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I'm trying to learn the Cox Proportional Hazards Model on my own, and found this link that describes it in clear terms. But when I get to Formula (5) ($S(t) = \exp(−H(t))$) I can't figure out where that's coming from. On the author's previous page, he shows that the survival function equals $S(t) = \exp(−H(t))$ if we assume an exponential distribution, but in Cox we don't assume that.

Is $S(t) = \exp(−H(t))$ something that works for any hazard distribution? I can't think of a way to prove/disprove this, and the intuition isn't making sense for me.

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  • $\begingroup$ I derive standard survival models in notes here. $\endgroup$
    – Charlie
    May 31, 2013 at 18:57

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All of these terms are standard in actuarial science and all of them apply to all distributions (but when I have seen these terms in studying for exams, we're almost always talking about distributions that are defined only for nonnegative reals). $H(t)$ is the cumulative hazard function, and for any distribution is defined as $$H(t) = \int_0^t h(x) \,dx.$$ Notice the name makes perfect sense with this definition, since we are "adding" up the hazard function up to a certain point to get the cumulative hazard function. Now, since $$f(t) = F'(t) = -S'(t)$$ then we have $$h(t) = \frac{f(t)}{S(t)} = \frac{-S'(t)}{S(t)} = -\frac{d}{dt} (\ln S(t)).$$ Finally, that means we have $$H(t) = \int_0^t -\frac{d}{dx} (\ln S(x)) \,dx = -\ln S(t)$$ since $S(0)$ is usually required to be 1 and thus $\ln S(0) = 0$.

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    $\begingroup$ Actually, the origin of these is in statistical survival analysis. I remember arguing for the adoption of terms like 'hazard' (along with other such terminology and tools from survival analysis, like KM and proportional hazards models and all the rest) rather than 'force of mortality' or more generally, something like 'force of decrement' and the sometimes less statistical tools that actuaries were mostly using (and certainly teaching) 3 decades or so back. All the tools and ideas are pretty standard in the actuarial field now, but that's not where it all comes from. $\endgroup$
    – Glen_b
    May 30, 2013 at 1:38
  • $\begingroup$ @Glen_b I am not saying they came from actuarial science, but they are standard in actuarial science. You need to learn these topics for Exams MLC/3L and C/4 (in the U.S.). And, that is how I know them. But, I do appreciate the lesson! $\endgroup$
    – GeoffDS
    May 30, 2013 at 2:34
  • $\begingroup$ Okay, no worries. Yes, I knew it was in those exams - it's standard in a number of other actuarial programs as well. $\endgroup$
    – Glen_b
    May 30, 2013 at 3:36
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Yes, it goes for any hazard function. The hazard function is defined as $$h(t)=\frac{f(t)}{S(t)}$$ where $f(t)$ is the probability density function with respect to time, & the survival function is $$S(t)=1-F(t)$$ where $F(t)$ is the cumulative distribution function. So integrate the first expression, & you get the cumulative hazard $$H(t)=-\log S(t)$$

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