# lec2 - Model 1 Let y hij = the response at measurement...

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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 off-diagonal 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 var-cov 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 416.5-392.7=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 variance-covariance models to these data by imposing some structure on R h , h = 1 , 2, the error var-cov matrix for boys and girls. • In models 0a and 0b, we fit heteroscedastic (SAS uses the term het- erogeneous) and non-heteroscedastic versions of the Toeplitz struc- ture for R h , h = 1 , 2. A Toeplitz var-cov matrix is banded. In the non-heteroscedastic 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....
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lec2 - Model 1 Let y hij = the response at measurement...

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