PROC PHREG does not report an R² statistic. However, Allison (1995, pp. 247-249) discusses a "generalized" R² statistic that is based on the likelihood-ratio statistic (LRT) for testing the global null hypothesis. The LRT statistic is given in the Global Tests table in the PHREG displayed results and is the difference in the -2 log likelihood (-2logL) for the null model (without covariates) and the fitted model (with covariates), both of which are found separately in the Model Fit Statistics table. The formula Allison gives is:

R² = 1 - e^-(LRT/n)

where LRT = -2logL(0) - [-2logL(p)], n is the sample size, logL(0) is the log-likelihood for a null model with no covariates, and logL(p) is the log-likelihood for the fitted model with p covariates. Allison uses the Total from the Censored Summary table for n.

Note that this generalized R² does not have a "proportion of variation explained by the model" interpretation. Allison states, "R-square does not tell you anything about how appropriate the model is for the data" and "It's just a statistic between 0 and 1 that is larger when the covariates are more strongly associated with the dependent variable."

Hosmer and Lemeshow (1999, pp. 228–230) briefly discuss R² measures for the Cox model stating, "As shown in a detailed study by Schemper and Stare (1996), there is not a single, simple, easy to calculate, useful, easy to interpret measure for a proportional hazards regression model." They also state, "A perfectly adequate model may have what, at face value, seems like a terribly low R² due to a high percent of censored data" and they have difficulty in recommending "one measure over another." In any event, Hosmer and Lemeshow present the same R² statistic that is presented by Allison as "perhaps the easiest and best one to use."

Note that the generalized R² measure is discussed in the PROC LOGISTIC documentation as the "generalized coefficient of determination" due to Cox and Snell (1989). See the "Generalized Coefficient of Determination" section of the PROC LOGISTIC documentation for additional information and formulas. The Nagelkerke (1991) reference discusses properties and interpretation.

Allison, Paul D. 1995. Survival Analysis Using the SAS System: A Practical Guide. Cary, NC: SAS Institute Inc.

Cox, D. R., and E. J. Snell, 1989. The Analysis of Binary Data, Second Edition. London: Chapman and Hall.

Heinzl, H. 2000. "Using SAS to Calculate the Kent and O'Quigley Measure of Dependence for Cox Proportional Hazards Regression Model." Computer Methods and Programs in Biomedicine, 63(1), 71–76.

Hosmer, D. W., Jr., and S. Lemeshow. 1999. Applied Survival Analysis: Regression Modeling of Time to Event Data. New York: John Wiley and Sons Inc.

Kent, J., and J. O'Quigley. 1988. "Measures of Dependency for Censored Survival Data." Biometrika 75:525–534.

Nagelkerke, N. J. D. 1991. "A Note on a General Definition of the Coefficient of Determination." Biometrika 78:691–692.

Schemper, M., and J. Stare. 1996. "Explained Variation in Survival Analysis." Statistics in Medicine 15:145–153.

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