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Suggested method for data with assumed non-proportional hazards
We are planning a study on longitudinal healtcare data. Exposure is starting a certain medication ("A"). Controls are those in the dataset not starting A, matched on age, sex and some other variables. Outcome is the presence of a laboratory value below a certain threshold on the first test after initiating A (ie binary outcome of interest). The time to the first testing of this laboratory value can vary considerably. In consequence, we assume that hazards will be non-proportional (time to event depends on when testing takes place rather than being related to risk).
What other methods than Cox proportional hazards model could be recommended in this context?
Requirements on the model include:
Can be used with non-proportional hazards (which we assume a priori).
Censoring (eg death of the patient or end of follow up).
Multivariable adjustment using both categorical and continuous predictors.
Can model competing risks (other outcomes related to A that invalidate the out