Multi-centre survival analysis 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.

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*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)?
Thank you!

If you have corresponding covariate values from all of the institutions, then one approach would be to build a Cox model with all the cases while including the institutions as a type of random effect. That could be done with a gamma frailty term in the original coxph() modeling in R, or a Gaussian random effect in the coxme package. That allows for a distribution of center-specific hazards beyond those associated with the covariates, providing at least a start at addressing the issues in questions 1 and 3.
With respect to question 2, if you expect that most events of interest happen in the first year and you have reasonably complete follow-up through at least 1 year, then you could do a binomial regression with respect to outcomes at 1 year. Note that your expectation does not necessarily invalidate the proportional hazards (PH) assumption; your assumption has to do with the baseline hazard, and it's possible that PH still holds at later times around a lower overall hazard.
For further investigation of center-specific bias in terms of treatment selection, you could consider evaluating the propensity of receiving treatment A versus B as a function of covariates, including the institution as a predictor. That's often done with a logistic regression, but the more flexible propensity modeling provided by the twang package can be helpful. You also could then incorporate the propensity of having received treatment A versus B into your survival model to try to correct further. See the many posts on this site about propensity-scores.
