for which the survival is known? – Aniko Nov 23 '14 at 0:09
• My data set is typical. My response is the survival time for $n$ individual over $t$ years, and I have for each individual their age, sex, and income. In addition to that, I've the the survival time for each age as mentioned in my question. Now I need to fit cox model and use my baseline data about age. Hopefully this answers your question. – user9292 Nov 23 '14 at 3:17

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.