MATH 6643 Summer 2012 Applications of Mixed Models/Students/smithce/Model4

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Combined Model

$mathach_{ij} = \underbrace{{\color{Red}\gamma_{00}} + {\color{Red}\gamma_{01}Sector_j} + {\color{Red}\gamma_{02}ses.m_j} + {\color{Blue}\gamma_{10}}ses_{ij} + {\color{Blue}\gamma_{11}Sector_j}ses_{ij}}_{Fixed} + \underbrace{{\color{Red}u_{0j}} + {\color{Blue}u_{1j}}ses.d_{ij} + r_{ij}}_{Random}$

Fixed Portion of the Model Equivalent to FE model for Model 2 (LINK).

${\color{Red}\gamma_{00}} + {\color{Red}\gamma_{01}}Sector_j + {\color{Red}\gamma_{02}}ses.m_j + {\color{Blue}\gamma_{10}}ses_{ij} + {\color{Blue}\gamma_{11}}Sector_jses_{ij}$

Random Portion of the Model Non-equivalent RE model as compared to Model 2 (LINK).

${\color{Red}u_{0j}} + {\color{Blue}u_{1j}}ses.d_{ij} + r_{ij}$
${\color{Red}u_{0j}} + {\color{Blue}u_{1j}}(ses_{ij} - ses.m_{j}) + r_{ij}$

fitca <- lme( mathach ~ ses * Sector + ses.m, dd, random = ~ 1 + ses.d | id )  #c# Model 4 #c#


Linear mixed-effects model fit by REML
Data: dd

 AIC BIC logLik 23889.77 23945.66 -11935.88

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

 StdDev Corr (Intercept) 1.7465424 (Intr) ses.d 0.6444623 0.398 Residual 6.0649774

Fixed effects: mathach ~ ses * Sector + ses.m

 Value Std.Error DF t-value p-value (Intercept) 13.841 0.327 3602 42.362 0.00E+00 ses 1.529 0.246 3602 6.218 0.00E+00 SectorPublic -1.830 0.458 77 -3.997 1.00E-04 ses.m 3.034 0.594 77 5.106 0.00E+00 ses:SectorPublic 1.354 0.324 3602 4.175 0.00E+00

Correlation:

 (Intr) ses SctrPb ses.m ses 0.121 SectorPublic -0.734 -0.114 ses.m -0.157 -0.21 0.244 ses:SectorPublic -0.068 -0.728 0.155 0.014

Standardized Within-Group Residuals:

 Min Q1 Med Q3 Max -3.104 -0.734 0.022 0.751 2.851

Number of Observations: 3684
Number of Groups: 80

L <- list( 'Effect of ses' = rbind(
"Within-school" =  c( 0,1,0,0,0),
"Contextual"    =  c( 0,0,0,1,0),
"Compositional" =  c( 0,1,0,1,0)))
wald( fitca,L )


 numDF denDF F.value p.value Effect of ses 2 77 40.835 <.00001

 Estimate Std.Error DF t-value p-value Lower 0.95 Upper 0.95 Within-school 1.529 0.246 3602 6.218 <.00001 1.047 2.011 Contextual 3.034 0.594 77 5.106 <.00001 1.851 4.216 Compositional 4.562 0.593 77 7.690 <.00001 3.381 5.744