Lecture 22: Random effects modelsIndependence AssumptionHow to deal with it?Nurse staffing in ICU exampleRandom effects modelingAdding in the random effectWhat does this look like? Linear RegressionFitting Random Effects Models in RRandom Effects?InterpretationStataApplied exampleLecture 22:Random effects modelsBMTRY 701Biostatistical Methods IIIndependence AssumptionAll of the regression assumptions we’ve discussed thus far assume independenceThat is, patients (or other ‘units’) have outcomes that are unrelatedBut what if they are?•the same person is measured multiple times•people from the same house are studied•people treated in the same hospital are studied•different tumors within the same patient are evaluatedIn all of those examples, the independence assumption ‘falls apart’How to deal with it?Two main approaches:Random effects model:•include a ‘random intercept’ to account for correlation•individuals who are ‘linked’ (i.e., from same house, hospital, etc.) receive the same interceptGeneralized estimating equations (GEE)•model the correlation as part of the regression•two part modeling:mean modelcovariance modelNurse staffing in ICU exampleHospitals in MD from 1994-1996, discharge dataAll patients with abdominal aortic surgery (AAS)Goal: evaluate the association between the nurse-to-patient ratio in the ICU for risk of medical and surgical complications after AAS.Data:•patient outcomes (complications)•nurse:patient ratioIssue: patients treated within the same hospital are likely to have correlated outcomesRandom effects modeling),0(~)(logit)(logit21010NbNursebyNurseyjijjijiiStandard logisticmodelRandom effectslogistic modelAdding in the random effectConditional on the random effect, the observations within a hospital are independenceHence, independence is restored!Even so, random effects are considered ‘nuisance parameters’•we generally don’t care about them•they are necessary, but not interestingOur primary interest is still in β1What does this look like? Linear Regression0 2 4 6 8 102 4 6 8xyFitting Random Effects Models in Rlibrary(nlme)re.reg <- lme(y ~ x, random=~1|hospid)o.reg <- lm(y~x)bi <- re.reg$coefficients$random$hospidb0 <- re.reg$coefficients$fixed[1]b1 <- re.reg$coefficients$fixed[2]par(mfrow=c(1,1))plot(x,y)abline(o.reg)for(i in 1:20) {lines(0:10, b0+b1*(0:10) + bi[i], col=2)}abline(o.reg, lwd=2)Random Effects?Histogram of bibiFrequency-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.00 1 2 3 4 5InterpretationRecall “nuisance” parametersIn most cases, we do not care about random intercepts“Fixed” effects are interpreted in the same way as in a standard regression modelStataxtreg: random effects linear regressionxtlogit: random effects logistic regressionxtpoisson: random effects poisson regressionstcox, ...shared(id): random effects Cox regressionAlso, ‘cluster’ option in many regression commands in StataApplied