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The hazard function for the Cox proportional hazard model is written as follows.

$$\lambda(t|X_i) = \lambda_0(t)\exp(\beta_1X_{i1} + \cdots + \beta_pX_{ip}) = \lambda_0(t)\exp(X_i \cdot \beta)$$

I would like to know how come the time, $t$, is not really used in prediction (this link states that the hazards ratio does not depend on time). I know that one of the assumption is that the effects of the covariates are constant over time, but the $t$ being used as an input would imply that it has some influence in the prediction.

Furthermore, in this example, there are 2 subplots: one with the cumulative hazard and one with the survival predicted for an individual. What is the y-axis in each of these or how do I interpret these plots? The second plot seems clear to me: it is the probability of survival over time. The first plot does not seem so clear: is it the relative risk ratio over time (meaning at 50 years, the relative risk ratio is 6 times the baseline hazard)?

Any help is appreciated.

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Time can and must be used in predictions of failure or survival derived from a Cox model.

The graphs show the cumulative hazard and survival. The y-axis on the 1st graph is the cumulative hazard. The value of the cumulative hazard can exceed "1" as it is the sum of all risks prior to and including the time of interest. The instantaneous hazard has no upper boundary and can exceed 1. The y-axis of the second graph is the survival estimate- it would be more conventional for this graph to begin at "1" and have a lower bound of "0".

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  • $\begingroup$ Can the instantaneous hazard can exceed 1 even for a single individual? It would make sense to interpret it for a cohort, but I don't think it make sense to interpret it at an individual level. Can it make sense? In lay terms, if I had to explain it to myself or someone else, how do I say it? $\endgroup$ – Jane Wayne Jun 5 '17 at 22:34
  • $\begingroup$ I think where I am misunderstanding the interpretation is that the value is a ratio of the hazard at time t over the baseline hazard at time t. $\endgroup$ – Jane Wayne Jun 5 '17 at 22:46
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    $\begingroup$ The instantaneous hazard is only interpretable for the cohort under analysis. You can think of the point estimate of the hazard ratio as the mean of all hazard ratios for all time points weighted by the number of persons in the analysis at any given time period. $\endgroup$ – Todd D Jun 5 '17 at 23:25

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