Effect in linear model versus effect in mixed model

Question

Consider a dataset with 3 observations pertaining to 5 patients. This can be modeled in several ways, two of which are that

$$ X_{ij} = \xi_i + Y_j + \epsilon_{ij}, $$ $i = \{1,..,3\}$, $j = \{1,...,5\}$ where $\xi$ is the mean value associated with each observation, $Y_j$ is a $\mathcal{N}(0, \nu^2)$ distributed variable corresponding to random patient effects, and $\xi_{ij}$ is the $\mathcal{N}(0,\sigma^2)$ fixed residual, and the other is simply that $$ X_{ij} = \xi_i + \gamma_j + \epsilon_{ij}, $$ i.e., that there's a different mean value for each patient, and then the fixed noise effect.

Question: If I fit both models and am interested in the "effect" a patient has, in the first model I get a variance, while the other gives me an estimated individual effect. Is there a way I can compare the effects of patients estimated by the two models? I don't quite see how I can compare a mean value parameter to a variance parameter....?

Answer 1

You can compare the "BLUPS" or "conditional modes" in the mixed model, which are the predicted (not estimated) values of $Y_j$, with the fixed-effect values in the other model. In R, you would typically fit the fixed-effect model with lm() (or nlme::gls()) [and extract the coefficients with coef()] and the mixed model with lme4::lmer or nlme::lme [and extract the BLUPs/conditional modes with ranef()].