Understand the assumptions behind the linear mixed model design.

A reinsurer fits a random-intercept model for loss ratios across 30 cedents observed over 5 years. The residual plot fans out and the random-intercept variance sits at the REML boundary of zero. Which assumptions just broke?

A linear mixed model writes the response as a fixed part plus a random part plus noise.

$$y=X\beta +Zu+\epsilon$$

$X$ is $n\times p$, $Z$ is $n\times q$, $\beta$ is the $p$-vector of population parameters, $u$ is the $q$-vector of cluster-level deviations, and $\epsilon$ is the $n$-vector of within-cluster noise.

Two Gaussian assumptions carry the entire model.

$$u\sim N(0,G),\epsilon \sim N(0,R)$$

Both have mean zero. The fixed intercept inside $X\beta$ absorbs the population level; random effects encode deviations from that level. A nonzero estimated mean for $u$ signals a coding error.

The implied marginal distribution of $y$ is:

$$y\sim N(X\beta ,ZG{Z}^{\top }+R)$$

KEY: The marginal covariance $V=ZG{Z}^{\top }+R$ is what REML and ML actually fit. Different $(G,R)$ pairs can yield the same $V$, which is why some random-effect structures are unidentifiable.

Common mistakes

• Forgetting the mean-zero part. Candidates list "normal random effects" without mean zero. The zero mean is what lets the fixed intercept identify the population mean.
• Treating "linear" as covariate-linear. Polynomial and spline terms are valid because linearity is in $\beta$. Rejecting an LMM because the curve bends is the canonical wrong call.
• Asserting $\text{Cov}(u,\epsilon )=0$ with nothing behind it. The assumption is not identifiable from the fitted $V=ZG{Z}^{\top }+R$, so credit comes from the indirect argument rather than a formal test: residual spread flat across predicted $u_i$, plus a design reason the cluster effect and the within-cluster noise should be unrelated.

Bottom line

• Model form: $y=X\beta +Zu+\epsilon$, fixed effects $\beta$, random effects $u$, residuals $\epsilon$.
• Distributional core: $u\sim N(0,G)$ and $\epsilon \sim N(0,R)$, both Gaussian, both mean zero, mutually independent ($\text{Cov}(u,\epsilon )=0$).
• "Linear" is in the parameters, not the covariates; splines, polynomials, and interactions are allowed.
• Marginal model: $y\sim N(X\beta ,ZG{Z}^{\top }+R)$, the form ML and REML fit directly.

Exam shortcut

When the prompt asks for "the assumptions," always include zero mean and Gaussian for both $u$ and $\epsilon$. Most candidates remember one; full credit requires both. When a residual-vs-fitted plot funnels, name homoscedasticity of $R$ as the violation, not the random-effect distribution. When the random-effect variance estimate hits the zero boundary, suspect fixed-part mis-specification or residual heteroscedasticity before concluding "no clustering."

The full lesson (about 1,918 words, 13 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• B1

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