In education research there is a type of multilevel model referred to as a slopes-as-outcomes model by Raudenbush and Bryk (2002). Raudenbush and Bryk's HLM software uses its own notation in describing multilevel models, but it is often easy to convert into lme4
notation. The slopes-as-outcomes model they describe however, does not seem to have any easy analog. Model 4 at this link shows a worked example using the HLM software. This example fits the following model:
$$
MATHACH_{ij} = β_{0j} + β_{1j} SES + r_{ij}
$$
$$
β_{0j} = γ_{00} + γ_{01}(SCHTYPE) + γ_{02}(MEANSES) + u_{0j}
$$
$$
β_{1j} = γ_{10} + γ_{11}(SCHTYPE) + γ_{12}(MEANSES) + u_{1j}
$$
This model is saying that for each school j
both an intercept term $B_{0j}$ and a slope for socioeconomic status (SES) $B_{1j}$ is estimated. However, both $B_{0j}$ and $B_{1j}$ are predicted by the variables MEANSES
and school type (SCHTYPE
).
In lme4
I would attempt to fit this model by using the following command:
library(lme4)
mod1 <- lmer(MATHACH ~ SES + MEANSES + SCHTYPE + (1 + SES | SCH)
In R I know this will return coefficients for SES
, MEANSES
, and SCHTYPE
as fixed effects, and an intercept and SES
slope for each school as part of the random effects. I also know that the slope and intercept for the schools are estimated conditional on the effects of SCHTYPE
and MEANSES
.
In HLM however, the reported outputs are different. As the example I linked to shows, HLM reports 2 coefficients for MEANSES
and SCHTYPE
— one each for the effect on the $j$ intercepts and $j$ SES
slopes. So instead of an output where the call to fixef(mod1)
looks like this:
fixef(fm1)
(Intercept) SCHTYPE MEANSES
251.40510 10.46729 9.7734
It would look like this:
fixef(fm1)
-- Intercepts --
(Intercept) SCHTYPE MEANSES
251.410 10.49 9.74
-- Slope SES --
(Intercept) SCHTYPE MEANSES
232.40 4.49 8.34
I am not sure how HLM derives these estimates. I am wondering if there is a way within lme4
to partial out the effect of a predictor like SCHTYPE
between the random intercepts and the random slopes. Or if HLM is simply doing some post-estimation regression of SCHTYPE
and MEANSES
on the BLUPS for the slopes and intercepts and reporting those coefficients alongside the general results for the model.
I'd like to replicate this type of analysis but in R to take advantage of R's superior handling of missing data and binomial responses.
This question has previously been raised, and the model form suggested was this:
lmer(mathach ~ sector + meanses + ses + sector:ses + meanses:ses +
(1+ses|school),data=dat)
However, I'm not sure that using this model one could interpret the sector:ses
interaction term as the prediction of the effect of sector
on the school-level ses
slopes. For the analysis I would like to do I'm interested in assessing the magnitude and direction of the impact of school-level indicators on the school intercept and slope estimates.
Best Answer
This may be a late answer but I will try to answer it however. In pages 117-130 of Raudenbush and Bryk (2002), as you rightly mentioned, the slope as outcome model has been represented as
$$ MATHACH_{ij} = β_{0j} + β_{1j} SES + r_{ij} $$
$$ β_{0j} = γ_{00} + γ_{01}(SCHTYPE) + γ_{02}(MEANSES) + u_{0j} $$
$$ β_{1j} = γ_{10} + γ_{11}(SCHTYPE) + γ_{12}(MEANSES) + u_{1j} $$
This translates to a a 2 level HLM model that can be written in lmer as
The authors explain the significant coefficients in Page 128 which are nothing but the interaction effects coefficients.
I came across a webpage that describes Bayesian meta-regression where they model the slopes of the hierarchical model, which is what you had in mind perhaps.