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I am considering a survival time analysis with death as the outcome. I first performed a time-dependent Cox regression with BMI as the independent variable (linear model). The likelihood ratio test confirmed a significant improvement from the null model. Proportional hazard assumption was evaluated using Schoenfeld's method, and significant deviations were confirmed. Therefore, a time-dependent Cox regression was performed again with BMI as a restricted cubic spline (df=3) in the model (spline model). The likelihood ratio test confirmed a significant improvement compared to the linear model. I evaluated this spline model with Schoenfeld and found no significant deviations. I am not sure if I can assess proportional hazard assumption by Schoenfeld's method in the time-dependent Cox model with restricted cubic spline. From this result, is it safe to say that there is no significant deviation of proportional hazard assumption in the spline model? I would be grateful if you could enlighten me.

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  • $\begingroup$ I've only seen examples of using the Schoenfeld residual PH assessment in models without time-dependent covariates. $\endgroup$ Jan 15, 2023 at 14:56

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With at most a single event possible per individual, the counting-process data form used to fit a Cox model with time-varying covariates "is just a mechanism, a trick almost, that allows the program to select the correct $x$ values for [an individual] at a given time" (Therneau and Grambsch, page 70). All that matters for such a Cox model is the set of covariate values in place among members of the risk set at each event time.

With that in mind, there's nothing wrong with using scaled Schoenfeld residuals to evaluate proportional hazards (PH) in your situation. Your identification of a violation of PH in your initial model with time-varying covariates led to a more realistic nonlinear functional form for association of the BMI covariate with outcome. That's good.

You do, however, need to be very careful in considering whether all that matters is the set of covariate values in place among members of the risk set at each event time. Death is more likely to be a function of risk factors integrated over time, rather than an instantaneous covariate value.

You also need to evaluate the direction of causality in such a model. For example, a cancer patient with cachexia might be developing progressively lower BMI until death. How much sense does it make to use BMI as a predictor of death in that case?

Such assumptions are probably more important than the PH assumption.

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  • $\begingroup$ I understand that using scaled Schoenfeld residuals to evaluate proportional hazard (PH) is not a problem in my situation. Thank you for your further detailed advice. It is very helpful. $\endgroup$
    – Totti
    Jan 16, 2023 at 10:28

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