Cox Regression Residuals

In multiple linear regression, a residual is the difference between the observed and predicted value of the dependent variable based on observed values of the independent variables. Unfortunately, there is no simple counterpart in Cox regression. Instead, the following types of “residuals” are used with Cox regression for such purposes as identifying potential outliers:

• Cox-Snell residuals
• Martingale residuals
• Deviance residuals
• Schoenfeld residuals
• Scaled Schoenfeld residuals

Cox-Snell Residuals: The Cox-Snell residual at time t_k is

As remarked elsewhere, we generally use the Breslow estimate of H₀(t_k), namely

Martingale Residuals: The martingale residual at time t_k is

where

Thus martingale residuals can take a value between -∞ and 1. A large negative martingale residual indicates a high-risk subject who still had a long survival time.

The martingale residuals sum to zero and in large samples they are uncorrelated with one another and have an expected value of zero.

Deviance Residuals: The deviance residual at time t_k is

where sign(c) = 1 if c > 0, sign(c) = -1 if c < 0 and sign(0) = 0. High values of  indicate potential outliers. For a model with a good fit these residuals are symmetric around zero but they don’t necessarily sum to zero.

Schoenfeld Residuals: For any subject s who dies at time t_j and any covariate x_i, we define the Schoenfeld residual (aka partial residual) by

In large samples, these residuals are uncorrelated with one another and have an expected value of zero.

Scaled Schoenfeld Residuals: These are defined by

where d = = the total number of deaths and [c_il] is the r × r covariance matrix for the Cox regression coefficients.

You can plot these residuals against time to test whether the proportional hazards assumption holds. If the assumption holds, then these residuals will be randomly distributed about the x-axis.

Under construction