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I have a couple questions about Cox survival regression:

1) Is it true that the hazard function h(t) is not available (even WITHOUT time dependent covariates)- and if not, is it because the baseline hazard is not defined (only the cumulative hazard is and you cant go from the cumulative hazard to the hazard)?

2) When you have time dependent covariates is it not possible to estimate the survival curve for a chosen set of covariate values (I'd like to plot prototypical values)? Can anyone explain why not?

B_Miner
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1) Yes that's correct. There is no intercept in the CoxPH - it is one of the reasons why it's usually complemented by Kaplan-Meier or cumulative incidence curves

2) I've had some efforts into looking into time dependant covariates and it seems to be a mine-field, I've summarised most of the things I've learned in my previous question. If you leave the cox regression then you could try poisson regression or the Laplace regression (quantile regression) that might be easier to work with. One important part is that even though the PH might be violated it might not be important unless there is a clear trend in your data. What you get is an average and that might be good enough - T. Thernaugh mentions in his book that it's not always that you have to address this issue.

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Max Gordon
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  • Time-dependent covariates do not necessarily imply a time-dependent coefficient (and thus a non-proportional hazard ratio). @B_Miner: it's true that with time-dependent covariates the ability to predict survival curves from Cox model is usually lost. See for example the paragraph _Prediction and Time-Dependent Covariates_ from here http://www.bios.unc.edu/~lin/publications/1999/FisherLin99.pdf (Time-dependent covariates in proportional-hazards regression model; Fisher and Lin; Annu. Rev. Public Health. 1999. 20:145–57) – boscovich Apr 18 '12 at 17:02
  • @andrea: Is your distinction between these that the latter (time-dependent coefficient) is an interaction with time? I was wondering if the answer to my second question was because of the first - that because the survival curve is cumulative, you'd need the hazard curve to know how to "accumulate" the survival as the covariates changed? – B_Miner Apr 18 '12 at 19:50
  • Regarding the link, the argument of forecasting the covariate values in time seems weak - it is of course hypothetical but that is the point of simulating the "what-if". I do think I apprciate the logic of the second argument however-> – B_Miner Apr 18 '12 at 19:53
  • : Second, if we know the future value of some covariates (e.g. blood pressure), the existence of the values implies that the subject has not reached a death endpoint. This fact implies that one cannot estimate the survival curve from the observed and future values of a quantity such as blood pressure. The existence of a positive value would imply that the subject was still alive. Knowing the covariate would imply knowledge at each stage of the vital status. – B_Miner Apr 18 '12 at 19:53
  • @andrea Thank you for your clarification, time dependant coefficients and variables are truly two different things. – Max Gordon Apr 18 '12 at 20:00
  • @B_Miner I've seen time dependent variables and I don't think it's that difficult to implement in the calculations, I know the cmprsk-package has it in it's crr function. I think that the logic isn't that knowledge of the future implies that the subject must be alive, I rather think of it as during the summer it's usually warmer - hence I know that it will occur in the future just as I know menopaus will occur in x years in 50 % of my subjects (of course if they stay alive that long). – Max Gordon Apr 18 '12 at 20:01
  • @B_Miner: yes: a time-dependent coefficient is a consequence of an interaction between a covariate (which of course could be constant over time, i.e. time-_in_dependent) and (a function of) time. – boscovich Apr 19 '12 at 06:52