Get the Correct Hazard Ratio from SAS PROC PHREG Procedure

Cox proportional hazards model is a commonly used model in providing hazard ratio to compare survival times of two population groups. The exponentiated linear regression part of the model describes the effects of explanatory variables on hazard ratio. PROC PHREG is a SAS procedure that implements the Cox model and provides the hazard ratio estimate. The estimate is interpreted as the percent change in the hazards of the two population groups given an increase of one unit in a given explanatory variable and conditional on fixed values of all other explanatory variables. When the increase in the given explanatory variable is not equal to one unit, the hazard ratio estimate should be interpreted with caution. This paper illustrates this issue via an example with a multi-level treatment group variable as an explanatory variable. The paper provides three options (with sample codes) to obtain the correct hazard ratio when the increase in the explanatory variable is not equal to one unit: 1> Computing from the regression coefficient estimates of PROC PHREG output, 2> Recoding the values of the explanatory variable such that the increase is equal to one unit, 3> Using the CLASS statement to specify the explanatory variable in PROC TPHREG (experimental) procedure. This paper is not limited to any particular operating system. It is intended for users with some survival analysis and basic SAS data steps knowledge. INTRODUCTION In survival analysis, the hazard function is a useful way to describe the distribution of survival times. The hazard ratio is the ratio of the hazard functions between two population groups. If the hazard ratio estimate is less than one, this means that the hazard function for the first group is smaller than that for the second group. Cox proportional hazards model (Cox model) is a commonly used semi-parametric model in computing the hazard ratio estimates. It is a general proportional hazards model which does not require to specify the underlying survival function. The model assumes a parametric form (exponentiated linear regression form) for the effects of the explanatory variables and an unspecified non-parametric form for the underlying survival function. Thus, the Cox model can provide estimate on the hazard ratio without knowing the underlying survival function. With the parametric assumption that the relationship between the explanatory variables and the log hazard is linear, and further assumption that the effects of the explanatory variables are the same at all values of time (that is, the explanatory variables are not time dependent), a typical form of the Cox model is (Note: Equations in this section are from Survival Analysis Using the Proportional Hazards Model, Instructor-based Training Course Notes. 2005. Cary, NC: SAS Institute Inc.): } ... { 0 2 2 1 1 ) ( ) ( ik k i i X X X i e t h t h β β β + + + = Baseline hazard function Linear function of explanatory variables Taking the logarithm of both sides of the above equation, the model becomes ik k i i i X X X t h t h β β β + + + + = ... ) ( log ) ( log 2 2 1 1 0 The hazard ratio between group A and group B is: