I have conducted a crossed-effect logistic regression using Stata's meqrlogit command and have been asked to provide the equation notation. I have participants (ID) who each rated 8 activities (act) and I am looking at the impact of 3 independent variables (iv1, iv2, iv3) on the outcome (use). Can anyone assist with what the equation would look like for this model?
meqrlogit use iv1 iv2 iv3 || _all: R.act || ID:
This is a logistic mixed effects model and it's equation can be written as:
$$\\log\left[ \frac { P( \text{use} = \text{1} ) }{ 1 - P( \text{use} = \text{1} ) } \right] =\mathbf{X \beta} + \textbf{Zb}$$
where $\mathbf{X}$ is the model matrix for the fixed effects, $\mathbf{\beta}$ is the fixed effects coefficient vector, $\textbf{Z}$ is the model matrix for the random effects, $\textbf{b}$ is the random effects vector and $\epsilon$ is the error term vector.
Another way to write this, using multilevel modelling-type notation, would be:
$$ \log\left[ \frac { P( \text{use} = 1 ) }{ 1 - P( \text{use} = 1 ) } \right] = \alpha_{j[i],k[i]} + \beta_{1}(\text{iv1}) + \beta_{2}(\text{iv2}) + \beta_{3}(\text{iv3}) \\ \text{, for ID j = 1,} \dots \text{,J} \\ \text{, act k = 1,} \dots \text{,K} \\ $$