# Equation notation for crossed-effect mixed models

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:

$$\log\left[ \frac { P( \text{use} = 1 ) }{ 1 - P( \text{use} = 1 ) } \right] = \alpha_{j[i],k[i]} + \beta_{1}(\text{iv1}) + \beta_{2}(\text{iv2}) + \beta_{3}(\text{iv3}) \\ \text{, for ID j = 1,} \dots \text{,J} \\ \text{, act k = 1,} \dots \text{,K} \\$$

• It's a logistic model, so it's a glmm, not a lmm. May 3 at 19:53
• @PSellaz Ah, yes you are right. I will amended the answer. May 3 at 21:07
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