I have 15 years of data of patients (2001-2015). T0 differs between them, i.e. the starting year for the survival period is strongly heterogenous. I want to model their survival rate, i.e. what is the probability that they survive after T0 each year? More so, I want to see how the survival rate differs by one or more characteristics of the patient. What would be the best way to model and estimate this?
My first option would be a survival function/hazard function. However, I am uncertain how to deal with (1) differing starting periods, i.e. T0 could be 2004 for one patient, 2012 for another, (2) different right-censoring times, i.e. if T0 is 2012, I only have 3 years of data to observe death compared to 12 years if T0 is 2003 and (3) varying probability rates dependent on a patient characteristics, which I am unsure how to incorporate and even if it is feasible.
My second option which looks more feasible in my opinion is a logit model where the dependent variable is death in a certain year or not, and the independent variable interacts time after T0 with the patient characteristic I am interested in. But if this is a valid option, I am not sure why I would even consider a survival or hazard function.