# What is the formula to calculate the scaled schoenfeld residuals for a coxPH model?

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!

• See ?residuals.coxph for information on how to get these from R. – AdamO Jan 12 '18 at 17:56

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!

• 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. – Federico Tedeschi Mar 13 '19 at 14:57

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 $$t_k$$.