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Linear and generalized linear mixed effects models

# Linear and generalized linear mixed effects models - 11...

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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 between-group sampling model for group-specific means, in addition to a within-group 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 within-group 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 within-group heterogeneity. 11.1 A hierarchical regression model Let’s 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 school-specific 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 between-school 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 10.1007/978-0-387-92407-6 11, c Springer Science+Business Media, LLC 2009

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196 11 Linear and generalized linear mixed effects models intercept of the regression line can be interpreted as the school-level average math score. -2 -1 0 1 2 20 40 60 80 SES math score ● ● 5 10 15 20 25 30 40 45 50 55 60 65 sample size intercept
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