Suppose that one investigates the two sources of variability with data $y_{ij}$ acquired from $j$th subject under $i$th session ($i=1,2; j=1,2..., n$). A linear mixed-effects model can be formulated as follows,
$$y_{ij} = \alpha_0 + b_i + c_j + \epsilon_{ij}$$
where $\alpha_0$ is a constant, $b_i$ and $c_k$ are the random effects for $i$th session and $j$th subject respectively, and $\epsilon_{ij}$ is a residual term. With assumptions of $b_i ~\sim N(0, \tau_1^2)$, $c_i ~\sim N(0, \tau_2^2)$, and $\epsilon_{ij} ~\sim N(0, \sigma^2)$, the intraclass correlation (ICC) values for sessions and subjects can be defined respectively as
$$ICC_{session} = \frac{\tau_1^2}{\tau_1^2+\tau_2^2+\sigma^2}, ICC_{subject} = \frac{\tau_2^2}{\tau_1^2+\tau_2^2+\sigma^2}.$$
And the above ICC values can be obtained through lmer in R package lme4. I have two questions:
Is there a way to test the significance of the above ICC values in the context of LME model, similar to the $F$-stat for ICC defined under random-effects ANOVA?
Intuitively the bigger the variability between the two sessions (more different between the two sessions), the higher the session ICC. But how is this intuition consistent with the notion that ICC measures reliability, reproducibility, or consistency? Is the session reliability (or reproducibility, consistency) about any two subjects' responses coming from the same session, not about the difference between the two sessions?
