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Unformatted text preview: 11 Linear and generalized linear mixed effects models In Chapter 8 we learned about the concept of hierarchical modeling, a data analysis approach that is appropriate when we have multiple measurements within each of several groups. In that chapter, variation in the data was rep resented with a betweengroup sampling model for groupspecific means, in addition to a withingroup sampling model to represent heterogeneity of ob servations within a group. In this chapter we extend the hierarchical model to describe how relationships between variables may differ between groups. This can be done with a regression model to describe withingroup variation, and a multivariate normal model to describe heterogeneity among regression coefficients across the groups. We also cover estimation for hierarchical gen eralized linear models, which are hierarchical models that have a generalized linear regression model representing withingroup heterogeneity. 11.1 A hierarchical regression model Lets return to the math score data described in Section 8.4, which included math scores of 10th grade children from 100 different large urban public high schools. In Chapter 8 we estimated schoolspecific expected math scores, as well as how these expected values varied from school to school. Now sup pose we are interested in examining the relationship between math score and another variable, socioeconomic status (SES), which was calculated from parental income and education levels for each student in the dataset. In Chapter 8 we quantified the betweenschool heterogeneity in expected math score with a hierarchical model. Given the amount of variation we ob served it seems possible that the relationship between math score and SES might vary from school to school as well. A quick and easy way to assess this possibility is to fit a linear regression model of math score as a function of SES for each of the 100 schools in the dataset. To make the parameters more interpretable we will center the SES scores within each school separately, so that the sample average SES score within each school is zero. As a result, the P.D. Hoff, A First Course in Bayesian Statistical Methods , Springer Texts in Statistics, DOI 10.1007/9780387924076 11, c Springer Science+Business Media, LLC 2009 196 11 Linear and generalized linear mixed effects models intercept of the regression line can be interpreted as the schoollevel average math score.21 1 2 20 40 60 80 SES math score 5 10 15 20 25 30 404550556065 sample size intercept...
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This note was uploaded on 11/24/2010 for the course STAT 201a taught by Professor Wu during the Spring '10 term at Pasadena City College.
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