In the linear mixed model, when we fit the model using lmer in the lme4 package in R, we will get the results like:

Random effects:
Groups   Name Variance  Std.Dev. Corr       
subject  X21  8.558e+00 2.925380            
         X22  2.117e-03 0.046011 -1.00      
         X23  2.532e-05 0.005032  1.00 -1.00
Residual      1.453e+00 1.205402            
Number of obs: 100, groups:  subject, 20 

As you may notice I got the -1 and 1 in the correlation matrix of the random effects. I read some of the posts like What to do with random effects correlation that equals 1 or -1? and Random effect equal to 0 in generalized linear mixed model, or Why do I get zero variance of a random effect in my mixed model, despite some variation in the data?

These posts say that the 0 in the covariance matrix or -1 and 1 in the correlation matrix mean that the optimization algorithm hit "a boundary": correlations cannot be higher than +1 or lower than -1. Even if there are no explicit convergence errors or warnings.

I agree with that, but what if I want to simply consider two extreme cases, like say we have the mixed effect model as $$y=X\beta+Z\alpha+\epsilon$$ where $\beta$ is the fixed effect and $\alpha$ is the random effects, and $$\epsilon \sim N(0, V) \text{ and }\alpha \sim N(0, \Omega)$$ One extreme case would be that $\Omega$ is the zero matrix, i.e. $\alpha$ is a zero vector, which implies there is no random effect, so responses even within a subject are independent. Does that mean the other extreme case would be all of the elements in the correlation matrix of the random effects are all 1 or -1, i.e. the random effects are perfect linearly related? Then how do we explain this regarding to the model?



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