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 Lets 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 = i + 1 i x ij + ij (1) Stage 2: i = z i + i 1 i = z 1 i 1 + i (2) Where i , i and i assumed to be independent, following normal distribution. 2. Fixed effects model : include an indicator variable for subject id in the regression. y ij = + a i + 1 x ij + 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 = + b i + 1 x ij + ij (4) Where b i are random intercepts, both 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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