# Generalized linear (logit) mixed-effects model with the random (crossed) effects drawn from a bivariate normal distribution

I am trying to implement a generalized mixed-effects model specified as:

Dependent variable $$y = \log(\frac{p}{1 - p})$$ where $$p$$ is a quantity measured for a pair of individuals ($$i$$ and $$j$$).

$$E(y_{ij} \mid X_{ij}, a_i, b_j) = \beta X_{ij} + a_i + b_j,$$

where $$X_{ij}$$ represents my covariates (predictor or independent variables) for individuals $$i$$ and $$j$$, $$\beta$$ represents my fixed effects regression coefficients, and $$a_i$$ and $$b_j$$ represent the random effects (crossed effects) for the two individuals involved (every data point represents some feature values or measurements for a pair of individuals and we want to control for the idiosyncratic contributions of the individuals involved in a pair by conditioning on these individuals via the random crossed effects).

$$(a_i, b_i) \sim MVN(0, \Sigma), \quad \Sigma = \ \left[ {\begin{array}{cc} \sigma_a^2 & \rho\sigma_a\sigma_b \\ \rho\sigma_a\sigma_b & \sigma_b^2 \\ \end{array} } \right].$$

This model is fit using the Laplace approximation, and the authors of the paper whose model I am following use the lme4 package in R (I am open to both R and a way to do this in Python).

I am unable to understand how to implement these crossed effects for the individuals with the generalized mixed-effects model. I have my $$y$$ and $$X$$ in place and have the IDs for each pair and each individual making up a particular pair. I feel like I need to use the individual IDs for my random effects here, but I am not sure how. Most default examples seem to add a random intercept or random slopes corresponding to the covariates specified in $$X$$ but this does not seem to be the case here.

If anyone has any pointers about how to implement the above-specified model using lmer, it would help a lot. I hope I specified the model clearly enough and would be happy to provide an application scenario or point to the paper which describes and uses this model if that helps.

• This is a very good question. However, I'm not sure if it's necessarily a question for StackExchange, unfortunately. The question does seem to center around programming. You don't appear to have any statistical issues you need clarified. It may be worth asking the authors of the package lme4. – Weiwen Ng Oct 22 '19 at 19:45
• Thanks @WeiwenNg! I might try that route too. I think this is in keeping with the many conceptual questions asked too since I do need the formulation for the problem and any further pointers relating to how to implement it in lme4 using Laplace approximation would be an additional help. – Pranav Goel Oct 22 '19 at 22:30