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I am new to survival modeling and the Cox-PH model is quite simple with $$ \lambda(t|x;\beta) = \lambda_0(t) \exp(x'\beta). $$ where $x$ does not depend on time. I'm wondering if it is also possible to include time and functions thereof as covariates in order to make less strict parametric assumptions regarding the baseline hazard. Consider the following $$ \lambda(t|x;\beta,\alpha) = \lambda_0(t) \exp(x'\beta + \alpha_1 t + \alpha_2 t^2 + \alpha_3 \log(t) + \ldots). $$

This seems equivalent to picking a different baseline hazard $$ \lambda(t|x;\beta,\alpha) = \lambda_1(t ; \alpha) \exp(x'\beta) $$ where $$ \lambda_1(t ; \alpha) = \lambda_0(t) \exp(\alpha_1 t + \alpha_2 t^2 + \alpha_3 \log(t) + \ldots), $$ but you can fit it using all the normal software for fitting the Cox-PH model.

Is this a reasonable thing to do?

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You need a different likelihood function to handle time-dependent covariates. The best explanation I've seen is at https://cran.r-project.org/web/packages/survival/vignettes/timedep.pdf which is a vignette for the R survival package (one of many incredible vignettes that comes with this package by Therneau).

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  • $\begingroup$ I see. Is there a standard background article on what changes with time-dependent covariates? Thanks! $\endgroup$
    – ted
    Jan 29, 2017 at 4:03
  • $\begingroup$ I'm not clear on what you are asking. $\endgroup$ Jan 29, 2017 at 14:04
  • $\begingroup$ An article that discusses what procedures or assumptions change for the cox ph model once you add time dependent covariates. $\endgroup$
    – ted
    Feb 5, 2017 at 5:31
  • $\begingroup$ The pdf file I provided above. $\endgroup$ Feb 5, 2017 at 14:33
  • $\begingroup$ @ted think this might be what you are looking for math.ucsd.edu/~rxu/math284/slect7.pdf $\endgroup$
    – jeffmax
    Jun 22, 2017 at 21:05

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