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I am using a Cox proportional hazards model to estimate the probability of an event (failure of a mechanical device, say) occurring at discrete, finite times: $t = 0, 1, 2, \ldots$. In Cox's original paper, Regression Models and Life Tables, he says that, if you let $T$ be the random variable representing the time to failure of a given member of your population, and $T$ is discrete, then $$ \lambda(t) = Pr(T = t|T \geq t). $$ So it seems to me that, if I estimate $\lambda_0(t)$ in the canonical Cox model formula $$ \lambda(t) = \lambda_0(t) \exp \left( \sum \beta_i X_i \right) $$ as a "baseline" probability that failure occurs (that is, a sort of average of the probability of the event occurring at time $t$ when the $X_i$ are all zero), I can obtain an estimate of the probability of the event occurring at a time $t$ in the future for a device with risk characteristics $X_i$ by just plugging in the values of $X_i$ and the value of $t$ into the formula. Is this the correct way to use a Cox model, or am I not thinking hard enough?

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It's too bad no one ever had anything to say about this, but the short answer is: I wasn't thinking hard enough! Moreover, I was wrong about a few things.

First of all, hazards modeling is concerned with the hazard function. If $T$ is a continuous random variable representing failure time of a subject, then the hazard function is defined as follows: \begin{equation}\label{first} \lambda(t)=\lim_{\Delta t \downarrow 0} \frac{\mathbb P(t \leq T<t+\Delta t | T \geq t)}{\Delta t} \end{equation}

In a Cox model done the right way, if we want to get the probability density function $f_T(t)$, it can be found by taking the product $$ f_T(t) = \lambda(t)S(t), $$ where $S(t)$ is the survival function, given by $$ S(t) = \exp \left( -\int_0^t \lambda(s)ds \right). $$ What I don't really understand is how, precisely, these three equations need to be modified if $T$ is discrete. In Cox's paper, he says that the hazard function for discrete $T$ is (contrary to what I said in my original question) $$ \lambda (t) = \sum_{j = 1}^N \mathbb P(T = t_j|T\geq t_j)\delta(t-t_j), $$ where $\{t_j\}_{j=1}^N$ is the set of points where $T$ has mass. Is there some way to obtain this from the first equation above?

I'm getting ready to post another question about this. If it gets answered, I'll come back here and link to it. Stay tuned...

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