Where: - \( Y_{ij} \) is the outcome for the \( i \)-th individual in the \( j \)-th group. - \( \beta_0 \) and \( \beta_1 \) are the fixed effects coefficients. - \( X_{ij} \) is the predictor variable. - \( u_j \) is the random effect for the \( j \)-th group, assumed to follow a normal distribution with mean 0 and variance \( \sigma_u^2 \). - \( \epsilon_{ij} \) is the residual error, assumed to follow a normal distribution with mean 0 and variance \( \sigma^2 \).
