# Interpretation of intercept in random-effects only LMM

I am using LME4 to fit models for a repeated measures study in psychology. Before jumping in to my fixed effects, I decided to start by comparing different random effects structures. I fit a number of models with differing random effects structures, and (thought!) I understood the output. But I noticed (see model output below) that each model yields a different overall intercept. I thought the intercept in the models would be the Grand Mean (M = 59.2994) since no fixed effects were specified. But in that case, shouldn't the intercept be the same for all models?

Here are the four random-effects only models I'm comparing: (subject is a random factor, the fixed factors are prefaced with f_ and refer to aspects of the 'treatment' subjects received).

#RANDOM EFFECTS STRUCTURES
randoms <- list(
"A" = lmer(r_strength ~ (1|subject), data = df_blocks, REML = FALSE),
"B" = lmer(r_strength ~ (1 + f_graph | subject), data = df_blocks, REML = FALSE),
"C" = lmer(r_strength ~ (1 + f_direction | subject), data = df_blocks, REML = FALSE),
"D" = lmer(r_strength ~ (1 + f_version | subject), data = df_blocks, REML = FALSE)
)


And here is a table (using modelsummary) of the estimates:

Note that model B is the maximal theoretically-justified random effect structure. My planned next step was to compare models adding fixed effects and their interactions, but I don't want to push forward until I really understand the output I have :)

(Thanks in advance! I'm a PhD student and long-timer lurker, first time poster. I have learned so much from this community and am so grateful for the time perfect strangers devote to helping others do better statistics! )

• 0. Welcome to the CV.SE community. 1. Fun little question (+1). $x_0$ has a "special meaning" and it is not something to simply marginalise out (under REML or ML estimation). Please see my answer below for more details. Mar 22 at 1:00
• Thank you so much! Mar 23 at 6:50

The fixed intercept (effectively a columns of $$1$$'s in our design matrix $$X$$) is not "just another $$x_j$$" variable. That is because it is estimated at the bottom/$$0$$-th level of our LMM and supersedes any grouping factor we might have. As such, $$\beta_0$$ corresponds to all variables $$x_{ij}$$ being set to 0; in models B/C/D where f_graph, f_direction, f_version are respectively used this changes the interpretation of $$\beta_0$$. And this is why the r_strength ~ (1|subject) leads to the estimate that is closest to the grand mean; in that case $$\beta_0$$ is reflecting the "grand mean" as it marginalises out the random subject-specific variation that is itself hypothesised to be in the form of: $$N(0, \sigma_{\text{Subject}})$$.