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I'm learning survival analysis and currently have questions about the Cox proportional hazards model. The following figure wants to show that for a certain smoking subject, the hazard of a 10-year increase in age is still proportional. I'm wondering why the numerator is not $h_i(t+10)$, which is also $\lambda_0(t+10)e^{{\mathbf{\beta}^\intercal}x}$?

enter image description here

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  • $\begingroup$ I have one idea. The "t" stands for the time until another death happens. Therefore, as long as the subject is alive, the baseline hazard rate is still an exponential distribution without any changes in the parameter. Hence t could not be changed to t+10. $\endgroup$ – Yujian Mar 2 '18 at 19:16
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To extend the answer from Aksakal a bit, note that there are two different uses of time in this analysis. One is the t (described by Aksakal and used in the cited equations) for time at which survival is calculated following entry into the study, the other is time in terms of age of the individual at entry into the study.

The "10-year increase in age" being analyzed refers to a difference between 2 individuals in age at entry to study; the hazard is for comparing individual i who, say, started at age 70 against one, j, who started at age 60. If the proportional hazards assumption holds, then the hazard ratio between starting at age 70 versus age 60 will be the same at all times t following entry into the study.

Using $t+10$ for individual j in the equation, as proposed by the OP, is something completely different: it would be comparing the hazard for individual i at a time t (following entry into the study) against the hazard for individual j at a time 10 units later, $t+10$. For that comparison you would need to know the baseline hazard, not just the coefficients, and the displayed ratio would in general depend on the value of t chosen.

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  • $\begingroup$ Thank you! Now I understand the 10-year increase is used for the comparison of two subjects who have a 10-year age gap, not one subject's two states that have a 10-year distance. So it means that the hazards between two subjects at the same time point would always be "proportional". But it might not be "proportional" if the hazards are from different time intervals, even for the same subject. $\endgroup$ – Yujian Mar 3 '18 at 1:49
  • $\begingroup$ @Yujian : precisely. $\endgroup$ – EdM Mar 3 '18 at 2:15
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It's time to survive from time $t=0$. You can calculate the different quantities, such as rate of death (hazard) at time t, or cumulative probability to survive from time $t=0$ to time t, from time $t=s$ to $t$ etc.

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