I am modelling a survival analysis over a rather long follow-up period (10 years). My exposure is time-invariant and clearly violates the proportional hazards assumptions so Cox Proportional Hazards regression models are not an option. I was wondering about alternatives to conduct my analyses. Please find below some key points:

Stratification is not possible because it is my main variable of interest that violates the assumption and I need to compare between groups

Time-varying models are not possible given the nature of my main variable of interest

I initially thought about time-partitioned model (splitting follow-up time and interacting time with my main variable of interest) but I am not sure that is a good idea because when I plot the KM curves it is all crossing – so I struggle to find a good time interval for splitting

Given these considerations I have thought about employing flexible parametric models. Howevers, from my understanding they make strong assumptions about the shape of the curve – which is something I cannot be certain of. Would a flexible parametric model with restricted cubic spline what I am looking for? But How can I define the number of knots? And how about the distribution?

Could you please provide some inputs and examples? What do you suggest?

I use Stata MP 15 as statistical software

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#### Best Answer

I disagree that AFT and PO are necessarily the right next steps. It depends on what you are interested in learning from the model. If you are interested in estimating a hazard ratio, understand that, the idea that there is one hazard ratio that applies over a period of time, implies to some extent that hazards must be proportional.

On the other hand, in many applications there are much more informative summaries of survival analyses than hazard ratios, which don't inherently have a PH assumption baked in. For example, you can calculate risk differences and risk ratios at domain-relevant time-points. These are typically more intuitive and easier to interpret correctly than hazard ratios. RDs and RRs are still available using stratified Cox models (assuming your exposure is categorical). For an overview of these ideas, you can have a look at this reference: https://pubmed.ncbi.nlm.nih.gov/25660080/

Now, if you insist on summarizing your data using hazard ratios, and hazards are not proportional, you can examine how the hazard ratio is changing over time using **interactions between time and your time-invariant covariate of interest.** This is a valid use of Cox models under non-proportional hazards and can be quite informative – explained in this paper: https://pubmed.ncbi.nlm.nih.gov/12915864/. On the other hand, if the violation of proportionality is not too extreme, a single hazard ratio can still be a reasonable summary of the data – explain in this paper: https://pubmed.ncbi.nlm.nih.gov/32167523/.

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