Free CAS MAS-II (Modern Actuarial Statistics II) Linear Mixed Models Practice Questions
Linear mixed models on CAS Exam MAS-II cover random-effects assumptions, hierarchical and nested grouping structures, REML and ML estimation, BLUP predictions, residual and Q-Q diagnostic interpretation, variance-component identification, and intraclass correlation (CAS).
Sample Questions
Question 1
Easy
Which of the following is NOT a standard assumption of a linear mixed model with random intercepts?
Solution
E is correct. The standard assumptions of a Gaussian linear mixed model with random intercepts are: (1) the random effects satisfy ; (2) the residuals satisfy 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.
Question 2
Medium
An actuary reports that a random-intercept linear mixed model fit to loss ratios grouped by territory has an estimated intraclass correlation coefficient that is close to zero. Which of the following is the most appropriate interpretation?
Solution
B is correct. The intraclass correlation coefficient equals 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.
Question 3
Hard
An actuary wishes to use a likelihood ratio test to compare two nested linear mixed models that share the same random-effects structure but differ in their fixed-effect specifications. Which of the following best describes the most appropriate estimation choice and the reason for it?
Solution
B 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.