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 restricted maximum likelihood (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.

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

$X$ 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 $\varepsilon$ is the $n$-vector of within-cluster noise.

Two Gaussian assumptions carry the entire model.

$$u \sim N(0, G), \quad \varepsilon \sim N(0, R)$$

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:

$$y \sim N\!\big(X\beta,\;ZGZ^{\top} + R\big)$$

KEY: The marginal covariance $V = ZGZ^{\top} + R$ is what REML and maximum likelihood (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 $\text{Cov}(u, \varepsilon) = 0$ with nothing behind it. The fitted $V = ZGZ^{\top} + R$ assumes zero cross covariance, so justify independence from the design and use residual-spread plots as indirect diagnostics. Random slopes change $Z$ and $G$; a non-diagonal $R$ permits correlation among residuals. Neither change by itself relaxes independence between $u$ and $\varepsilon$.

Bottom line

• Model form: $y = X\beta + Zu + \varepsilon$, fixed effects $\beta$, random effects $u$, residuals $\varepsilon$.
• Distributional core: $u \sim N(0, G)$ and $\varepsilon \sim N(0, R)$, both Gaussian, both mean zero, mutually independent ($\text{Cov}(u, \varepsilon) = 0$).
• "Linear" is in the parameters, not the covariates; splines, polynomials, and interactions are allowed.
• Marginal model: $y \sim N(X\beta,\;ZGZ^{\top} + R)$, 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 $\varepsilon$. 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 2,998 words, 20 min read) adds 8 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• B1

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