L18handout - PUBH 7430 Lecture 18 J. Wolfson Division of...

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Unformatted text preview: PUBH 7430 Lecture 18 J. Wolfson Division of Biostatistics University of Minnesota School of Public Health November 10, 2011 Linear mixed effects model Yij = xij β + zi bi + ij • Yij = Observation j from cluster i • xij = Vector of covariates for observation j from cluster i • β = Vector of fixed effect coefficients for elements of xij • zi = Vector of covariates from cluster i • bi = Vector of random effect coefficients for elements of zi • ij = Residual error term for observation j from cluster i Fixed vs. random effects Yij = xij β + zi bi + ij • Fixed effects (β ) are components of the mean model with values that are fixed but unknown. We want to estimate their value. • Random effects (bi ) are components of the mean model with values that are assumed to be drawn from some distribution (usually Normal). We can obtain predictions of these random effects, but we generally focus on estimating their variance. Fixed vs. random effects Yij = xij β + zi bi + • Though the residual ij is also a random quantity whose variance we are interested in, we don’t typically refer to it as a random effect. • A model with both fixed and random effects is often called a mixed or mixed effects model, but is sometimes also called a random effects model. ij Partitioning the variance One can think of random effects models as partitioning sources of variance. • In regression models, estimating fixed effects of interest comes down to averaging differences between pairs of observations. • Different models give different weights to these pairwise differences. • Random effects model is one way of assigning these weights. Partitioning the variance • Focus on Var (Yij | Xij ), variance of observation j in cluster i • In standard linear model, we assumed that all observations have the same variance (given covariates Xij ): Var (Yij | Xij ) = σ 2 • Since observations are assumed independent, all pairs of observations are treated “equally” • If σ 2 is large, then lines connecting pairs of points may vary greatly in slope/intercept ⇒ regression coefficients will be hard to estimate precisely. Partitioning the variance Now, suppose instead that we acknowledge that observations are clustered, and each cluster has a different intercept. Then, variability in Yij is partitioned into two sources: 2 Var (Yij | Xij ) = C 2 + σ1 where • C 2 is the variance of the cluster means (intercepts) 2 • σ1 is the variance around that cluster’s mean line Partitioning the variance With this model, pairs of points are treated differently depending on whether they are: • Within the same cluster: Given more weight since they 2 have less variability (Var (Yij | Xij , bi ) = σ1 ) • In different clusters: Given less weight since they have 2 more variability (Var (Yij | Xij ) = C 2 + σ1 ) Note: Pairs of observations in different clusters contribute some information; more efficient use of data than just fitting separate regression lines to each cluster and then averaging across clusters Wallabies: Ear length vs. Age q q q 700 q q qq q q q q 650 qq q q q q q q 600 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q q q q q q q 0 q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q Ear length q q q qq q q q q qq q qq q q qq q q q q q q q q q q 50 100 150 200 250 Age in days since one year old 300 350 Random effects Ear length vs. Age, Animal 45 q q 650 0.000 600 0.004 Ear length 700 0.008 q −100 −50 0 50 100 q q q q q q qq q q q qq qq q q q q qq q q qqq q qq q q q qq q q q q q q q q qq q q q q q q q q qq q q q q qq q q q q q q q q qq q q qq q q qq q q q q q q q q qq q q q q qq q q qq q q q q q qq q q q q qqq q qqq q q q q q q q qq q qq q q q qq q qq qq q qq q q qq q qq q q qq q q qq qq q qq q qq q q q qqq qq q q q qqq q qq q q q q q q q q 0 50 100 150 200 250 300 350 Age in days since one year old Random effects Ear length vs. Age, Animal 47 q q 0.000 650 600 0.004 Ear length 700 0.008 q −100 −50 0 50 100 q q q q q q qq q q q qq qq q q q q qq q q qqq q qq q q q qq q q q q q q q q qq q q q q q q q q qq q q q q q q q qqq q q q q qq q q qq q q qq qq q q q q q q qq q q q q q qq q q qq qq q qq q q q q q qqq q qqq q q q q q q qq q qq q q qq q q qq qq q q qq qq q q qq q q q qq q q qq qq q qq q qq q q q qqq qq q q q qqq q qq q q q q q q q q 0 50 100 150 200 250 300 Age in days since one year old 350 Random effects Ear length vs. Age, Animal 81 q q 650 0.000 600 0.004 Ear length 700 0.008 q −100 −50 0 50 100 q q q q q q q q q qq qq q q q qq q q q qq qqq q q qq q q qq q q q qq q q qq qq q q q q q q q qq q q q q qq q q q q q q q q qq q q qq q q qq q q q q q q q q qq q q q q qq q q qq q q q q qq q q q q q qqq q qqq q q q q q q qq q qq q q qq q q qq qq q q qq qq q q q qq q q qq q q qq qq q qq q qq q q q qqq qq q q q qqq q qq q q q q q q q q q 0 50 100 150 200 250 300 350 Age in days since one year old Random effects Ear length vs. Age, Animal 87 q q 0.000 650 600 0.004 Ear length 700 0.008 q −100 −50 0 50 100 q q q q q q qq q q q qq qq q q q q qq q q qqq q qq q q q qq q q q q q q q q qq q q q q q q q q qq q q q q qq q q q q q q q q qq q q qq q q qq q q q q q q q q qq q q q qq qq q q q q qq q qq q q q q q q qqq q qq q q q q q q q qq q qq q q qq q q q q qq q q qq q q q q q qq q q qq q q qq q q qq q qq qq q q q qqq q qq q q qqq q qq q q q q q q q q 0 50 100 150 200 250 300 Age in days since one year old 350 Partitioning the variance Reweighting observations in this way can give more precise estimates of fixed effects: Linear model for wallaby data • Residual variance σ 2 = 1500.8 ˆ ˆ • SE (β1 ) = 0.027 Random intercepts model for wallaby data ˆ • Variance of intercepts C 2 = 1206.9 • Residual variance σ1 = 155.25 ˆ2 ˆ • SE (β1 ) = 0.0094 Random slopes • So far, we have only dealt with random intercepts • But random slopes are also possible Example: HIV treatment • Baxter et al. (2000) AIDS 15: F83-F93 • Objective: Compare short-term reduction in HIV viral load for those using vs. not using genotypic anti-retroviral testing (GART) in management of patients with failing antiretroviral therapy • Design: Viral load measured at baseline, 4, 8, and 12 weeks • Question: Does GART reduce viral load across duration of the study? Random slopes Observations/notes • Viral loads within individuals are correlated across time • Subjects start study with different viral loads (recall: not randomized study, so start level may be systematically different between groups) • Subjects may also have different “inherent” HIV viral load trajectories across time Random slopes Linear random effects model for HIV patients Yij = β0 + β1 timeij + β2 trtij + β3 timeij × trtij + b0i + b1i timeij + ij where • Yij is the log of viral load for person i measured at time j • trtij , timeij , and ij are the treatment assignment, measurement time, and error for the individual • b0i is a person-specific intercept • b1i is a person-specific slope • • ˆ b0i < 0 → person i started off with a lower viral load ˆ b1i < 0 → person i ’s viral load showed a steeper decline across time 10 q q q q Log(viral load) 5 q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q −5 q 2 4 6 Time (weeks) 8 10 ...
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