## Estimating HLM Models Using Stata: Part 1

Estimating HLM Models Using Stata: Part 1 The data used in this tutorial can be downloaded from (here)[https://github.com/jeralbri/tutorial-data/tree/master/data]. What follows replicates the results from Raudenbush and Bryk’s (2002, herafter R&B) cannonical text on hierarchical linear models (see especially chapter 4). The Empty Model As a first step, R&B begin with an empty model containing no covariates. $Y_{ij} = \beta_{0j} + e_{ij}$ Each school’s intercept, $$\beta_{0j}$$, is then set equal to a grand mean, $$\gamma_{00}$$, and a random error $$u_{0j}$$.

## Estimating HLM Models Using Stata: Part 2

Estimating HLM Models Using Stata: Part 2 The Means-as-Outcomes Model After estimating the empty model, R&B develop a Means-as-Outcomes model in which a school-level variable, meanses, is added to the model for the intercept. This variable reflects the average student SES level in each school. Recall that, when fitting the empty model, we began with the following equation: $Y_{ij} = \beta_{0j} + e_{ij}$ Now the intercept can be modeled as a grand mean, $$\gamma_{00}$$, plus the effect of the average SES score, $$\gamma_{01}$$, plus a random error, $$u_{0j}$$.

## Estimating HLM Models Using Stata: Part 3

Estimating HLM Models Using Stata Part 3 Random Coefficient Model Next, R&B present a model in which student-level SES is included instead of average SES, and they treat the slope of student SES as random. One complication is that R&B present results after group-mean centering student SES. Group-mean centering means that the average SES for each student’s school is subtracted from each student’s individual SES. The following creates a new, group mean centered SES variable.
Estimating HLM Models Using Stata: Part 4 Intercepts- and Slopes-as-Outcomes R&B present a final model that includes one further generalization of the random coefficients model. Start again with the level-1 model. $Y_{ij} = \beta_{0j} + +\beta_{1j}(\text{SES}_{ij}) + e_{ij}$ Next, model the intercept, $$\beta_{0j}$$, as a function of school-level characteristics mean SES and school type (public or private). \begin{aligned} \beta_{0j} &= \gamma_{00} + \gamma_{01}(\text{Mean SES}_j) + \gamma_{02}(\text{School Type}_j) + u_{0j} \\ \beta_{1j} &= \gamma_{10} + \gamma_{11}(\text{Mean SES}_j) + \gamma_{12}(\text{School Type}_j) + u_{1j} \end{aligned}