Coefficients in discrete proportional hazards model

I am aware of the rationale behind taking the exponentiated coefficients in a proportional hazards model as representing change in hazard per unit change in the corresponding predictor. This also implies that it should be possible to obtain a hazard greater than 1. However, with the inverse of the cloglog link function being $1-e^{-e^{\nu}}$, it doesn't seem possible for the hazard to actually exceed 1. What am I missing?

Thank you.

• Do you have discrete or continuous time models? In discrete time, you model probability to leave the state, and that is obviously < 1. In continuous time you model the exit rate, and usually you parametrize the hazard rate as $\theta(x, t) = \exp(\beta' x) \cdot \phi(t)$. This can be > 1. I don't understand the link function in this context. – Ott Toomet Aug 13 '16 at 0:53
• I have a model in discrete time. The dependent variable for each subject is zero until either they are censored - thus being coded zero in their final entry - or experience the event - thus being coded 1. This yields a binary response variable, to which I am fitting a binomial regression model with the cloglog link function in line with recommendations (eg Singer & WIllett, 2003) for data that occurs in continuous time but is available at discrete points. – Marko Aug 13 '16 at 1:07
• PS Happy to be referred to an existing answer. I have tried my best to find one, but they seem to be mostly about the continuous-time case. – Marko Aug 13 '16 at 20:35
• Damned, I think I answered yesterday... – Ott Toomet Aug 13 '16 at 21:11

In continuous time hazard rate is the exit probability per unit time (given still in the initial state). It may exceed one. In discrete time, you look at probability to exit during the time interval, and obviously, as a probability, it must be $\le 1$. I think you have to be clear of this distinction to understand the problem.
It seems to me that you model the (continuous time) hazard rate as $e^\nu$ (constant in time). In this case the probability to have exited the initial state by $t$ is $Pr(T < t) = 1 - e^{-e^\nu t}$ (this is exponential CDF). If your hazard rate is not constant, you will have an integral instead of $-e^\nu t$.