I see models for "mixed effects" (i.e., models with fixed as well as random factors) specified in the literature in two ways, and I'd like to understand the conditions under which they are equivalent...
Model Formulation 1:
$$ Y = \ X \beta + \ Zb +\epsilon $$ $$ \epsilon \thicksim N(0,\sigma^2 I) $$ $$ b \thicksim N(0,\Sigma)$$ $$ Cov(\epsilon,b)=0 $$ Here the random effects are specified explicitly. The distribution for $ Y $, which can be written $Y \thicksim N(X \beta+\ Z b, \sigma^2 I)$, is sometimes called "conditional on $b$."
Model Formulation 2:
$$ Y \thicksim N(X \beta, \Gamma) $$ $$ \Gamma = Z \Sigma Z' + \sigma^2 I $$
Here, the random effects are specified implicitly via $Z$ and the elements of covariance matrix $ \Gamma $, the expression for which can be derived from the assumptions in Model 1. This formulation is like that used for "Generalized Least Squares."
These two different formulations are sometimes used interchangeably. E.g., Rencher and Schaalje, 2008. On page 480, the mixed model specification is like Model 1, whereas on page 486, expression (17.3), it is like Model 2. In the latter case, it is being used for the exposition of residual maximum likelihood (REML).
My concern is, in using Model 2 instead of Model 1, is one giving up some degrees of freedom in the calculations? Or are these two formulations completely equivalent, i.e., both leading to the same results for $\Sigma$ and $\beta $ as a function of the observed data for $Y$, $X$, and $Z$?