Methods and formulas for the tests for the proportional hazards assumption for Fit Cox Model in a Counting Process Form

Select the method or formula of your choice.

The tests of the proportional hazards assumption in Minitab Statistical Software use the approach from Grambsch and Therneau (1994)1.

DF

The degrees of freedom are 1 for each estimated coefficient. For the overall test, the degrees of freedom are the sum of the degrees of freedom for the coefficients.

Chi-square

Global test for all predictors

The tests assesses the null hypothesis that the slopes for all of the terms in the model are zero for the relationship between the scaled Schoenfeld residuals and a time-scaled transformation of the event times against the alternative hypothesis that at least one slope is nonzero.

The test statistic for the simultaneous test of all predictors in the model has the following form:

Under the null hypothesis that the assumption of proportional hazards is true for all predictors, asymptotically follows a chi-square distribution with p degrees of freedom.

Test for individual predictors

The tests assess the null hypothesis that the slope is zero for the relationship between the scaled Schoenfeld residuals for a term and a time-scaled transformation of the event times against the alternative hypothesis that the slope is nonzero.

The test statistic with respect to an individual predictor has the following form:

Under the null hypothesis that the assumption of proportional hazards is true for the predictor, asymptotically follows a chi-square distribution with 1 degree of freedom.

Notation

TermDescription
the event times
dthe number of event times
pthe number of estimated parameters in the model
A time scale transformation of the event times. The following functions are in Minitab Statistical Software:
  • Identity
  • Natural log
  • Left-continuous Kaplan-Meier survival function
  • Rank function
the variance-covariance matrix of
Sthe d × p matrix of the Schoenfeld residuals
S*the d × p matrix of the scaled Schoenfeld residuals
a d-dimensional vector whose components are the centered values of where
the sample variance of the components of
the sample mean of the components of
the standard error of the jth component of the vector of estimated parameters from the Cox proportional hazard model
the correlation coefficient between the scaled Schoenfeld residuals for the jth predictor and the time-transformed functional values,

P-value

The p-value has the form:

where is a random variable that follows a chi-square distribution with degrees of freedom. is the test statistic.

1 Grambsch, P.M. and Therneau, T.M. (1994). Proportional hazards tests and diagnostics based on weighted residuals. Biometrika 81(3), 515-526.