Methods and formulas for the regression diagnostics in Fit Cox Model with Fixed Predictors only

The Cox-Snell residual for subject i with response time has the following form:

where is the Breslow's estimator of the baseline cumulative hazard rate:

Recall that is a step function with jumps at the observed event times. The size of the jump at time has the following form:

The calculation of the Cox-Snell residual depends on the tie handling method. For the Breslow approximation, the Cox-Snell residual has the following form:

For the Efron approximation, the Cox-Snell residual has the following form:

where has the following form:

for

where is the first event time at which subject i is in the risk set and is the last event time at which subject i is in the risk set.

The Martingale residual for subject i has the following form:

where is the Cox-Snell residual and depends on the tie handling method. Additionally, is an indicator for if subject i is censored, such that if subject i experienced the event and otherwise.

The deviance residual for subject i is a transformation of the Martingale residual:

where is the Martingale residual for subject i.

The Schoenfeld residual vector is a p-component vector. For subject i with event time t the Schoenfeld residual vector has the following form:

where is the weighted average of the covariates over the risk set at time t. The weighted average has the following form:

where is an indicator variable that has the value 1 if subject i is at risk at time t and 0 otherwise, which is equivalent to if and otherwise.

If the subject does not experience the event at time t, the vector contains missing values.

The calculation of the Schoenfeld residual vector depends on the tie handling method. For the Breslow approximation, the Schoenfeld residual vector has the following form:

where

For the Efron approximation, the Schoenfeld residual vector has the following form:

where

the function has the same definition as for the Cox-Snell residual:

and

for

The scaled Schoenfeld residual vector has the following form:

where is the observed number of uncensored survival times and is the Schoenfeld residual vector.

The calculation of the score residual vector depends on the approximation method for ties in the event times. For the Breslow approximation, the score residual vector has the following form:

where

For the Efron approximation, the score residual vector has the following form:

where , and have the same definitions as for the Schoenfeld residual vector:

and

for

Other names for DFBeta include the weighted score residual, the scaled score residual, and the standardized score residual. DFBeta represents the difference between the coefficient vectors when subject i is not in the estimation of the coefficients:

Minitab calculates an approximation of DFBeta from Cain and Lange (1984)¹ with the following form:

where is the score residual vector. For details on the calculation of , go to Methods and formulas for the coefficients and regression equation for Fit Cox Model with Fixed Predictors only.