Both can be correct, but most likely you should use (start, stop). It has to do with how you define the baseline hazard. If the baseline hazard should be a function of time since 0 then use (start, stop). If the hazard functions should be reset, such it is a functio of time since last stop time then you can use your model 2.  In far most cases, model 1 is the right way to go.    For your second question about how the hazard ratio is calculated, the simple answer is that it is the maximum point of the Cox's partial likelihood function. The hazard ratio has no closed analytical form. Except in simple cases with very few events. In your case the estimated hazard ratio is the maximum point of the function (x/(1+2*x))*(1/(1+x)) which is sqrt(1/2). I dont get exactly same number as you mention btw, but I have tried run the SAS code and the solution is indeed sqrt(0.5)=0.707. ... View more