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

2. 2 Recall yesterday’s example Drug exposedSeizure history, no drugsControls # kids 31296506 Major malformation, growth retard or small head 34 (11%)3 (3%)21 (4%) Relative Risk (relative to controls) 2.62.75- Odds Ratio (relative to controls) 2.82.75- 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. 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. 4 Analysis via Poisson Regression 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 proc genmod; class drug; model one3=drug/dist=poisson; run;

5. 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. 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. 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. 8 Inference for an estimated risk ratio Suppose the data are represented as a 2x2 table. We can show that Disease? YesNo Exposedab Unexposedcd

9. 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. 10 Analysis via Poisson Regression 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” proc genmod; class drug; model one3=drug/dist=poisson; run;

11. 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. 12 MLE-based inference for the log incidence rate The variance of the estimated hazard is given by taking the negative inverse of the expected information: This implies Var(LRR)=1/a+1/c which is close to the previous formula when sample sizes are large

13. 13 Variance of a standardized rate If w 1 ….w K are considered fixed and known, and I s is estimated from a study population, then If the w k are person years and the disease rate is computed under a Poisson assumption, then

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

15. 15 Log odds ratios & logistic regression

16. 16 Compare with Logistic Regression output 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 proc genmod descending; class drug; model one3=drug/dist=binomial; run;

17. 17 Example – coffee and cancer Cancer? YesNo Coffee347555 No Coffee2088 Odds Ratio = Log Odds Ratio = Var(estimated LOR)= 95% confidence interval for true LOR:

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

19. 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. 20 Excess risk estimated by So Var(ER) is estimated by sum of variance of two estimated rates