Quantifying Disease-exposure association

1. Quantifying Disease-exposure association Readings • Jewell Chapter 4 • Rothman and Greenland Chapter 4

2. Motivation • Mothers who smoke are more likely to have a low-birthweight baby • Drinking arsenic-contaminated water will increase your risk of lung cancer • Children exposured to moulds and other home allergens are more likely to develop asthma • People with high cholesterol are likely to have higher blood pressure and are more likely to develop heart disease Today’s lecture will focus on how to quantify what “more likely” means.

3. Measuring association for binary outcomes Suppose we are interested how disease prevalence varies with exposure or other factors. The Relative Risk (RR) for an outcome D associated with an exposure E is given by: • RR=1 implies no association • For rare events, RR can be large even when probabilities look very close (e.g. Jewell Table 3.5)

4. Notes about relative risks • Epidemiologists tend to like it because of its direct interpretation • Consistent with a multiplicative model. E.g. Pr(suicide|single parent)=2.5*Pr(suicide|two parent) • Must be positive (probabilities must be positive) • Has limited upper range (since probabilities cannot exceed 1). This can make modeling very difficult.

5. Notes: OR=1 implies no association OR must be positive, but has no upper limit Useful for common diseases Very close to RR for rare diseases Odds ratios

6. Examples of odds ratios • Suicide example: OR=2.50 • Home Allergen P1 = Pr(Asthma|Maternal History)= P0 = Pr(Asthma|No Maternal History)= OR = RR =

7. Odds Ratio vs Relative Risk Notes: • For rare disease, Pr(not D)~1, regardless of exposure, so OR~RR (famous Cornfield result) • If RR > 1, then

8. Measuring association for disease incidence • If incidence is characterized by cumulative incidence or incidence proportion, then these are probabilities and we can use odds ratios or relative risks as before • If incidence is characterized by instantaneous rates, then the ratio of these instantaneous probabilities (relative hazard) is simply the hazard ratio.

9. Disease incidence associations under constant hazards If h0(t) is constant (and equal to h0), then I0(t) =1-exp(-ht)~h0t (if h and/or t is small) So RR(t) will be close to RH(t) in this case. R-Code to generate Jewell Figure 4.1 fun = function(h0,RH,t){ h1 = h0*RH i0 = 1-exp(-h0*t) i1 = 1-exp(-h1*t) RR = i1/i0 OR = i1*(1-i0)/(1-i1)/i0 c(RH,RR,OR)} t=seq(1,20); y=matrix(0,ncol=3,nrow=length(t)) for (i in 1:length(t)){y[i,]=fun(.007,2,t[i])} matplot(t,y,type="l",xlab="Time (years)", ylab="Measures of Association")

10. How to chose a measure of effect? • Epidemiologists may often have a preference for relative risk, since it has a straightfoward interpretation. • Since no model is ever “correct”, choose a measure that makes sense for the study design • Odds ratio when prevalence of a binary outcome is of interest • Hazard ratio/Risk Ratio when time to event or disease incidence rates are of interest.

11. Odds ratios & logistic regression Let X be a covariate taking the value 1 if subject is exposed, 0 otherwise and suppose Straightforward algebra establishes that β1 is interpretable as the log odds ratio

12. Logistic Regression (cont’d) • Natural modeling framework for bringing in additional covariates if prevalence of a binary outcome is of interest • Cornfield result also suggests using for rare binary outcomes • Can derive the previous variance formula for the estimated odds ratio using standard likelihood calculations

13. 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

14. Example - Antiepileptic Drugs

15. 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

16. Analysis via Logistic Regression 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 - arsenic 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

18. 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 vrisk = tapply(y$at.risk,y$conc,sum) # SUM PYR BY CONC crate = cancer/vrisk # COMPUTE HAZARD BY CONC rrisk = crate/crate[1] # COMPUTE RELATIVE RISK cbind(cancer,vrisk,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

19. Arsenic (cont’d) For environmental regulation, want exposure with risk more than 1% above background: P(D)-P(0)=.01 This is using the concept of additive risk or excess risk

20. Attributable Risk What percentage of risk is associated with a specific risk factor. In epilepsy example, risk was 4% for controls, 11% for drug exposed. Attributable risk depends on prevalence of exposure in the general population. From Jewell Section 4.7:

21. Attrributable risk (cont’d) • An exposure must have either an extremely large relative risk or be very prevalent in order to explain a high proportion of the incidence of a specific health outcome. • In case of antiepileptic drugs, attributable risk is likely to be very small.