Assuming we have data for two groups from a time-to-event analysis with no censoring, there are two modeling approaches to comparing the group association with the risk of the event. Option one is a Cox proportional hazards model where a linear model is estimated having:

$$ \log \lambda (X|Z)= \log \lambda(X|Z=0) + \beta X$$

that is estimated by maximizing the partial likelihood

$$ \mathcal{L}(\beta; X, t) = \prod_{j=1}^K \frac{\lambda(X_j|Z_j)}{\sum_{\mathcal{l} \in\mathcal{R}\lambda(X_j|Z_\mathcal{l}) }} $$

More detail found here.

Alternately, one can consider the Poisson likelihood for an event and a log-linear model for a constant-in-time hazard function and an offset to designate the observation time, $t$ and each failure having the Poisson count variable equal to 1.

$$\log \lambda (X) = \alpha + \beta X - \log(t)$$

and $$ \mathcal{L}(\alpha, \beta, X) = \prod_{i=1}^n \lambda(X_i) \exp(-\lambda(X_i))$$

The question I have is under what conditions of the hazard function, or sample, is the $\beta$ that maximizes the Cox partial likelihood equal (or asymptotically equivalent to) to the $\beta$ that maximizes the Poisson likelihood? How do we prove this?

Assuming we have data for two groups from a time-to-event analysis with no censoring, there are two modeling approaches to comparing the group association with the risk of the event. Option one is a Cox proportional hazards model where a linear model is estimated having:

$$ \log \lambda (X|Z)= \log \lambda(X|Z=0) + \beta X$$

that is estimated by maximizing the partial likelihood

$$ \mathcal{L}(\beta; X, t) = \prod_{j=1}^K \frac{\lambda(X_j|Z_j)}{\sum_{\mathcal{l} \in\mathcal{R}_j\lambda(X_j|Z_\mathcal{l}) }} $$

More detail found here.

Alternately, one can consider the Poisson likelihood for an event and a log-linear model for a constant-in-time hazard function and an offset to designate the observation time, $t$ and each failure having the Poisson count variable equal to 1.

$$\log \lambda (X) = \alpha + \beta X - \log(t)$$

and $$ \mathcal{L}(\alpha, \beta, X) = \prod_{i=1}^n \lambda(X_i) \exp(-\lambda(X_i))$$

The question I have is under what conditions of the hazard function, or sample, is the $\beta$ that maximizes the Cox partial likelihood equal (or asymptotically equivalent to) to the $\beta$ that maximizes the Poisson likelihood? How do we prove this?