Variance explained by a subset of random effects Given a fitted linear mixed model:
$$
y = X\beta + Zu + \epsilon
$$
where $y \in \mathbb{R}^n$, $X \in \mathbb{R}^{n \times p}$ is fixed effects design matrix, $\beta \in \mathbb{R}^p$ fixed effects, $Z \in \mathbb{R}^{n \times m}$ random effect matrix, and $ u \sim N(0, \sigma^2_z)$ , $e \sim N(0, \sigma^2_eI_n)$
Once fit, one can extract the random effects (or essentially the prediction of random effects) (given $y$, $\sigma^2_z$ and $\sigma^2_e$) and it would be something like $u \sim N(m, \Sigma)$ say through conditional distribution $E(u|y)$ and $Var(u|y)$
The question is: How can one estimate the percent variance explained $p$ by a subset of columns of $Z$, say $S$, without refitting the model and using the informtion provided by the fit above which are $\sigma^2_z$, $\sigma^2_e$ and  $u \sim N(m, \Sigma)$

I can think of the following approaches:
Approach 1: I have seen the following approach in the literature (particularly in genomics):
$$ p = \frac{w_2\sigma^2_z}{w_0\sigma^2_z + w_1\sigma^2_e}$$
where $w_1 = 1$ and $w_0 = \frac{1}{n-1}tr(CZZ^T)$ where $C = I_n - \frac{1}{n} 1_n1_n^T$ is the centering matrix. And the reason for doing this is $w_0$ is basically expected value of sample variance of $z \sim N(0, ZZ^T)$. Now as for the $w_2$ and assuming $S$ is a column subset of $Z$, I have $w_2$ being evaluated this way: $w_2 = tr(CSS^T)$
Approach 2: I also see the possibility of applying the same subsetting strategty to the blups. i.e. $u_s \sim N(\mu, V)$ where $u_s$ is the vector made up of elements of $u$ corresponding to the $S$ subset of $Z$. Since have are talking about the affine transformation then $b \sim N(Su_s, SVS^T)$ is the prediction vector of the subset. Now one can move forward with calculation of coefficient of determination and so on.

To estimate the proportion of variance explained by a subset of columns of $Z$, a proper approach would be to compare the residual error variance of the regressions with the full model and the reduced model after dropping the corresponding variables of those columns. But based on the restriction that refitting the model is not permitted, Approach 1 is sensible. There, in the denominator, the expectation of the residual sum of squares is taken. This residual sum of squares is the sum of due to the error component $\epsilon$ and the variance of the random effect $u$. On the numerator, you have the expectation of the explained sum of squares corresponding to the random effect $u$.
