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I have managed to work out how to manually calculate the Schoenfeld residual for a coxPH model and i am getting the same answers as R.

However - i have absolutely no idea how to calculate the Scaled Schoenfeld residuals.

If anyone has the formula for these Scaled Schoenfeld residuals it would be greatly appreciated!

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  • $\begingroup$ See ?residuals.coxph for information on how to get these from R. $\endgroup$ – AdamO Jan 12 '18 at 17:56
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On page 228 in this book on Survival Analysis by Liu et. al. the formula for the scaled Schoenfeld residual for observation i and covariable m is described as:

$$ r_{im}^{\text{scaled schoen}} = \frac{r_{im}^{schoen}(\beta_m)}{V(\beta_m, t_i)} $$

Where $V(\beta_m, t_i)$ describes the variance of the coefficient for the variable $X_m$. This is estimated via the observed Fisher-Information-Matrix that you presumably already calculated in the Fisher-Scoring-step for the estimation of the coefficients (negative 2nd derivative of Partial log-Likelihood w.r.t. all coefficients).

I think, intuitively this makes sense, as you scale the Schoenfeld residuals by the variance of the coefficient of interest. I am looking forward to your response on that - hope this short paragraph helps!

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  • $\begingroup$ If I well understand, such standardization is the one proposed by Pettitt and Bin Daud (1990). As noted here:(jstor.org/stable/pdf/…) it is only valid in case of if the covariates are uncorrelated at each time point. PETTITT, A. N. & BIN DAUD, I. (1990). Investigating time dependence in Cox's proportional hazards model. Appl. Statist. 39, 313-29. $\endgroup$ – Federico Tedeschi Mar 13 at 14:57
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The scaled Schoenfeld residual for time $t_k$ is given by multiplication of the vector of Schoenfeld residuals at time $k$ by the inverse weighted covariance matrix of the (partial likelihood) estimate of the regression coefficients. See here:

(pdf)

at pages 39-42.

Here:

(pdf) you can find the formula:

$r_{k}^*(\beta) = V^{-1}(\beta, t_k)r_k(\beta)$, where it is highlighted that everything depends on the $\beta$ vector of coefficients from the Cox regression, $r_k$ is the Schoenfeld residual for events at time $t_k$, $r^*$ expresses scaled Schoenfeld residuals, $V^{-1}(t_k)$ is the covariance matrix of covariates for events at time $t_k$.

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