MATH 6643 Summer 2012 Applications of Mixed Models/Students/smithce/Model1

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Model 1

Level 1 Model

mathach_{ij} = {\color{Red}\beta_{0j}} + {\color{Blue}\beta_{1j}}ses_{ij} + r_{ij}

Level 2 Model (Between School Model)

{\color{Red}\beta_{0j}} = {\color{Red}\gamma_{00}} + {\color{Red}\gamma_{01}Sector_j} + {\color{Red}u_{0j}}
{\color{Blue}\beta_{1j}} = {\color{Blue}\gamma_{10}} + {\color{Blue}\gamma_{11}Sector_j} + {\color{Blue}u_{1j}}

Combined Model (by substitution)

mathach_{ij} = {\color{Red}\gamma_{00}} + {\color{Red}\gamma_{01}Sector_j} + {\color{Red}u_{0j}} + ({\color{Blue}\gamma_{10}} + {\color{Blue}\gamma_{11}Sector_j} + {\color{Blue}u_{1j}})ses_{ij} + r_{ij}
mathach_{ij} = \underbrace{{\color{Red}\gamma_{00}} + {\color{Red}\gamma_{01}Sector_j} + {\color{Blue}\gamma_{10}}ses_{ij} + {\color{Blue}\gamma_{11}Sector_j}ses_{ij}}_{Fixed} + \underbrace{{\color{Red}u_{0j}} + {\color{Blue}u_{1j}}ses_{ij} + r_{ij}}_{Random}

Fixed Portion of the Model

{\color{Red}\gamma_{00}} + {\color{Red}\gamma_{01}}Sector_j + {\color{Blue}\gamma_{10}}ses_{ij} + {\color{Blue}\gamma_{11}}Sector_jses_{ij}

Random Portion of the Model

{\color{Red}u_{0j}} + {\color{Blue}u_{1j}}ses_{ij} + r_{ij}


fit <- lme( mathach ~ ses * Sector, dd, random = ~ 1 + ses | id, control = list(msMaxIter=200, msVerbose=T))
# Note that id refers to schools not students!


Linear mixed-effects model fit by REML
Data: dd

AICBIClogLik
2391423964-11949


Random effects:
Formula: ~1 + ses | id
Structure: General positive-definite, Log-Cholesky parametrization

StdDevCorr
(Intercept)2.151 1(Intr)
ses0.355 20.973 3
Residual6.0754


Fixed effects: mathach ~ ses * Sector

ValueStd.ErrorDFt-valuep-value
(Intercept)13.9850.387360236.10
ses1.6760.22836027.30
SectorPublic-2.3770.52678-4.50
ses:SectorPublic1.3980.30736024.50


Correlation:

(Intr)sesSctrPb
ses0.204
SectorPublic-0.736-0.15
ses:SectorPublic-0.152-0.7440.252


Standardized Within-Group Residuals:

MinQ1MedQ3Max
-3.0761-0.73780.02460.75652.7849


Number of Observations: 3684
Number of Groups: 80


Notes

1. Var({\color{Red}u_{0}}) = 2.151^2
2. Var({\color{Blue}u_{1}}) = 0.355^2
3. Correlation({\color{Red}u_{0}},{\color{Blue}u_{1}}) = 0.973
4. Var({\color{Black}r_{ij}}) = 6.075^2
5 {\color{Red}\gamma_{00}} = 13.98 Intercept for Catholic schools
6 {\color{Blue}\gamma_{10}} = 1.67 Slope of ses for Catholic schools
7 {\color{Red}\gamma_{01}} = -2.37 Change in Intercept for Public schools (e.g. Intercept for Public = 13.98 - 2.37 = 11.61)
8 {\color{Blue}\gamma_{11}} = 1.39 Change in Slope for Public schools (e.g. ses slope for Public = 1.67 + 1.39 = 3.06)

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