I will be referring here to Nakagawa and Schielzeth (2013). As those authors state, $R^2$ for OLS regression could be defined as follows:
$$R^2 = \frac{\sum^n_{i=1}(\bar{y} - \hat{y_i})^2}{\sum^n_{i=1}(y_i - \bar{y})^2} = \frac{var(\hat{y_i})}{var(\hat{y_i})}$$
and so, if we define mixed model like this:
$$y_{ij} = \beta_0 + \sum^p_{h=1}\beta_hx_{hij} + \alpha_{j} + \epsilon_{ij}$$
then $R^2$ could be defined as follows:
$R^2$ for fixed effects ($R^2_m$):
$$R^2_m = \frac{\sigma^2_f}{\sigma^2_f + \sigma^2_\alpha + \sigma^2_\epsilon}$$
where:
$$\sigma^2_f = var\left( \sum^p_{h=1}\beta_hx_{hij}\right)$$
$R^2$ for random effects ($R^2_c$):
$$R^2_m = \frac{\sigma^2_f + \sigma^2_\alpha}{\sigma^2_f + \sigma^2_\alpha + \sigma^2_\epsilon}$$
However, this seems to be the same as ICC:
$$ICC = \frac{\sigma^2_\alpha}{\sigma^2_\alpha + \sigma^2_\epsilon}$$
and so:
$$R^2_c - R^2_m \approx ICC$$
This puzzled me a bit. Could you correct me if I am wrong? Why ICC is the same as $R^2$ for mixed models? If those two measures are equivalent, why are they not used like this? (or if they are, could you provide a reference?) Could you explain why they are equivalent?
Reference: Nakagawa, S. & Schielzeth, H. (2013). A general and simple method for obtaining $R^2$ from generalized linear mixed-effects models. Methods in Ecology and Evolution, 4, 133–142.
