# What is the meaning of “t” in Cox hazard proportional model?

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 $\lambda_0(t+10)e^{{\mathbf{\beta}^\intercal}x}$? • 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. – Yujian Mar 2 '18 at 19:16

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.
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.