Covariate & other Standard PH model Lognormal frailty model Gamma frailty model Retransformation method Statistics coefficient Hazard ratio coefficient Hazard ratio coefficient Hazard ratio coefficient Hazard ratio Veteran status -0.241 0.786 -0.241 0.786 -0.392 0.676 -0.241 0.786 Age_70 0.064*** 1.066 0.064*** 1.066 0.101*** 1.106 0.064*** 1.066 Vet × Age_70 0.046** 1.047 0.046* 1.047 0.082* 1.085 0.046** 1.047 Female_cnd -0.556*** 0.573 -0.556*** 0.573 -0.911*** 0.402 -0.556*** 0.573 Educ_cnd -0.016 0.985 -0.016 0.985 -0.027 0.973 -0.016 0.985 Married_cnd -0.168 0.846 -0.168 0.846 -0.286 0.751 -0.168 0.846 Intercept -7.092***   -7.092***   -11.493***   -7.092***   Random effect     0.000   0.642***   0.480***   Mean frailty score     1.000   1.000   1.271   Shape coefficient 1.265***   1.265***   1.915***   1.265***   -2 log likelihood 4246.70   4246.70   4250.9   4246.70   Note: The parameter “shape” is tested by * 0.05<p<0.10; **0.01<p<0.05; ***p<0.01.

Table 2: Results of four hazard rate models on the mortality of older Americans between 1993 and 1997: Fixed-effects and frailty models (n=2,000).