I have to calculate intraclass correlation to quantify intra and inter-reliability. However, I am not totally sure whether the classical two-way ICC is appropriate because of the correlation among sample units.
In the experiment, each patient has more than one vessel that will be evaluated by 2 readers. The sample unit is the vessel.
The classical two-way ICC is calculated based on the model: $$ y_{ij} = \mu + s_i + r_j + \epsilon_{ij} $$ where $$ s_i \stackrel{i.i.d.}{\sim} N(0, \sigma^2_s) \mbox{ is a random effect for the vessel i;}\\ r_j \stackrel{i.i.d.}{\sim} N(0, \sigma^2_r) \mbox{ is a random effect for the rater j;} \\ \epsilon_{ij} \stackrel{i.i.d.}{\sim} N(0, \sigma^2) \mbox{ is a random effect for the measurement ij,} $$ then $$ ICC = \frac{\sigma^2_s}{\sigma^2_s + \sigma^2_r + \sigma^2}. $$
However, the random effects $s_i$ are not independent since they are from the same patient. I think it is necessary to include a random effect for the patients. In this way, the random effects $s_i$ could be considered independent as described below:
$$ y_{kij} = \mu + p_k + s_i + r_j + \epsilon_{kij} $$ where $$ p_k \stackrel{i.i.d.}{\sim} N(0, \sigma^2_p) \mbox{ is a random effect for the patient k;}\\ s_i \stackrel{i.i.d.}{\sim} N(0, \sigma^2_s) \mbox{ is a random effect for the vessel i;}\\ r_j \stackrel{i.i.d.}{\sim} N(0, \sigma^2_r) \mbox{ is a random effect for the rater j;} \\ \epsilon_{kij} \stackrel{i.i.d.}{\sim} N(0, \sigma^2) \mbox{ is a random effect for the measurement kij,} $$ then $$ ICC = \frac{\sigma^2_s}{\sigma^2_p + \sigma^2_s + \sigma^2_r + \sigma^2}. $$
Does it make sense?