I have a design in which each participant is measured at three days. On each day, participants are measured twice.
The dataset is structured as follows: $$ \begin{array}{c|c|c|c} y & \text{Subject ID} & \text{Day} & \text{Repetition} \\ \hline \ldots & 1 & 1 & 1 \\ \ldots & 1 & 1 & 2 \\ \ldots & 1 & 2 & 1 \\ \ldots & 1 & 2 & 2 \\ \ldots & 1 & 3 & 1 \\ \ldots & 1 & 3 & 2 \\ \ldots & 2 & 1 & 1 \\ \ldots & 2 & 1 & 2 \\ \ldots & 2 & 2 & 1 \\ \ldots & 2 & 2 & 2 \\ \ldots & 2 & 3 & 1 \\ \ldots & 2 & 3 & 2 \\ \ldots & \ldots & \ldots & \ldots \\ \end{array} $$
I want to calculate two things:
- Between-day reliability
- Within-day reliability
I though about fitting a linear mixed model for crossed random effects of the form: $$ y_{ijl} = \beta_0 + u_i + v_j + w_l + \epsilon_{ijl} $$ where $u_i\sim \mathrm{N}(0, \sigma^{2}_{\text{ID}})$ is a random effect for the participants and $v_j\sim \mathrm{N}(0, \sigma^{2}_{\text{Day}})$ is a random effect for the days and $w_l\sim \mathrm{N}(0, \sigma^{2}_{\text{Rep}})$ is a random effect for the repetition within each day. $\epsilon_{ijl}\sim \mathrm{N}(0, \sigma^{2})$ is the error term.
I would then estimate the intraclass correlation coefficients (ICC) for the between-day and within-day reliability as follows: $$ \mathrm{ICC}_{between} = \frac{\sigma^{2}_{\text{ID}} + \sigma^{2}_{\text{Rep}}}{\sigma^{2}_{\text{ID}} + \sigma^{2}_{\text{Day}} + \sigma^{2}_{\text{Rep}} + \sigma^{2}} $$ and $$ \mathrm{ICC}_{within} = \frac{\sigma^{2}_{\text{ID}} + \sigma^{2}_{\text{Day}}}{\sigma^{2}_{\text{ID}} + \sigma^{2}_{\text{Day}} + \sigma^{2}_{\text{Rep}} + \sigma^{2}} $$
Given that the number of days (3) and repetitions (2) is very small, the estimation of random effects for these terms is questionable (see also the answer here).
Hence I thought about including the day and repetition as fixed effects.
Question: How would I calculate the within- and between-day reliability when the day and repetition are modelled using fixed effects instead of random effects?
