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Lecture 22: Random effects models BMTRY 701 Biostatistical Methods II

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Independence 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’

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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 model

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Nurse staffing in ICU example Hospitals in MD from , 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 outcomes

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Random effects modeling Standard logistic model Random effects logistic model

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Adding 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 β 1

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What does this look like? Linear Regression

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Fitting Random Effects Models in R library(nlme) re.reg <- lme(y ~ x, random=~1|hospid) o.reg <- lm(y~x) bi <- re.reg$coefficients$random$hospid b0 <- re.reg$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)

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Random Effects?

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Interpretation 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 model

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Stata 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 Stata

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Applied example rnals_publications/ecp/sepoct01/pronovost.htm rnals_publications/ecp/sepoct01/pronovost.htm

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