Methods and formulas for the model summary for Fit Cox Model in a Counting Process Form

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Log-likelihood
R-sq
AIC, AICc, and BIC

For the summary statistics, let be the Breslow partial likelhood function or the Efron partial likelihood function evaluated at β. The function depends on the specification for the analysis. The calculations for the model without terms use . The calculations for the model with terms use where is the (partial) maximum likelihood estimate of .

Log-likelihood

Without terms	With terms

R-sq

The R² statistic exists only for the model with terms. Minitab Statistical Software calculates the standardized version of the Cox & Snell Generalized R² proposed in Nagelkerke (1991)¹.

where

is the sample size and

has the following form:

In the expression for R², the numerator is the Cox & Snell generalized R² statistic and is the maximum possible value for that statistic.

AIC, AICc, and BIC

Without terms

Without terms, the equations for the different information criterion reduce to the following equality:

With terms

For censored survival models, Minitab Statistical Software calculates the information criteria with the number of observed events as the sample size. See Volinksy & Raftery (2000)² for information on this adjustment. With terms, the information criteria have the following equations:

where is the number of parameters in the model and is the number of observed events.

¹ Nagelkerke, N.J.D. (1991). A note on the general definition of the coefficient of determination. Biometrika, 78(3), 691-692. https://doi.org/10.1093/biomet/78.3.691

² Volinsky, C.T. & Raftery, A.E. (2000). Bayesian information criterion for censored survival models. Biometrics 56(1), 256-262. https://doi.org/10.1111/j.0006-341X.2000.00256.x