If a Cox PH Model is specified we should always check the Proportionality assumption. It's obvious that, if it is violated the coefficient for the covariate this is the case for is not the efficient estimate, but only an average over time and might therefor be misleading. But what about the other coefficients in the model, which fulfill the PH assumption? I specified a Cox PH model and one of my control variables does violate the assumption. However I'm having some troubles fitting the time-dependent version. As this variable is not really of interest for me, can I skip the estimation with the time-dependent coefficient, or does the violation of the PH assumption by one variable also have an effect on the coefficients of the other variables?
I haven't readily found a clear, succinct answer on this. Therneau and Grambsch in Modeling Survival Data imply that there is bias in the other coefficients, when they talk on page 145 about stratifying a continuous covariate to deal with a violation of proportional hazards (PH):
Quantitative variables can always be discretized, but it is not always obvious how to do so. A too coarse categorization leaves residual bias in the beta coefficients for the regressors and a too fine division loses efficiency. (Emphasis added.)
As discussed in an answer to your related question, however, the real problem might be mis-specification of the functional form of the relationship between your continuous covariate and outcome that then shows up as an apparent violation of PH. If PH is still violated after trying better modeling of that functional form, you should also consider (a) whether the non-proportionality is so large as to make a substantive difference in your results, or (b) if it is just an artifact of a few outlier data points. From page 142 of Therneau and Grambsch, with respect to (a):
Suppose the Schoenfeld residual plot or other diagnostic technique gives strong evidence of nonproportionality for one or more covariates. What should one do? The first two questions to ask are "does it matter" and "is it real;" it will often turn out that nothing is required. A "significant" nonproportionality may make no difference to the interpretation of a data set, particularly for large sampIe sizes.
They later go on to an example with respect to (b) in which an apparent violation of PH was "due entirely to the two very early event times that show up as outliers" (page 145).
You would be wise to test for bias in your own data, comparing regression coefficient values in the model that violates PH in the continuous covariate against a model that stratifies by that covariate. Consider doing that and reporting your results here as an answer to this question, to provide at least an example for others of what can happen in this circumstance.