I have question regarding survival analysis. I am looking at observational studies comparing patient survival who received treatments A or B at different centres (>10), for the same condition and have been followed-up for different time periods (between 1 and 10 years). I have the duration of follow-up for all participants (or time to death), as well as data for some covariates.
I am wondering what the best way to compare the effect of treatments A and B on survival. My current plan was to attempt an analysis pooling the log hazard ratios/standard errors of the Cox model for each centre, as often done with other meta-analyses.
Is it an issue that there are quite a lot of varying ratios of participants receiving treatment A or B depending on the centre (these are observational studies), varying sample sizes and that the follow up periods vary by centre, particularly when combining all centres? Or is there a better analysis strategy?
The Cox regression model assumes proportional hazards, but most likely adverse effects of the treatment would occur in the first year. Is there a way to account for this in the model, or elsewhere? I understand the Cox regression would consider survival across the entire follow-up period, but would it be worth, for this reason, to compare survival at 1 year as well?
I suppose this approach would be superior to simply indiscriminately pooling all participants and running a singular Cox regression on it, due to centre-specific biases (given the treatment is surgical) and population/selection biases. Is there a way I could estimate this bias to see whether it actually exists (in case that the demographics are all comparatively similar between centres/groups)?