Methods and formulas for the coefficients and regression equation for Fit Cox Model in a Counting Process Form

The semi-parametric Cox proportional hazards model uses the predictor values for an individual, , to predict the risk score, . The equation has the following general form:

where is the vector of estimated coefficients. The estimated coefficients can include values for higher order terms, such as the squares of continuous predictors. The estimated risk score is valid for the entire period of a study and does not depend on time. In the output, the equation has the following form where a separate equation appears for different levels of categorical factors:

Let the log-partial likelihood function for the Cox proportional hazards model be . The vector that maximizes the partial likelihood function, , gives the estimated coefficients for the model. To find , set the partial derivatives of the log-partial likelihood function equal to zero and solve the equations for . Minitab Statistical Software uses the Newton-Raphson iteration method to solve the equations. See Murray (1972)¹ for a description of the Newton-Raphson iterative method.

The vector of partial derivatives of the log-partial likelihood function depends on if the response variable includes tied event times. If the response variable includes ties, then the estimation uses either the Efron approximation or the Breslow approximation. If the response variable has no ties, all 3 methods provide the same estimates. The fewer ties are in the data, the closer the results of of the two approximation methods are. The more ties are in the data, the more the Efron approximation improves on the Breslow approximation.

The calculations use the following definition:

Term	Description
	the vector of covariate values that corresponds to the sample unit with the event time

Data without ties

The calculations for data without ties use the following definitions:

Term	Description
	the number of event times
	the risk set at time , which is the set of all sample units that have yet to fail prior to time
	a counting variable for the number of parameters in the model, where is the number of parameters in the model

The partial likelihood function for the Cox proportional hazards model with no ties has the following form:

The log-partial likelihood function has the following form:

The vector of partial derivatives with respect to the components of has the following form:

so that the partial derivative for a particular coefficient, , has the following form:

Data with ties

The calculations for data with ties use the following definitions:

Term	Description
	the number of events at time
	the set of all sample units that have the event at time
	the risk set at time , which is the set of all sample units who have yet to fail prior to time

Additionally, let

Efron approximation for data with ties

The approximated partial likelihood function has the following form:

The log-partial likelihood function has the following form:

so that the partial derivative for a particular coefficient, , has the following form:

Breslow approximation for data with ties

The approximated partial likelihood function has the following form:

The log-partial likelihood function has the following form:

so that the partial derivative for a particular coefficient, , has the following form:

The standard errors of the coefficients are the positive square roots of the diagonal elements of the variance-covariance matrix of the parameter estimates. The variance-covariance matrix has the following form:

where the observed information matrix, depends on if the response variable includes tied event times. If the response variable includes ties, then the estimation uses either the Efron approximation or the Breslow approximation. If the response variable has no ties, all 3 methods provide the same estimates. The fewer ties are in the data, the closer the results of of the two approximation methods are. The more ties are in the data, the more the Efron approximation improves on the Breslow approximation.

Data without ties

The (k, l) element of the observed Fisher's information matrix has the following form:

where the (k, l) element of the Hessian matrix for the partial log-likelihood function has the following form:

Efron approximation for data with ties

The (k, l) element of the observed Fisher's information matrix has the following form:

where the (k, l) element of the Hessian matrix for the partial log-likelihood function has the following form:

where

and

Breslow approximation for data with ties

The (k, l) element of the observed Fisher's information matrix has the following form:

where the (k, l) element of the Hessian matrix for the partial log-likelihood function has the following form:

where

and

The test statistic has the following form:

where is the estimated standard error of the coefficient . The value of is the positive square root of the k^th diagonal element of .

An approximate 100(1 – α) confidence interval for the coefficient has the following form:

where is the upper α percentile point of the standard normal distribution.

For a model that includes a categorical variable with s levels as a stratification variable, the regression coefficients are constant across strata. The estimation of the regression coefficients in the stratified model has the same process as for the proportional hazards model without stratification. For the stratified model, the log-partial likelihood function has the following form:

where is the log partial likelihood within stratum j. Sum the derivatives across each stratum to obtain the partial likelihood equations. The derivatives across each stratum are the same as the derivatives for the proportional hazards model without stratification. The Breslow and Efron methods apply accordingly.