Dear community,
I guess my questions are pretty basic, but I didn’t manage to figure them out by myself. They all around individual formulas from a hierarchical model.
Say I have a model with three continuous variables and several observations for each subject.
library(tidyverse)
library(datawizard)
library(brms)
set.seed(1234)
Data < data.frame(DV = rnorm(80),
A = rnorm(80),
B = rnorm(80),
C = rnorm(80),
Subj = as.factor(rep(c(1:5),16))
)
Data2Model < Data %>%
group_by(Subj) %>%
mutate(
A = standardise(A), # Center variables per subject
B = standardise(B),
C = standardise(C)
) %>%
ungroup()
Model <
brm(
data = Data2Model,
formula = DV ~ A + B + C + (A+B+CSubj),
chains = 4,
cores = 4
)
As far as I understand, I can get the individual formulas for each subject using this:
coef(Model) %>% as.data.frame() %>% select(contains("Estimate"))
# Subj.Estimate.Intercept Subj.Estimate.A Subj.Estimate.B Subj.Estimate.C
# 1 0.022104393 0.0009816632 0.13212726 0.47990893
# 2 0.012057911 0.0131331115 0.06024109 0.07725161
# 3 0.034164734 0.0717136684 0.06014023 0.33624930
# 4 0.017360800 0.0356747686 0.05482032 0.18452892
# 5 0.004253234 0.0328254212 0.13192382 0.01854751
The formula for Subject 1 can be described as:
and for Subject 2:
My questions are:

Is there a measure for how these formulas are different from one another? Is that, in practice, the adjusted ICC?

Is there a way to inspect whether the individual formulas given by the HLM predict better than if I would extract the formulas from 5 single regression models (one for each subject)? I would like to measure what is the contribution of the shrinkage and what are the benefits of the use of the HLM to compute the coefficients?

It is more like a crossvalidation question (actually, maybe they all are). I am interested in how much the individual formulas are individualised. How can I inspect whether the formula’s coefficients of subject 1 will better predict the data of subject 1 and not the coefficients formula of subject 2 for the data of subject 1?
Thanks a lot,
SH