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) Living ArrangementN# suicidesPr(suicide) Single parent Two parents Relative Risk2.5

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 Odds ratios 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

6 Examples of odds ratios Suicide example: OR=2.50 Home Allergen P 1 = Pr(Asthma|Maternal History)= P 0 = Pr(Asthma|No Maternal History)= OR = RR = Maternal historyNo asthmaAsthma No23034 Yes

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 h 0 (t) is constant (and equal to h 0 ), then I 0 (t) =1-exp(-ht)~h 0 t (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 β 1 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 Relative RiskOdds Ratio Control-- Siezure History Drug Exposed Drug exposedSeizure history, no drugsControls # kids Major malformation, growth retard or small head 34 (11%)3 (3%)21 (4%)

15 Analysis via Poisson Regression Std 95% Conf Chi- DF Estimate Error Limits Square Pr > ChiSq Intercept <.0001 DRUG DRUG DRUG Scale proc genmod; class drug; model one3=drug/dist=poisson; run;

16 Analysis via Logistic Regression Standard 95% Conf Chi- Parameter DF Estimate Error Limits Square Pr > ChiSq Intercept <.0001 DRUG DRUG DRUG Scale proc genmod descending; class drug; model one3=drug/dist=binomial; run;

17 Example - arsenic Consider the high village only, as well as the baseline group corresponding to all of SW Taiwan. Arsenic # cancer pyr rate/ rrisk

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) <2e-16 factor(conc)

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.