Proportional hazards assumption Cox's model vs AFT

Question

I'm modelling survival data with Cox's model and AFT model (weibull) with only treatment as dependent variable. From the KM plot of the survival probability i saw that there isn’t a different trend by the two groups and that the distance between the two curves rests constantly over time. (the two curves cross each other but rest "crossing" all the follow-up time) That suggest to us that the PH assumption is met for the treatment variable.

For testing the proportional hazards assumption in another way I add a time dependent variable for the treatment in the Cox's model, and it results not significant, so seems the assumption is met for the treatment variable.

The problem is that if I try to add the same time dependent variable to AFT weibull model the variable result significant and the GOF of the model result incremented.

I have to make a confrontation between the two models so, I consider the assumption met and use only the treatment variable for all the two models or I add the time dependent variable?

Answer 1

You are doing a lot of hypothesis testing, which is generally not a good way to decide on a model. Ideally, a single model would be pre-specified or you could do model averaging between candidate models. The described data-dependent way of picking a model will invalidate inference from the model that is picked at the end.

In general, it may be that the parametric model is more powerful than the semi-parametric model. Alternatively, it could be that the Weibull model does not fit the data all that well, but that adding the interaction makes the model a little more flexible so that it fits the data better. In that case, you may decide that it's more of an artifact of forcing a Weibull model on the data.

One way of getting an approximate idea of what is going on would be to plot the survivor curves (or CDFs) from all models (Cox with and without time interaction, as well as Weibull with and without time interaction) together with the Kaplan-Meier curves for each treatment group. This will tell you whether one of these models is fitting the data terribly.