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The variable year was 'centered' using a centering constant of 1967 because this minimized correlation between the random slopes and intercepts.
Since coefficient and standard error estimates did not change substantially between the models using independent, exchangeable, and unstructured covariance between the random slopes and intercepts, independent covariance was specified in the final models for simplicity.
We then used a forward selection procedure (Bryk and Raudenbush 1992) involving three steps: (1) adding fixed explanatory variables (including interaction terms between them); (2) adding cross-level interaction terms; and (3) adding random intercepts, random slopes, and covariances between the random slopes.
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The level-three random intercepts represent (potentially unknown) differences between ICUs and the random slopes for APACHE III allow the dependence of in-hospital mortality on patient severity to vary between hospitals.
cResults from Model 2, when no significant between-city heterogeneity was detected (without the random slopes for cities, g3 i ).
The random components measure the degree to which the random slopes and random intercept vary across prefectures.
The bottom panel of the table presents the random slopes of the partnership statuses.
In the G variance covariance matrix, the (sigma_{int}^{2}) and (sigma_{slope}^{2}) represent variance components for the random intercept and slope, respectively; the ((sigma_{int,slope}}^{2})) are covariance components representing the correlation between the random intercept and the random slope.
The relationships between the random intercept and the random slope were not significant (1.01 0.95,1.08) and there was no heterogeneity across institutions within countries.
A covariance term between individual intercepts and slopes is included in the Random slope-model, hence d.f.
Age and sex slopes fixed to allow convergence in the random slope and intercept model.
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