ABSTRACT
Poisson regression is compared with exponential proportional hazards regression. Then ordinary Poisson regression is compared with Cox proportional hazards regression. If rates are from a Poisson process (meaning they do not vary), the two methods produce the same results. A piecewise Poisson model, split on all the failure times, can exactly reproduce the Cox model; they are mathematically the same. Poisson models with quadratic splines, restricted cubic splines, and fractional polynomials can be used to allow the baseline hazard to change over time. Royston-Parmer models are introduced. Finally, methods for relaxing the proportional hazards assumption are discussed for the Cox model, the Poisson model, and the Royston-Parmer model, allowing estimated rate ratios to vary over time. Advantages and disadvantages of these models are reviewed.
