More on exposure/response associations

1. More on exposure/response associations Readings • Jewell Chapters 3 & 7

2. Recall yesterday’s example • Questions: • We saw how to use logistic regression to estimate the odds ratios. Is there something analogous for Risk Ratios? • How much uncertainty is there in these estimates? How can we construct a confidence interval?

3. Risk ratios & Poisson regression Let X be binary exposure indicators as before, and suppose Y~Poisson(h(X)), where Hence, • Natural framework when data comes in terms of # events and person-time • Can also use to analyze rare binary events

4. Analysis via Poisson Regression procgenmod; class drug; model one3=drug/dist=poisson; run; Std 95% Conf Chi- DF Estimate Error Limits Square Pr > ChiSq Intercept 1 -3.1820 0.2182 -3.6097 -2.7543 212.63 <.0001 DRUG 1 1 0.9654 0.2775 0.4214 1.5093 12.10 0.0005 DRUG 2 1 -0.2837 0.6172 -1.4934 0.9260 0.21 0.6457 DRUG 3 0 0.0000 0.0000 0.0000 0.0000 . . Scale 0 1.0000 0.0000 1.0000 1.0000

5. Illustrating Poisson regression with the arsenic data Consider the high village only, as well as the baseline group corresponding to all of SW Taiwan. Arsenic # cancer pyr rate/100000 rrisk 0 3159 14689807 934 4 8341

6. Example - arsenic setwd("C:\\Work\\Reference_Materials\\Datasets\\Arsenic\\SWTaiwan") y=read.table("mlun.sw.dat",header=T) # READ IN DATA y$lar=log(y$at.risk/100000) # COMPUTE LOG OF PYR cancer = tapply(y$events,y$conc,sum) # SUM NUMBER CANCER BY CONC prisk = tapply(y$at.risk,y$conc,sum) # SUM PYR BY CONC crate = 100000*cancer/prisk # COMPUTE HAZARD BY CONC rrisk = crate/crate[1] # COMPUTE RELATIVE RISK cbind(cancer,prisk,crate,rrisk) # PRINT IN NICE FORMAT OK = y$conc==0 | y$conc>900 # SUBSET OF LOW AND HIGH result = glm(events ~ factor(conc), family="poisson",offset=lar,subset=OK,data=y) summary(result) Estimate Std. Error z value Pr(>|z|) (Intercept) 3.06827 0.01779 172.452 <2e-16 factor(conc)934 0.80201 0.50026 1.603 0.109

7. Arsenic (cont’d) • Can compute relative risk for each village and then plot to visualize dose response. • In practice: • Need to include age as well (more next week) • Model dose response as part of Poisson regression (also more next week)

8. Inference for an estimated risk ratio Suppose the data are represented as a 2x2 table. We can show that

9. Example - Epilepsy For Relative Risk associated with Seizure history versus control, we have: a=3, b=93, c=21, d=485 Hence, And the correspond standard error is 0.607 Lets compare with Poisson regression output

10. Analysis via Poisson Regression procgenmod; class drug; model one3=drug/dist=poisson; run; Std 95% Conf Chi- DF Estimate Error Limits Square Pr > ChiSq Intercept 1 -3.1820 0.2182 -3.6097 -2.7543 212.63 <.0001 DRUG 1 1 0.9654 0.2775 0.4214 1.5093 12.10 0.0005 DRUG 2 1 -0.2837 0.6172 -1.4934 0.9260 0.21 0.6457 DRUG 3 0 0.0000 0.0000 0.0000 0.0000 . . Scale 0 1.0000 0.0000 1.0000 1.0000 95% Confidence intervals correspond to estimate +/- 1.96*Std Error If the confidence interval for the log relative risk excludes zero, we say that there is a “significant association”

11. Example - Epilepsy Now lets do Relative Risk associated with Drugs versus control, we have: a=34, b=278, c=21, d=485 Hence, And the correspond standard error is 0.268 Compare again Poisson regression output

12. The variance of the estimated hazard is given by taking the negative inverse of the expected information: MLE-based inference for the log incidence rate This implies Var(LRR)=1/a+1/c which is close to the previous formula when sample sizes are large

13. Variance of a standardized rate If w1….wK are considered fixed and known, and Is is estimated from a study population, then If the wk are person years and the disease rate is computed under a Poisson assumption, then

14. Inference for an estimated odds ratio Suppose the data are represented as a 2x2 table. We can show that

15. Log odds ratios & logistic regression

16. Compare with Logistic Regression output procgenmod descending; class drug; model one3=drug/dist=binomial; run; Standard 95% Conf Chi- Parameter DF Estimate Error Limits Square Pr > ChiSq Intercept 1 -3.1396 0.2229 -3.5765 -2.7028 198.41 <.0001 DRUG 1 1 1.0384 0.2876 0.4748 1.6020 13.04 0.0003 DRUG 2 1 -0.2944 0.6275 -1.5243 0.9355 0.22 0.6390 DRUG 3 0 0.0000 0.0000 0.0000 0.0000 . . Scale 0 1.0000 0.0000 1.0000 1.0000

17. Example – coffee and cancer Odds Ratio = Log Odds Ratio = Var(estimated LOR)= 95% confidence interval for true LOR:

18. Small sample adjustments Estimators and their estimated variances may be inaccurate for small samples. Adjusted logodds ratios is:

19. Variance formulae for excess risk • Excess risk estimated by So Var(ER) is estimated by sum of variance of two estimated rates Variance formulae for attributable risk is more complicated (Jewell p 84)

20. Excess risk estimated by So Var(ER) is estimated by sum of variance of two estimated rates