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Table 12.1 Results of a Proportional Hazards Regression Analysis of Data in Table 11.4 Regression Standard Covariate Coefficient Error p Value exp(coefficient) x (age) 1.01 0.46 0013 2.75 x (cellularity) 0.35 0.44 0.212 1.42 The 95% confidence intervals for b (age) and b (cellularity) are 1.01< 1.96 (0.46) or (0.11, 1.91) and 0.35 < 1.96 (0.44) or (90.51, 1.21), respectively. Consequently, the 95% confidence intervals for the relative risks are (e, e) or (1.12, 6.75) and (e\, e) or (0.60, 3.35), respectively. The small number of patients (30) may have contributed to the large standard errors of b and b and consequently, the wide confidence intervals. The lower bound of the confidence interval for age is only slightly above 1. This suggests that the importance of age should be interpreted carefully. In general, if the number of subjects is small and the standard errors of the estimates are large, the estimates may be unreliable. When the two covariates are considered simultaneously, the risk for a patient with x : 1 and x : 1 relative to patients with x : 0 and x : 0 can be estimated. The relative risk is estimated as exp(1.01 ; 0.35) : 3.90 for a patient who is over 50 years of age and whose cellularity is 100%, compared to patients who are younger than 50 and whose cellularity is less than 100%. Using the same data set ‘‘C:!AML.DAT’’ defined in Example 11.3, the following SAS code can be used to obtain the results in Table 12.1. data w1; infile ‘c:!aml.dat’ missover; input t cens x1 x2; run; proc phreg; model t*cens(0) : x1 x2 / rl; run; If BMDP 2L is used, the following code is applicable. /input file : ‘c:!aml.dat’ . variables : 4. format : free. /print cova. /variable names : t, cens, x1, x2. /form time : t. status : cens. response : 1. /regress covariates : x1, x2. 306 If SPSS is used, the following code suffices. data list file : ‘c:!aml.dat’ free / t cens x1 x2. coxreg t with x1 x2 /status : cens event (1) /print : all. Example 12.2 In a study (Buzdar et al., 1978) to evaluate a combination of 5-flourouracil, adramycin, cyclophosphamide, and BCG (FAC-BCG) as adjuvant treatment in stage II and III breast cancer patients with positive axillary nodes, 131 patients receiving FAC-BCG after surgery and radiation therapy were compared with 151 patients receiving surgery and radiation therapy only (control group). Cox’s regression model was used to identify prognostic factors and to evaluate the comparability of the two treatment groups. The model was fitted to the data from 151 patients to determine the variables related to length of remission. The possible prognostic variables considered were age (years), menopausal status (1, premenopausal; 0, other), size of primary tumor (2, :3 cm; 4, 3—5 cm; 7, 95cm), state of disease (2, stage II; 3, stage III), location of surgery (1, M. D. Anderson Hospital; 0, other), number of nodes involved (2, :4; 7, 4—10; 12, 910), and race (1, Caucasian; 2, other). The covariates were selected by the forward selection method outlined in Section 11.9. Three variables — number of nodes involved, state of disease, and menopausal status — were selected for use in the model, all related significantly (p : 0.1) to disease-free time. The regression equation including these variables only is log h G (t) h (t) : 0.111(number of nodes 9 6.16) ; 0.8122(stage 9 2.39) ; 0.872(menopausal 9 0.26) Table 12.2 gives the details of the fit. Relative risk was taken as h G (t)/h (t), the ratio of the risk of death per unit of time for a patient with a given set of prognostic variables to the risk for a patient whose prognostic variables were average in value. The relative risk for each variable was calculated by considering favorable or unfavorable values of that variable, assuming that other variables were at their average value. Note that the risk of relapse per unit time for a patient with 12 positive nodes is 3.04 (ratio or risk) times that for a patient with only two positive nodes. The risk of relapse per unit time for a stage III patient was 2.25 times that of a stage II patient. The Cox’s regression model was also fitted to the combined group of FAC-BCG and control patients, including type of treatment (0, control; 1, FAC-BCG), menopausal status, size of primary tumor, and number of involved nodes as potential prognostic variables. The regression equation with three 307 Table 12.2 Patient Characteristics Related to Disease-Free Time in Cox’s Regression Model Fit to Control Patients Maximum Relative Risk? Prognostic Regression Significance Log Ratio of Variable Coefficient Level (p) Likelihood Favorable Unfavorable Risks Number of nodes 0.1110 :0.01 9257.407 0.63 1.91 3.04 Stage 0.8122 0.016 9254.533 0.73 1.64 2.25 Menopausal status 0.8720 :0.1 9250.576 0.80 1.91 2.39 Source: Buzdar et al. (1978). Reprinted by permission of the editor. ? Favorable variables: number of nodes : 2, stage II, postmenopausal. Unfavorable variables: number of nodes :12, stage III, premenopausal. Table 12.3 Patient Characteristics Related to Survival, Treatment Included Maximum Relative Risks? Prognostic Regression Significance Log Ratio of Variable Coefficient Level (p) Likelihood Favorable Unfavorable Risks Treatment 91.8792 :0.01 9201.200 0.37 2.42 6.55 Menopausal 0.9644 0.01 9197.719 0.73 1.91 2.62 status Size of 0.1611 0.05 9195.865 0.72 1.61 2.24 primary tumor Source: Buzdar et al. (1978). Reprinted by permission of the editor. ? Favorable variables: treatment— FAC-BCG, postmenopausal, size of primary tumor 2 cm. Unfavorable variables: no adjuvant treatment, premenopausal, size of primary tumor 7 cm. significant (p : 0.05) variables obtained was as follows: log h G (t) h (t) :91.8792(treatment9 0.47) ; 0.9644(menopausal status 9 0.33) ; 0.1611(size of primary tumor 9 4.04) Table 12.3 gives the details of the fit. The most important variable in predicting survival time was the type of treatment (FAC-BCG favorable); other signifi- cantly important variables were menopausal status and size of primary tumor. The risk of death per unit of time for a patient receiving no adjuvant treatment (control group) was 6.55 times that for a patient receiving the treatment, showing that FAC-BCG can prolong life considerably. 308 Example 12.3 Suppose that demographic, personal, clinical, and labora- tory data are collected from an interview and physical examination of 200 participants in a study of cardiovascular disease (CVD). These participants, aged 50—79 years and free of CVD at the time of the baseline examination, are then followed for 10 years. During the follow-up period, 96 of the 200 participants develop or die of CVD. We use this set of simulated data to illustrate further the use of the proportional hazards model in identifying important risk factors. Table 12.4 gives a subset of the simulated data of 68 participants. The event time T of interest is CVD-free time, which is defined as the time in years from baseline examination to the first time that a participant was diagnosed as having CVD or confirmed as a CVD death. CVD includes coronary heart disease (CHD) and stroke. The covariates of interest are age (AGE), gender (SEX : 1 if male and :0 if female); smoking status (SMOKE : 1 if current smoker, and 0 otherwise); body mass index (BMI : weight in kilograms divided by height in meter squared); systolic blood pressure (SBP); logarithm of ratio of urinary albumin and creatinine (LACR); logarithm of triglycerides (LTG); hypertension status (HTN : 1if SBP . 140 mmHg or DBP .90 mmHg or under treatments of hypertension, and :0 otherwise); and diabetes status (DM: 1 if fasting glucose. 126 mg/dL or under the treatments of diabetes, and :0 otherwise). For the CVD outcome of interest, we let DG denote the type of CVD. DG : 0 if the participant is free of CVD at the end of the study or confirmed as a non-CVD death (thus the CVD-free time is censored), :1 if the participant had a stroke, :2 if the participant had a CHD, and :3 if the participant had other CVDs. It is of interest to compare the risk of CVD among the three age groups: 50—59, 60—69, and 70—79. We create two dummy variables: AGEA : 1 if aged 50—69, :0 otherwise; and AGEB: 1 if aged 60—69, and :0 otherwise. Thus for a 70 to 79-year-old, AGEA : 0 and AGEB : 0. We also create a variable to denote the censoring status: CENS : 0 if t is censored, and : 1 if uncensored. To illustrate the different methods to handle ties, we fit the Cox proportional hazards model with the following six covariates: AGEA, AGEB, SEX, SMOKE, BMI, and LACR. The approximated partial likelihood function defined in (12.1.15)—(12.1.17) as well as the exact partial likelihood function (Delong et al., 1994) are applied. As noted in Sections 11.3 and 11.4, the exponential and Weibull regression models are also proportional hazard models. Therefore, for comparisons we also fit an exponential and a Weibull regression model with the same covariates to the data. The estimated re- gression coefficients obtained from the proportional hazards model with approximated discrete, Breslow, Efron, and exact partial likelihood functions as well as those from the exponential and Weibull regression models are given in Table 12.5. All of the estimates based on the Cox model and an approxi- mated partial likelihood function are very closed to those based on the exact partial likelihood. Those based on Efron’s approximation are almost identical to those (different only at the fourth decimal place) based on the exact partial 309 Table 12.4 A Subset of the Simulated Data for a Cardiovascular Disease Study in Example 12.3? ID T CENS DG AGEA AGEB SEX SMOKE BMI SBP LACR LTG AGE HTN DM 1 7.4 0 0 0 0 0 0 31.78 141 4.23 3.94 77.8 1 0 2 7.9 0 0 0 0 0 0 25.02 124 4.31 4.66 76.9 0 1 3 6.4 0 0 0 0 0 1 26.05 111 4.38 4.27 76.3 0 0 4 7.1 0 0 0 0 0 1 26.92 140 1.11 4.51 72.2 1 0 5 6.0 0 0 0 0 0 1 34.30 146 1.19 4.82 76.0 1 0 6 6.5 0 0 0 0 0 1 31.76 142 1.20 4.88 74.5 1 0 7 8.3 0 0 0 0 1 0 25.01 154 3.53 4.10 70.7 1 1 8 7.9 0 0 0 0 1 0 28.21 136 3.73 4.12 75.2 1 0 9 7.6 0 0 0 1 0 0 28.13 127 2.92 4.24 64.9 0 0 10 8.4 0 0 0 1 0 0 25.68 118 2.47 4.41 60.2 0 0 11 7.4 0 0 0 1 0 0 34.34 118 2.37 4.46 64.4 0 1 12 7.7 0 0 0 1 0 0 28.92 127 3.58 4.55 68.8 1 1 13 6.9 0 0 0 1 0 1 24.68 100 2.11 4.33 64.4 0 0 14 7.2 0 0 0 1 0 1 21.93 121 3.39 4.64 60.8 0 1 15 6.3 0 0 0 1 0 1 29.47 98 1.96 4.69 64.4 0 0 16 7.4 0 0 0 1 0 1 28.65 150 2.59 4.95 61.6 1 0 17 4.5 0 0 0 1 1 0 32.28 128 2.99 4.73 65.3 0 0 18 7.0 0 0 0 1 1 0 29.21 117 2.17 4.91 65.7 0 1 19 2.8 0 0 0 1 1 0 28.82 136 4.04 4.92 65.4 0 1 20 7.2 0 0 0 1 1 0 30.58 121 2.84 4.94 64.5 0 1 21 7.4 0 0 1 0 0 0 27.83 95 1.85 4.44 52.0 0 0 22 5.2 0 0 1 0 0 0 26.61 128 2.87 4.51 50.7 0 0 23 7.7 0 0 1 0 0 0 30.32 96 2.41 4.60 52.5 0 1 24 7.8 0 0 1 0 0 0 30.41 130 1.45 4.73 55.9 0 0 25 7.6 0 0 1 0 0 1 29.98 140 1.88 4.51 53.4 1 0 26 7.9 0 0 1 0 0 1 26.00 118 2.34 4.53 51.0 0 0 27 7.3 0 0 1 0 0 1 29.05 110 1.44 4.67 50.6 0 0 310 28 8.2 0 0 1 0 0 1 27.21 131 2.50 4.68 57.7 0 0 29 3.8 0 0 1 0 1 0 36.97 141 4.60 4.25 58.7 1 0 30 6.9 0 0 1 0 1 0 29.44 115 2.89 4.26 53.6 0 1 31 6.1 0 0 1 0 1 0 33.85 154 3.48 4.48 51.2 1 0 32 7.2 0 0 1 0 1 0 32.13 122 2.92 4.48 55.2 0 0 33 8.4 0 0 1 0 1 1 27.52 135 2.39 4.42 53.7 0 0 34 5.0 0 0 1 0 1 1 30.64 114 1.39 4.45 54.9 1 0 35 6.5 0 0 1 0 1 1 29.94 120 2.96 4.49 50.7 0 0 36 6.4 0 0 1 0 1 1 29.89 115 1.68 4.52 51.3 0 0 37 2.6 1 1 0 0 0 0 30.88 189 5.38 4.72 73.9 1 1 38 2.7 1 1 0 0 0 1 25.05 200 3.37 4.86 77.2 1 1 39 2.7 1 1 0 0 1 0 26.80 130 2.31 5.10 73.5 0 0 40 3.3 1 1 0 0 1 1 21.67 111 3.53 4.18 71.1 0 0 41 2.9 1 1 0 1 0 0 36.83 114 2.64 4.52 68.2 0 0 42 0.2 1 1 0 1 0 1 21.49 125 4.61 4.69 67.3 0 0 43 2.1 1 1 0 1 1 0 31.05 131 1.38 4.48 69.1 0 0 44 6.8 1 1 0 1 1 1 26.78 134 4.36 4.90 61.0 1 0 45 5.7 1 1 1 0 0 0 35.78 132 9.93 5.11 52.5 0 1 46 1.1 1 1 1 0 0 1 28.44 134 3.54 4.32 55.7 0 0 47 6.6 1 1 1 0 1 0 24.38 124 4.16 4.00 51.8 0 1 48 1.3 1 1 1 0 1 1 34.13 126 5.87 3.95 53.1 0 1 49 4.6 1 2 0 0 0 0 43.23 128 5.08 5.25 72.2 0 1 50 6.3 1 2 0 0 0 1 38.67 126 5.16 4.50 76.8 1 1 51 2.0 1 2 0 0 1 0 34.49 130 2.69 3.95 76.7 1 1 52 4.2 1 2 0 0 1 1 20.78 127 4.40 4.54 73.1 0 0 53 3.6 1 2 0 1 0 0 28.40 118 5.43 4.66 69.3 1 1 54 3.2 1 2 0 1 0 1 28.73 154 1.94 5.24 68.9 1 1 55 4.5 1 2 0 1 1 0 44.25 97 2.01 4.40 68.6 0 1 56 4.5 1 2 0 1 1 1 32.46 141 0.74 4.39 63.5 1 0 57 6.1 1 2 1 0 0 0 39.72 118 2.39 3.93 52.6 0 1 (Continued overleaf ) 311 Table 12.4 Continued ID T CENS DG AGEA AGEB SEX SMOKE BMI SBP LACR LTG AGE HTN DM 58 3.0 1 2 1 0 0 1 27.90 117 7.45 5.61 56.0 0 1 59 2.1 1 2 1 0 1 0 27.77 119 7.03 4.71 54.3 0 1 60 1.3 1 2 1 0 1 1 31.03 151 3.94 4.43 59.2 1 1 61 4.9 1 3 0 0 0 0 25.22 129 6.69 3.90 75.4 1 0 62 2.5 1 3 0 0 0 1 45.29 130 2.46 4.40 75.7 0 1 63 3.8 1 3 0 0 1 0 25.03 188 6.25 5.63 71.7 1 1 64 5.0 1 3 0 1 1 0 46.76 96 3.93 4.12 65.6 1 0 65 1.5 1 3 0 1 1 1 28.53 126 3.09 4.65 68.6 0 1 66 4.1 1 3 1 0 0 0 23.63 144 8.24 4.82 59.4 1 1 67 0.5 1 3 1 0 1 0 31.39 134 6.96 4.11 54.2 1 0 68 2.7 1 3 1 0 1 1 30.29 115 4.70 4.98 59.1 1 1 ? ID, participant id number; T, CVD event time (CVD-free time);CENS: 0 if censored, and :1 if uncensored; DG : 0ifnon-CVDat the end of the study or non-CVD death, :1ifstroke,:2 if coronary heart disease (CHD),and:3 if the other CVDs; AGEA : 1if aged 50—59 and :0otherwise;AGEB: 1ifaged60—69 and :0otherwise;SEX: 1ifmaleand:0 if female; SMOKE: 1ifcurrent smoker and 0 otherwise; BMI, body mass index; SBP, systolic blood pressure; LACR, logarithm of the ratio of urinary albumin and creatinine; LTG, logarithm of triglycerides; HTN: 1ifSBP.140 mmHg or DBP (diastolic blood pressure).90 mmHg and :0 otherwise; DM : 1 if fasting glucose.126 mg/dL and :0otherwise. 312 Table 12.5 Results from Fitting a Cox Proportional Hazards Model Based on Different Methods for Ties on the CVD Data Regression Coefficient Variable Breslow Discrete Efron Exact Exponential Weibull AGEA 91.3478 91.3662 91.3558 91.3560 91.2550 91.0436 AGEB 90.7709 90.7828 90.7753 90.7755 90.7107 90.5966 SEX 0.7134 0.7233 0.7187 0.7189 0.6862 0.5659 SMOKE 0.3762 0.3810 0.3776 0.3776 0.3440 0.2855 BMI 0.0253 0.0256 0.0255 0.0255 0.0233 0.0194 LACR 0.1735 0.1759 0.1739 0.1740 0.1658 0.1357 likelihood function. The estimated regression coefficients based on the two parametric models, particularly the exponential regression model, are also close to those based on the Cox hazards model. From the signs of the coefficients, we see that men, current smokers, and persons with high BMI and albumin—creatinine ratios have a higher hazard (risk) of CVD and shorter CVD-free time. The coefficients of the two age variables are both negative, indicating that persons in the younger age groups have a lower hazard (risk) of CVD. Suppose that ‘‘C:!EX12d2d1.DAT’’ contains eight successive columns, for T, CENS, AGEA, AGEB, SEX, SMOKE, BMI, and LACR, and that the numbers in each row are space-separated. The following code for the SAS PHREG and LIFEREG procedures can be used to obtain the results in Table 12.5. data w1; infile ‘c:!ex12d2d1.dat’ missover; input t cens agea ageb sex smoke bmi lacr; run; proc phreg; model t*cens(0) : agea ageb sex smoke bmi lacr / ties : breslow; run; proc phreg; model t*cens(0) : agea ageb sex smoke bmi lacr / ties : discrete; run; proc phreg; model t*cens(0) : agea ageb sex smoke bmi lacr / ties : efron; run; proc phreg; model t*cens(0) : agea ageb sex smoke bmi lacr / ties : exact; run; proc lifereg; Model a: model t*cens(0) : agea ageb sex smoke bmi lacr / d : exponential; Model b: model t*cens(0) : agea ageb sex smoke bmi lacr / d : weibull; run; 313 12.2 IDENTIFICATION OF SIGNIFICANT COVARIATES As noted earlier, one principal interest is to identify significant prognostic factors or covariates. This involves hypothesis testing and covariate selection procedures, similar to those discussed in Chapter 11 for parametric methods. The differences are that the Cox proportional hazard model has a partial likelihood function in which the only parameters are the coefficients associated with the covariates. However, statistical inference based on the partial likelihood function has asymptotic properties similar to those based on the usual likelihood. Therefore, the estimation procedure (discussed in Section 12.1) is similar to those in Section 7.1, and the hypothesis-testing procedures are similar to those in Sections 9.1 and 11.2. For example, the Wald statistic in (9.1.4) can be used to test if any one of the covariates has no effect on the hazard, that is, to test H : b G : 0. By replacing the log-likelihood function with the log partial likelihood function, the log-likelihood ratio statistic, the Wald statistic, and the score statistic in (9.1.10), (9.1.11), and (9.1.12) can be used to test the null hypothesis that all the coefficients are equal to zero, that is, to test H : b : 0, b : 0, , b N : 0 or H : b : 0 in (9.1.9). Similarly the forward, backward, and stepwise selection procedures discussed in Section 11.9.1 are applicable to the Cox proportional hazard model. The following example, using the SAS PHREG procedure, illustrates these procedures. Example 12.4 We use the entire CVD data set in Example 12.3 to demonstrate how to identify the most important risk factors among all the covariates. Suppose that the effects of age, gender, and current smoking status on CVD risk are of fundamental interest and we wish to include these variables in the model. In epidemiology this is often referred to as adjusting for these variables. Thus, AGEA, AGEB, SEX, and SMOKE are forced into the model and we are to select the most important variables from the remaining covariates (BMI, SBP, LACR, LTG, HTN, and DM), adjusting for age, gender, and current smoking status. The SAS procedure PHREG is used with Breslow’s approximation for ties (default procedure) and three variable selection methods (forward, backward, and stepwise). Two covariates, BMI and LACR, are selected at the 0.05 significance level by all three selection methods. The final model, in the form of (12.1.5), including only the four covariates that we purposefully included and the two most significant ones identified by the selection method, is 314 Table 12.6 Asymptotic Partial Likelihood Inference on the CVD Data from the Final Cox Proportional Hazards Model? 95% Confidence Interval Regression Standard Wald Relative Variable Coefficient Error Statistic p Hazards Lower Upper Final Model for the Cohort CV D Data AGEA 91.3558 0.2712 24.9910 0.0001 0.258 0.151 0.439 AGEB 90.7753 0.2618 8.7709 0.0031 0.461 0.276 0.769 SEX 0.7187 0.2193 10.7457 0.0010 2.052 1.335 3.153 SMOKE 0.3776 0.2208 2.9235 0.0873 1.459 0.946 2.249 BMI 0.0255 0.0124 4.2113 0.0402 1.026 1.001 1.051 LACR 0.1739 0.0446 15.2112 0.0001 1.190 1.090 1.299 b 9 b 90.580 4.9443 0.0262 0.560 b ; b 1.096 11.5409 0.0007 2.993 b 9 b 0.341 1.3001 0.2542 1.407 Hypothesis Testing Results (H : all b G : 0) Log-partial-likelihood ratio statistic 42.1130 0.0001 Score statistic 43.1750 0.0001 Wald statistic 41.3830 0.0001 ? The covariates, except AGEA, AGEB, SEX, and SMOKE, in the final model are selected among BMI, SBP, LACR, LTG, HTN, and DM. log h(t G ) h (t G ) : b AGEA G ; b AGEB G ; b SEX G ; b SMOKE G ; b BMI G ; b LACR G :91.3558AGEA G 9 07753AGEB G ; 0.7187SEX G ; 0.3776SMOKE G ; 0.0255BMI G ; 0.1739LACR G (12.2.1) The regression coefficients, their standard errors, the Wald test statistics, p values, and relative hazards (relative risks as they are termed by many epidemiologists) are given in Table 12.6. The estimated regression coefficients b G , i : 1, 2, . . . , 6, are solutions of (12.1.9) using the Newton—Raphson iterated procedure (Section 7.1). The estimated variances of b G , i : 1, 2, , 6, are the respective diagonal elements of the estimated covariance matrix defined in (12.1.13). The square roots of these estimated variances are the standard errors in the table. The Wald statistics are for testing the null hypothesis that the covariate is not related to the risk of CVD or H : b G : 0, i : 1, , 6, respect- ively. For example, the Wald statistic equals 10.7457 for gender with a p value 315 [...]... 0.0001 0.0008 0. 471 4 0.0 478 0. 070 6 1. 07 2.10 0.92 1.48 1.15 1.05 1.36 0 .72 1 0.99 1.1 3.23 1.16 2.19 1.35 (b) AGE SEX LACR LTG LACR* log(t;1) 0. 071 0 .74 1 90.0 87 0.395 0.143 0.014 0.220 0.120 0.199 0. 079 26.635 11.3 27 0.519 3.9 17 3.269 (c) AGE SEX LACR LTG AGE*log(t ;1) 0.038 0 .76 4 0.111 0.4 17 0.033 0.220 0.046 0.1 97 1.330 12.020 5.888 4.469 0.2488 0.0005 0.0152 0.0345 1.04 2.15 1.12 1.52 0. 97 1.39 1.02... statistical methods for the identification and 3 37 Table 12.9 Goodness-of-Fit Tests Based on Asymptotic Likelihood Inference in Fitting the CVD Data? Model Generalized gamma Log-logistic Lognormal Weibull Exponential Exponential LL LLR p BIC AIC 9198.842 9203.322 9206.0 17 9199.494 9203.061 9203.061 — — 14.3505 1.3046 8.4385@ 7. 1339A — — 0.0002 0.2534 0.01 47 0.0 076 92 17. 113 9218.983 9221. 678 ... Chi-Square Statistic p Relative Hazards Lower Upper 0.06 97 0 .75 28 0.1111 0.39 87 0.0136 0.2192 0.0459 0.1 976 26.1393 11 .78 93 5.8602 4. 072 2 0.0001 0.0006 0.0155 0.0436 1. 07 2.12 1.12 1.49 1.04 1.38 1.02 1.01 1.10 3.26 1.22 2.20 H : All coefficients equal zero Log-partial-likelihood ratio statistic Score statistic Wald statistic 44.002 44. 278 42.5 27 0.0001 0.0001 0.0001 Variable AGE SEX LACR LTG ? The... of 0.0010 and b : 0 .71 87 It indicates that after adjusting for all the variables in the model (12.2.1), gender is a significant predictor for the development of CVD, with men having a higher risk than women The relative hazard (or risk) is exp(b ), and for the covariate gender, it is exp(0 .71 87) : 2.052, which implies that men aged 50 79 years have about twice the risk of developing... longer than a given time for individuals with a given set of values for covariates The following is an example 325 Figure 12.3 Breslow estimate of survivorship functions at the averages of BMI and LACR from SEX*SMOKER subgroups in aged 70 79 participants from the fitted Cox proportional hazards model on the CVD data Example 12 .7 For the same model as... developing CVD in 10 years The 95% confidence interval for the relative risk is (1.335, 3.153), which is calculated according to (7. 1.8) For a continuous variable, exp(b ) represents G the increase in risk corresponding to a 1-unit increase in the variable For example, for BMI, exp (0.0255) : 1.026; that is, for every unit increase in BMI, the risk for CVD increases 2.6% To compare hazards among different... the data set ‘‘C:!EX12d2d1.DAT’’ defined in Example 12.3, the SAS code used for this example is the following data w1; infile ‘c:!ex12d2d1.dat’ missover; input t cens agea ageb sex smoke bmi lacr; run; proc phreg data : w1 noprint; model t*cens(0) : agea ageb sex smoke bmi lacr / ties : efron; baseline out : base1 survival : survival l: lowb u : uppb / method : p l; run; title ’K-P estimate of the survival. .. LACR for female nonsmokers Similarly, the specific covariate vectors for female smokers, male nonsmokers, and male smokers are, respectively, (0, 0, 0, 1, 31.19, 2. 67) , (0, 0, 1, 0, 28.19, 3.43), and (0, 0, 1, 1, 25 .76 , 3. 47) The estimated survival curves are shown in Figure 12.3 Similarly, Figures 12.4 and 12.5 give the estimated survival curves of the four groups in persons aged 60—69 years and 50—59... the CVD Data from the Cox Proportional Hazards Model with Time-Dependent Covariate 95% Confidence Interval for Relative Hazards Regressor Variable Regressor Coefficient Standard Error Wald Statistic p Relative Hazards Lower Upper (a) AGE SEX LACR LTG LTG* log(t;1) 0.068 0 .75 9 0.111 0.915 90.390 0.014 0.218 0.046 0.435 0.298 25.249 12.056 5 .78 1 4.420 1 .71 0 0.0001 0.0005 0.0162 0.0355 0.1910 1. 07 2.14 1.12... agea ageb sex smoke bmi lacr / ties : efron; baseline out : base1 survival : survival l : lowb u : uppb / method : ch; run; title ’Breslow estimate of the survival function and its lower and upper bands’; proc print data : base1; var t survival lowb uppb; run; The following SPSS code can be used to obtain the Breslow estimate of the survival function and its standard error at each uncensored observation . 90 .77 53 90 .77 55 90 .71 07 90.5966 SEX 0 .71 34 0 .72 33 0 .71 87 0 .71 89 0.6862 0.5659 SMOKE 0. 376 2 0.3810 0. 377 6 0. 377 6 0.3440 0.2855 BMI 0.0253 0.0256 0.0255 0.0255 0.0233 0.0194 LACR 0. 173 5 0. 175 9 0. 173 9. Cohort CV D Data AGEA 91.3558 0. 271 2 24.9910 0.0001 0.258 0.151 0.439 AGEB 90 .77 53 0.2618 8 .77 09 0.0031 0.461 0. 276 0 .76 9 SEX 0 .71 87 0.2193 10 .74 57 0.0010 2.052 1.335 3.153 SMOKE 0. 377 6 0.2208. 0 0 0 30.88 189 5.38 4 .72 73 .9 1 1 38 2 .7 1 1 0 0 0 1 25.05 200 3. 37 4.86 77 .2 1 1 39 2 .7 1 1 0 0 1 0 26.80 130 2.31 5.10 73 .5 0 0 40 3.3 1 1 0 0 1 1 21. 67 111 3.53 4.18 71 .1 0 0 41 2.9 1 1 0