I am trying to fit a coxph model in R. The study can be described as follows: I have a very large dataset, in counting process form, containing whether or not someone responded to a survey or not. The time variable is the consecutive number of months in which a response was received. The model predicts non-response, i.e. when someone does not respond. There are more than one record per id (itemcode in my dataset). There are no continuous covariates as of now. What is included in the model are seasonal effects--I would like to know how each season increases or lowers the hazard of non-response, relative to fall. The I have already stratified the model. The results are below:
Call: coxph(formula = Surv(start, cum.goodp, dlq.next) ~ winter + spring + summer + strata(sector) + cluster(itemcode), data = nr.sample.split) n= 651033, number of events= 42508 coef exp(coef) se(coef) robust se z Pr(>|z|) winter 0.26850 1.30800 0.01307 0.01283 20.92 <2e-16 *** spring -0.64040 0.52708 0.01385 0.01342 -47.72 <2e-16 *** summer 0.29188 1.33894 0.01414 0.01284 22.73 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 exp(coef) exp(-coef) lower .95 upper .95 winter 1.3080 0.7645 1.2755 1.3413 spring 0.5271 1.8972 0.5134 0.5411 summer 1.3389 0.7469 1.3057 1.3731 Concordance= 0.598 (se = 0.004 ) Rsquare= 0.009 (max possible= 0.636 ) Likelihood ratio test= 5864 on 3 df, p=0 Wald test = 4783 on 3 df, p=0 Score (logrank) test = 5634 on 3 df, p=0, Robust = 5015 p=0 (Note: the likelihood ratio and score tests assume independence of observations within a cluster, the Wald and robust score tests do not).
I then estimated a cox.zph function to test the PH assumption, the results of which are below:
rho chisq p winter -0.1283 691.45 0.00000 spring -0.1151 569.35 0.00000 summer -0.0163 9.36 0.00221 GLOBAL NA 1096.18 0.00000
Clearly, the PH assumption is not valid for any of the coefficients. Below is a plot of one, summer:
[![Plot of Beta(t) for coefficient "summer"]]
My question is: since the seasonal dummy variables are static by nature, and their coefficients clearly vary with the time variable, how much does it matter? I get that statistically, it means something, but does the violation of PH assumption invalidate the (intuitively appealing) result that non-response is more likely to happen in the summer and winter? If so, is there a way to handle this so that the PH assumption is not violated? I know about using the tt transform, but I can't seem to figure out the exact form for the function. Any advice, ideas or references would be greatly appreciated.