# 04LMM - PubH8452 Longitudinal Data Analysis Fall 2011...

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Unformatted text preview: PubH8452 Longitudinal Data Analysis - Fall 2011 Linear Mixed Models Linear Mixed Models Outline • Motivation • Random Intercept Model • Random Intercept and Random Slope Model • Linear Mixed Model (LMM) • Multilevel Mixed Effects Models • Optimization Algorithms • Inference for Fixed Effects • Inference for Variance Parameters • Inference about the Random Effects • Extending Linear Mixed Models 1 PubH8452 Longitudinal Data Analysis - Fall 2011 Linear Mixed Models Motivation Recall the orthodontic measurement data. One question of interest is the individual growth curve . Age (yr) Distance (mm) 16 18 20 22 24 26 28 89 11 13 10 89 11 13 6 89 11 13 9 89 11 13 1 89 11 13 5 89 11 13 2 89 11 13 7 89 11 13 8 89 11 13 3 89 11 13 4 89 11 13 11 Figure 1: Orthodontic distance measurements for girls 2 PubH8452 Longitudinal Data Analysis - Fall 2011 Linear Mixed Models Let’s consider only the girls for the moment. >Orthodont<- read.table ("orthodontic.dat",header=TRUE) > library (nlme) > Orth.new <- groupedData (distance ~ age | child,data = as.data.frame (Orthodont), + FUN = mean,inner = ~ age, labels = list( x = "Age", + y = "Distance" ),units = list( x = "(yr)", y = "(mm)") ) > OrthFem <- subset(Orth.new, male==0) > OrthFem[1:5,] Grouped Data: distance ~ age | child obs child age distance male 1 1 1 8 21.0 2 2 1 10 20.0 3 3 1 12 21.5 4 4 1 14 23.0 5 5 2 8 21.0 >library (lattice) >plot (OrthFem ) groupedData is a special data class in R library nlme designed for describing clustered data. Many convenient functions are defined for it. 3 PubH8452 Longitudinal Data Analysis - Fall 2011 Linear Mixed Models Let the random variable Y i denote the outcomes for the i th individual, measured at time t ij , i = 1 , . . . , m , j = 1 , . . . , n i . Thus Y i = ( Y i 1 , Y i 2 , . . . , Y in i ) . Three modeling strategies to characterize the individual growth curve: 1. Two-stage analysis : fit a linear regression line to each subject and analyze the subject-specific regression coefficient as responses in the second stage analysis. • Stage 1: y ij = β i + β 1 i x ij + ² ij (1) • Stage 2: β i = z i β + τ i β 1 i = z 1 i β 1 + υ i (2) Where ² i , τ i and υ i assumed to be independent, following normal distribution. 2. Fixed effects model : include an indicator variable for subject id in the regression. y ij = β + a i + β 1 x ij + ² i (3) Where a i is fixed effect for the i th subject. 3. Random effects model : include random effect in the regression. y ij = β + b i + β 1 x ij + ² ij (4) Where b i are random intercepts, both ² i and b i assumed to be independent, following normal distri- bution. 4 PubH8452 Longitudinal Data Analysis - Fall 2011 Linear Mixed Models Two-Stage Analysis > of.lis <- lmList (distance ~ I(age - 11), data = OrthFem) > coef (of.lis) (Intercept) I(age - 11) 10 18.500 0.450 6 21.125 0.375 9 21.125 0.275 1 21.375 0.375 5 22.625 0.275 2 23.000 0.800 7 23.000 0.550 8 23.375 0.175 3 23.750 0.850 4 24.875 0.4750....
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04LMM - PubH8452 Longitudinal Data Analysis Fall 2011...

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