Free CAS MAS-II (Modern Actuarial Statistics II) Linear Mixed Models Practice Questions

D is correct. The standard assumptions of a Gaussian linear mixed model with random intercepts are: (1) the random effects satisfy $u_j \sim N(0, \sigma_u^2)$; (2) the residuals satisfy $\epsilon_{ij} \sim N(0, \sigma_e^2)$ with constant variance; (3) the random effects are independent of the residuals; and (4) observations within a cluster are conditionally independent given the random effects. Orthogonality of the fixed-effect design columns is never required. Correlated predictors simply affect interpretation and standard errors. They do not invalidate the model.

B is correct. The intraclass correlation coefficient equals $\sigma_b^2 / (\sigma_b^2 + \sigma_e^2)$ and represents the share of total variance attributable to between-group differences. A near-zero ICC means the random intercept captures very little additional structure beyond what the residual variance already absorbs, which is evidence that the grouping factor may not be improving the fit relative to a single-level regression. The ICC is a structural diagnostic about variance components; it speaks neither to fixed-effect significance nor to the shape of the random-effects distribution.

D is correct. REML maximizes a restricted likelihood built from residual contrasts that eliminate the fixed effects, and the form of those contrasts depends on the fixed-effect design matrix. As a result, REML log-likelihoods from two models with different fixed-effect specifications are not on the same scale and cannot legitimately be subtracted to form a likelihood ratio statistic. Ordinary maximum likelihood estimates the full joint likelihood and remains comparable across nested fixed-effect specifications, so ML is the appropriate estimator when the hypothesis under test concerns fixed effects. REML is still preferred when the hypothesis concerns variance components and the fixed-effect specification is held constant.