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Recently, I thought of the following question: We are often taught about a "Cox Proportional Hazards Model" - this is able to model the hazard between different cohorts of patients, assuming that the hazard between any two groups of cohorts is always constant (I am not sure why this is such an important assumption - I would have thought that it would have been possible to make models when this assumption is not met).

I am aware that the Cox Proportional Hazards Model is able to estimate Survival Curves for each group of cohorts - thus : Why is the Cox Proportional Hazards Model based on "Hazard" instead of "Survival"? Does this have anything to do with the fact that perhaps (I don't know why) that Survival Curves usually tend not to be proportional to each other at all times?

Thanks!

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    $\begingroup$ could you define what it means for survival curves to be proportional? $\endgroup$
    – Ben
    May 17 at 16:01
  • $\begingroup$ Put otherwise, the hazard and the survival functions are intricately related. I've even heard analysts describe survival curves as parallel, to imply that the hazards are parallel - or that they are cloglog parallel. Anyway, the Cox model fundamentally models the log hazard function as a linear combination of predictors. If we wanted to get into the nitty gritty, why not call it Cox additive log hazards model? $\endgroup$
    – AdamO
    May 17 at 17:03
  • $\begingroup$ Survival functions are bounded between 0 and 1 (and endpoints are definitely attainable). If you had ${}S_2=kS_1$ for an $k\neq 1$, and $S_1$ does attain both bounds, what does that imply for $S_2$? $\endgroup$
    – Glen_b
    May 17 at 23:34

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Proportional survival rates depend on the overall prevalence of events, proportional hazards do not. Suppose you have two groups with different hazard rates, one where events occur in 10% of the population each year, and one where events occur in 20% of the population each year. The hazard ratio is 2, at any time point you should expect to see twice as many events in the second group as compared to the first. But the same is not true of survival, after 1 year you'd expect to see 90% and 80% survival, respectively. After another period of time, you'd find a different ratio of survival proportions. If the event rates were 20% and 40%, you'd still have the same doubling of events per time period, but the survival proportions would have a different ratio comparing 80% survival to 60% survival. The ratio of hazard rates ($x/y$) behaves a bit differently than the ratio of survival rates ($(1-x)/(1-y))$.

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