Methods and formulas for graphs that assess the proportional hazards assumption in Fit Cox Model with Fixed Predictors only

Andersen plots assess the appropriateness of the proportional hazards assumption for models that include stratification. The Andersen plot displays the estimated baseline cumulative hazard rate for the first stratum against the baseline cumulative hazard rates for the other strata. For details on the calculation of the baseline cumulative hazard rate, go to Methods and formulas for the survival function in Fit Cox Model with Fixed Predictors only.

Suppose that you have a Cox proportional hazards model with p predictors, . You can use an Arjas plot to determine whether to include a categorical predictor, , in the model. You can also check whether the proportional hazards assumption holds for predictor .

Suppose that has levels and that C(k) is the set of subjects in the grouping level k for predictor X, k = 1,…,K. The Arjas plot displays the total time on test of the estimated cumulative hazard rates up to time t, , against the cumulative number of observed events up to time t, . For further descriptions of Arjas plots, see Arjas (1988)¹ or Klein and Moeschberger (2003)².

The estimated cumulative hazard rate for subject j at time t has the following form:

where is a p-component vector of covariates for subject j and is the estimated baseline cumulative hazard rate. For details on the calculation of the baseline cumulative hazard rate, go to Methods and formulas for the survival function in Fit Cox Model with Fixed Predictors only. The calculations for and depend on whether the model has stratification.

Calculations without stratification

At each event time t for a level k of the categorical predictor X, the total time on test of the estimated cumulative hazard rates up to time t has the following form:

Also, the cumulative number of observed events up to time t has the following form:

where the equations use the following definitions:

Term	Description
	the response time for subject j
	an indicator for censoring where if the event happens and if the jth response time is right censored.
	an indicator for the event where if and 0 otherwise

Calculations with stratification

At each event time t for a level k of the categorical predictor X, the total time on test of the estimated cumulative hazard rates up to time t has the following form:

Also, the cumulative number of observed events up to time t has the following form:

where the equations use the following definitions:

Term	Description
	the estimated cumulative hazard function for stratum s
	the jth individual in stratum s
	the response time for subject j in stratum s
	an indicator for censoring where if the event happens and if the jth response time in stratum s is right censored.
	an indicator for the event where if and 0 otherwise