Outline An illustrative example • Elements of interpretation • Rank-hazard plots • Visualizing covariates in proportional hazards Model comparison with rank-hazard plots • Conclusion • model using R Juha Karvanen International CVD Epidemiology Unit Department of Health Promotion and Chronic Disease Prevention National Public Health Institute Finland Model Fitted model Data from the MORGAM Project • Call: Inclusion criteria coxph(formula = Surv(AGE1, CHDAGE1, status) ˜ BPM + RCHOL + BMI + • DSMOKER + strata(COHORT) + strata(region), data = f31[f31$basestat != Men from Finland, 30–65 years at baseline 1 & f31$exclusion == 0, ], subset = (SEX == 1 & RCHOL <= • rchollimit)) No cardiovascular disease at baseline • No hypercholesterolemia ( ⇒ very high RCHOL) • n=12698 (7 observations deleted due to missing) coef exp(coef) se(coef) z p No missing covariates • BPM 0.0163 1.02 0.00213 7.67 1.7e-14 RCHOL 0.2125 1.24 0.01559 13.63 0.0e+00 Response variable: The age at the first event of coronary heart • BMI 0.0152 1.02 0.00813 1.87 6.2e-02 disease (CHD) DSMOKER 0.5725 1.77 0.06119 9.36 0.0e+00 Covariates • exp(coef) exp(-coef) lower .95 upper .95 BPM, the mean of diastolic and systolic blood pressure (mmHg) BPM 1.02 0.984 1.01 1.02 • RCHOL 1.24 0.809 1.20 1.28 RCHOL, the ratio of total cholesterol to HDL cholesterol • BMI 1.02 0.985 1.00 1.03 BMI, body mass index ( kg / m 2 ) DSMOKER 1.77 0.564 1.57 2.00 • DSMOKER, daily smoker (1=yes, 0=no) • Rsquare= 0.029 (max possible= 0.679 ) Likelihood ratio test= 373 on 4 df, p=0 Cox’s proportional hazards model • Wald test = 415 on 4 df, p=0 Score (logrank) test = 414 on 4 df, p=0
Elements of interpretation Interpreting the results Study design p-value oriented interpretation • • The type of model • In the fitted model BPM, RCHOL and DSMOKER are statistically • Definition of the covariate • significant at 5 % risk level. Estimated model parameter • Covariates in ascending order by the p-values: RCHOL, • Unit of measurement • DSMOKER, BPM, BMI. Distribution of the covariate in the cohort (e.g. how common is • The p-values will change if covariates are added or removed ⇐ • smoking?) covariates in the model are correlated. Other interesting questions • What is the epidemiological relevance of these risk factors? • Is smoking a more serious risk factor of CHD than overweight in • the population? How the model will change if we add, remove or transform • covariates? How to visualize the fitted model? • First visualization attempt First attempt: BPM vs RCHOL Relative risk as a function of covariate values 10.0 10.0 10.0 5.0 5.0 5.0 relative hazard relative hazard 2.0 relative hazard 2.0 2.0 1.0 1.0 0.5 0.5 1.0 80 100 120 140 160 180 2 4 6 8 10 12 14 BPM RCHOL 0.5 Information on the covariate distributions still not fully utilized. 80 100 120 140 160 180 BPM
Second attempt: ranks R implementation Relative risk as a function of ranks of covariate values n<-length(x) x<-sort(x) 10.0 10.0 relativehazard<-exp(beta*(x-median(x))) BPM (median=111.5 mmHg) plot((1:n)/n,relativehazard,type="l",log="y") 5.0 5.0 relative hazard relative hazard 2.0 2.0 1.0 1.0 0.5 0.5 80 100 120 140 160 180 0.0 0.2 0.4 0.6 0.8 1.0 BPM cumulative covariate distribution Rank-hazard plot Rank-hazard plot Reference: medians Reference: normal upper limits 10.0 10.0 BPM (median=111.5 mmHg) BPM (reference=107.5 mmHg) RCHOL (median=4.64 ) RCHOL (reference= 5 ) BMI (median=26.17 kg/m^2) BMI (reference= 25 kg/m^2) DSMOKER DSMOKER 5.0 5.0 relative hazard relative hazard 2.0 2.0 1.0 1.0 0.5 0.5 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 cumulative covariate distribution cumulative covariate distribution
Rank-hazard plot Model comparison Model with logarithms of covariates 10.0 10.0 BPM (median=111.5 mmHg) BPM (reference=107.5 mmHg) 10.0 RCHOL (median=4.64 ) RCHOL (reference= 5 ) log(BPM) (median=4.714 ) BMI (median=26.17 kg/m^2) BMI (reference= 25 kg/m^2) log(RCHOL) (median=1.535 ) DSMOKER DSMOKER 5.0 5.0 log(BMI) (median=3.265 ) DSMOKER 5.0 relative hazard relative hazard 2.0 2.0 relative hazard 2.0 1.0 1.0 1.0 0.5 0.5 0.5 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 cumulative covariate distribution cumulative covariate distribution 0.0 0.2 0.4 0.6 0.8 1.0 cumulative covariate distribution Model comparison Model comparison Model without blood pressure 10.0 10.0 BPM (median=111.5 mmHg) log(BPM) (median=4.714 ) 10.0 RCHOL (median=4.64 ) log(RCHOL) (median=1.535 ) RCHOL (median= 4.64 ) BMI (median=26.17 kg/m^2) log(BMI) (median=3.265 ) BMI (median=26.17 kg/m^2) DSMOKER DSMOKER 5.0 5.0 DSMOKER 5.0 relative hazard relative hazard 2.0 2.0 relative hazard 1.0 1.0 2.0 0.5 0.5 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 cumulative covariate distribution cumulative covariate distribution 0.5 0.0 0.2 0.4 0.6 0.8 1.0 cumulative covariate distribution
Model comparison Model comparison Adding history of diabetes 10.0 10.0 BPM (median=111.5 mmHg) RCHOL (median= 4.64 ) 10.0 RCHOL (median=4.64 ) BMI (median=26.17 kg/m^2) BPM (median=111.5 mmHg) BMI (median=26.17 kg/m^2) DSMOKER RCHOL (median= 4.64 ) DSMOKER 5.0 BMI (median=26.17 kg/m^2) 5.0 DSMOKER 5.0 HISDIAB relative hazard relative hazard 2.0 2.0 relative hazard 1.0 2.0 1.0 0.5 0.5 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 cumulative covariate distribution cumulative covariate distribution 0.5 0.0 0.2 0.4 0.6 0.8 1.0 cumulative covariate distribution Model comparison Conclusion Rank-hazard plots visualize several covariates in the same plot. • Visualization may help interpreting the epidemiological relevance of • 10.0 10.0 the covariates. BPM (median=111.5 mmHg) BPM (median=111.5 mmHg) RCHOL (median=4.64 ) RCHOL (median= 4.64 ) BMI (median=26.17 kg/m^2) BMI (median=26.17 kg/m^2) Visualization is easy to implement in R. • DSMOKER DSMOKER 5.0 5.0 HISDIAB Future directions: How to visualize correlation between the covariates • in rank-hazard plots? relative hazard relative hazard 2.0 2.0 1.0 1.0 0.5 0.5 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 cumulative covariate distribution cumulative covariate distribution
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