Equivalence of Fixed Effects in Contextual Models with and without Random Slopes

Question

When estimating "contextual models" (i.e., models that contain level-1 predictors as well as their cluster means on level-2), the estimation of the fixed effects should be unaffected by the centering scheme (grand mean centering, cgm, vs. group mean centering, cwc), according to the (psychological) literature (see, e.g., Kreft et al., 1995, or Enders & Tofighi, 2007).

However, if I try this in R with lme4, this is (numerically) true only if the models are specified without random slopes.

Moreover, the well-known relationship for the context effect (or its variants), $\gamma_{10}^{cgm}+\gamma_{01}^{cgm}=\gamma_{01}^{cwc}$, also only holds in this case (see, e.g., Enders & Tofighi, 2007).

Enders and Tofighi (2007) write: Because of a lack of precision in the estimation process, the algebraic identities outlined by Kreft et al. (1995) may not hold exactly when substituting the estimated parameter values into the equations outlined in this section.

My questions are:

1. Why is this so?
2. Why is the imprecision only there for models without random slopes?
3. Which of the two is (cgm or cwc) is the "better" estimate?

Also note that the problem remains when using ML and not REML for estimation.

Example code (Example comes from here)

library(dplyr)
library(lme4)

AMIB <- read.csv(file = url("https://quantdev.ssri.psu.edu/sites/qdev/files/AMIBshare_daily_2019_0501.csv"), header = TRUE) %>% 
  select(id, negaff, pss) %>% 
  mutate(stress = 4 - pss) %>% 
  mutate(stress_trait = mean(stress, na.rm = TRUE), .by = id) %>% 
  mutate(stress_trait_c = stress_trait - mean(stress_trait)) %>% 
  mutate(stress_cgm = stress - mean(stress, na.rm = TRUE),
         stress_cwc = stress - stress_trait)

ri_cgm <- lmer(negaff ~ stress_cgm + stress_trait_c + (1 | id), data = AMIB)
ri_cwc <- lmer(negaff ~ stress_cwc + stress_trait_c + (1 | id), data = AMIB)

fixef(ri_cgm)
sum(fixef(ri_cgm)[-1])
fixef(ri_cwc)

rs_cgm <- lmer(negaff ~ stress_cgm + stress_trait_c + (stress_cgm | id), data = AMIB)
rs_cwc <- lmer(negaff ~ stress_cwc + stress_trait_c + (stress_cwc | id), data = AMIB)

fixef(rs_cgm)
sum(fixef(rs_cgm)[-1])
fixef(rs_cwc)

Output:

> fixef(ri_cgm)
   (Intercept)     stress_cgm stress_trait_c 
     2.4567674      0.8429788      0.1996167 
> sum(fixef(ri_cgm)[-1])
[1] 1.042596
> fixef(ri_cwc)
   (Intercept)     stress_cwc stress_trait_c 
     2.4587394      0.8429788      1.0425955 
> 
> fixef(rs_cgm)
   (Intercept)     stress_cgm stress_trait_c 
     2.4473444      0.7803662      0.2232565 
> sum(fixef(rs_cgm)[-1])
[1] 1.003623
> fixef(rs_cwc)
   (Intercept)     stress_cwc stress_trait_c 
      2.459474       0.775756       1.019996 

Literature

Kreft, I. G., De Leeuw, J., & Aiken, L. S. (1995). The effect of different forms of centering in hierarchical linear models. Multivariate behavioral research, 30(1), 1-21.

Enders, C. K., & Tofighi, D. (2007). Centering predictor variables in cross-sectional multilevel models: a new look at an old issue. Psychological methods, 12(2), 121.

Answer 1

The "better" estimate question is worth addressing separately from the precision questions. An excellent paper by Lüdtke and colleagues (2009) in Contemporary Educational Psychology explains why one might prefer grand- or group-mean centering in the context of using student reports about classroom instruction as predictors. As an applied example, it shows you how there is no absolute "better" estimate, rather there are "better" estimates given the question at hand and the quantity of interest.

As I see it, the group-mean centering (cwc in Enders & Tofighi, 2007) is about understanding the within person (longitudinal) or within group (cross-sectional) association between a predictor and the outcome. You isolate the dynamic aspect of the predictor from the trait-like aspect. By subtracting the person or group mean, this has an additional benefit of removing a potential correlation between the level 2 intercept (error term) and the uncentered or grand-mean centered version of the predictor. This is termed endogeneity in econometrics. See Hamaker & Muthén's 2020 paper for a nice explication of the relation between centering within groups/clusters in random effects models and fixed effect models. They also break down how the models with and without random slopes are not equivalent.

The problem with group-mean centering is mostly about interpretability. There is no common 0 for the cwc variable. The 0 value depends on which group an individual is in. In contrast, centering around a grand mean makes interpretability much easier - everyone has the same theoretical 0 value (when their score is equivalent to some constant, whether that is the mean or some value of the predictor). And as long as you add the person/cluster mean variable as a predictor to the model, you remove the potential for endogeneity.