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).

\[ \begin{aligned} \beta_{0j} &= \gamma_{00} + \gamma_{01}(\text{Mean SES}_j) + \gamma_{02}(\text{School Type}_j) + u_{0j} \\ \beta_{1j} &= \gamma_{10} + \gamma_{11}(\text{Mean SES}_j) + \gamma_{12}(\text{School Type}_j) + u_{1j} \end{aligned} \]

Substituting:

\[ \begin{aligned} Y_{ij} = &\gamma_{00} +\gamma_{01}(\text{Mean SES}_j) + \gamma_{02}(\text{School Type}_j) + \gamma_{10}(\text{SES}_{ij}) + \\ &\gamma_{11}(\text{Mean SES}_j)(\text{SES}_{ij}) + \gamma_{12}(\text{School Type}_j)(\text{SES}_{ij}) + \\ &u_{0j} + u_{1j}(\text{SES}_{ij}) + e_{ij} \end{aligned} \]

where \(\text{SES}_{ij}\) is again group-mean centered. \((\text{Mean SES}_j)(\text{SES}_{ij})\) and \((\text{School Type}_j)(\text{SES}_{ij})\) represent cross-level interactions estimating how much the effect of student-level SES on math achievement varies by school-level SES and school type. Remaining between-school variability in the outcome is captured with the \(u_{0j}\) random effect, and remaining between-school variability in the effect of student-level SES is captured with the random effect \(u_{1j}\).

The following code fits the model.

mixed mathach meanses schtype grp_cent_ses c.meanses#c.grp_cent_ses i.schtype#c.grp_cent_ses \\\
|| schid: grp_cent_ses , reml cov(un)

Performing EM optimization: 

Performing gradient-based optimization: 

Iteration 0:   log restricted-likelihood = -23252.888  
Iteration 1:   log restricted-likelihood = -23251.837  
Iteration 2:   log restricted-likelihood = -23251.834  
Iteration 3:   log restricted-likelihood = -23251.834  

Computing standard errors:

Mixed-effects REML regression                   Number of obs     =      7,185
Group variable: schid                           Number of groups  =        160

                                                Obs per group:
                                                              min =         14
                                                              avg =       44.9
                                                              max =         67

                                                Wald chi2(5)      =     746.33
Log restricted-likelihood = -23251.834          Prob > chi2       =     0.0000

------------------------------------------------------------------------------------------
                 mathach |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------------------+----------------------------------------------------------------
                 meanses |   5.339119   .3692995    14.46   0.000     4.615306    6.062933
                 schtype |   1.216672    .306386     3.97   0.000     .6161663    1.817177
            grp_cent_ses |   2.938764    .155086    18.95   0.000     2.634801    3.242727
                         |
c.meanses#c.grp_cent_ses |   1.038884   .2988887     3.48   0.001     .4530734    1.624695
                         |
  schtype#c.grp_cent_ses |
                      1  |  -1.642587   .2397809    -6.85   0.000    -2.112549   -1.172625
                         |
                   _cons |   12.11358   .1988076    60.93   0.000     11.72393    12.50324
------------------------------------------------------------------------------------------

------------------------------------------------------------------------------
  Random-effects Parameters  |   Estimate   Std. Err.     [95% Conf. Interval]
-----------------------------+------------------------------------------------
schid: Unstructured          |
               var(grp_ce~s) |   .1012193   .2138705      .0016096    6.365067
                  var(_cons) |    2.38187   .3717503       1.75415    3.234218
         cov(grp_ce~s,_cons) |   .1925536   .2044904     -.2082403    .5933475
-----------------------------+------------------------------------------------
               var(Residual) |   36.72119   .6261423      35.51425    37.96914
------------------------------------------------------------------------------
LR test vs. linear model: chi2(3) = 220.57                Prob > chi2 = 0.0000

Note: LR test is conservative and provided only for reference.

Note that the \(*\) operator in R automatically creates both the main effects and multiplicative terms for the interactions.