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.
mod_4 <- lmer(mathach ~ meanses*grp_cent_ses + schtype*grp_cent_ses + (1 + grp_cent_ses | schid), data = hsb)
summary(mod_4)## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: mathach ~ meanses * grp_cent_ses + schtype * grp_cent_ses + (1 +
## grp_cent_ses | schid)
## Data: hsb
##
## REML criterion at convergence: 46503.7
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -3.15921 -0.72319 0.01706 0.75439 2.95822
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## schid (Intercept) 2.3819 1.5433
## grp_cent_ses 0.1014 0.3184 0.39
## Residual 36.7211 6.0598
## Number of obs: 7185, groups: schid, 160
##
## Fixed effects:
## Estimate Std. Error df t value Pr(>|t|)
## (Intercept) 12.1136 0.1988 159.8921 60.931 < 2e-16 ***
## meanses 5.3391 0.3693 150.9689 14.457 < 2e-16 ***
## grp_cent_ses 2.9388 0.1551 139.3042 18.948 < 2e-16 ***
## schtype 1.2167 0.3064 149.5994 3.971 0.000111 ***
## meanses:grp_cent_ses 1.0389 0.2989 160.5528 3.476 0.000656 ***
## grp_cent_ses:schtype -1.6426 0.2398 143.3449 -6.850 2.01e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Correlation of Fixed Effects:
## (Intr) meanss grp_c_ schtyp mns:__
## meanses 0.245
## grp_cent_ss 0.080 0.020
## schtype -0.697 -0.356 -0.056
## mnss:grp_c_ 0.019 0.079 0.282 -0.028
## grp_cnt_ss: -0.056 -0.029 -0.694 0.082 -0.351Note that the \(*\) operator in R automatically creates both the main effects and multiplicative terms for the interactions.
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