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I am having a problem with checking the assumptions for a continuous variable in a proportional hazards model. If a variable were a factor with many levels, then I could use the logrank test or check whether the log(-log) transformations of survival curves are parallel. But what if a variable is continuous? Is that method still valid? Is Schoenfeld's test is a solution?

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If you have not assumed linearity for the continuous variables, or if linearity truly holds, then a next logical step is to assess proportionality of hazards using smoothed scaled Schoenfeld residual plots as implemented in the R survival package's cox.zph function. These plots show the estimated regression coefficient for a binary or continuous variable as a function of time. You hope for flatness in this relationship if PH holds. The function also provides a formal hypothesis test that is sometimes too sensitive against minor non-PH.

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