This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Model 1: Let y hij = the response at measurement occasion j , for the i th subject in gender group h . Model 1 includes a separate mean for each gender × time combination: y hij = β hj + ε hij where we assume ε hi = ( ε hi 1 , . . . , ε hi 1 ) T ∼ N ( , R ( θ )) ∀ h, i , where R is constant over h, i and is of the completely unstructured form ⇒ θ has 4 + ( 4 2 ) = 10 elements corresponding to the 4 diagonal elements of R and the 6 unique offdiagonal elements. • The REPEATED statement in PROC MIXED determines the form of R , the RANDOM statement specifies the random effects in the model (unlike PROC GLM, random effects appear only on the RAN DOM statement, not on the MODEL statement) and their varcov matrix D . • The S option of the MODEL statement prints the fixed effects esti mates (S for “solution”). • type=un specifies the unstructured form for R , subject=id specifies the cluster identifier, r=k and rcorr=k print the covariance matrix ( R ) and correlation matrix, respectively, corresponding to subject k (if =k is omitted SAS assumes k=1). 101 Model 0: In model 1 we assumed that R was the same for boys and girls. We relax this assumption in model 0 with the group=sex option on the repeated statement. That is, we assume ε hi = ( ε hi 1 , . . . , ε hi 1 ) T ∼ N ( , R h ( θ )). Notice that r=1,12 now asks for the R h matrix for subject 1 (the first girl) and for subject 12 (the first boy). • There does appear to be differences across gender in R h . Both AIC and a LRT of model 0 versus model 1, support model 0 as more appropriate for these data despite its large increase in parameters. – The AIC for model 0 is 448.7 vs 452.5 for model 1. – The LRT statistic is 416.5392.7=23.8 which is asymptotially χ 2 (10), gving p = . 0081. Models 0a–0e: In models 0a–0e, we retain the same mean structure and fit several simpler variancecovariance models to these data by imposing some structure on R h , h = 1 , 2, the error varcov matrix for boys and girls. • In models 0a and 0b, we fit heteroscedastic (SAS uses the term het erogeneous) and nonheteroscedastic versions of the Toeplitz struc ture for R h , h = 1 , 2. A Toeplitz varcov matrix is banded. In the nonheteroscedastic form (TYPE=TOEP), R ( θ ) = θ 1 θ 2 θ 3 θ 4 θ 1 θ 2 θ 3 θ 1 θ 2 θ 1 • In the heteroscastic form (TYPE=TOEPH), the diagonal elements of R above are allowed to differ, allowing for different variances at each age, and the correlation matrix corr( ε hi ) is assumed to be banded. – It is very common for longitudinal data to exhibit heteroscedas ticity over time, so heteroscedastic forms are always worth con sidering....
View
Full Document
 Fall '08
 Staff
 Variance, Longitudinal study, Repeated measures design, yhij

Click to edit the document details