Interpret output from a linear mixed model and make appropriate choices when evaluating modeling options.

A workers-comp analyst fits losses by claimant within 40 employers. Same industry, same coverage, but employers clearly differ. A linear mixed model lets you keep one fixed-effect coefficient per covariate while letting each employer have its own random intercept.

Why LMM exists. Ordinary least squares assumes independent residuals. Grouped data (claimants within employers, policies within agents, repeated measures within insured) violates that assumption. You can ignore grouping (biased SEs), add a fixed dummy per group (eats degrees of freedom, no shrinkage), or treat group as random (efficient, shrinks group estimates toward the population mean).

$$y = X\beta + Zu + \varepsilon,\quad u\sim N(0,G),\;\varepsilon\sim N(0,R)$$

The marginal distribution is $y\sim N(X\beta,\,V)$ with $V = ZGZ^{\top} + R$. The fixed effects $\beta$ describe how the average group responds to covariates. The random effects $u$ describe how each group deviates from that average.

A typical LMM printout has four blocks:

Fixed effects read like ordinary least squares (OLS): $\hat\beta_1 = 0.42$ means a one-unit increase in $x_1$ lifts predicted $y$ by 0.42, holding other covariates and the group fixed.

Common mistakes

• Comparing REML log-likelihoods across different fixed effects. REML likelihoods are functions of residual contrasts that depend on $X$. Refit with ML, or you will compute an LRT of 0 changes you cannot interpret.
• Using a plain $\chi^2_1$ for the variance-component LRT. This doubles your false-positive rate. The boundary mixture halves the $p$-value tail: $\chi^2_1 = 3.84$ at 5% becomes a critical value of 2.71 for the corrected test.
• Treating $\hat\sigma_u^2$ as a standard deviation. A printout of $\hat\sigma_u^2 = 0.49$ is variance. Group SD is $\sqrt{0.49} = 0.7$. Skipping the square root makes random effects look ten times wider than they are.

Bottom line

• Linear mixed model (LMM): $y = X\beta + Zu + \varepsilon$ with $u\sim N(0,G)$, $\varepsilon\sim N(0,R)$; fixed $\beta$ are population effects, random $u$ are group deviations, and marginal variance is $V = ZGZ^{\top}+R$.
• Variance components $G$ and $R$ drive the intraclass correlation $\rho = \sigma_u^2/(\sigma_u^2+\sigma_\varepsilon^2)$, the between-group share; high $\rho$ means strong within-group clustering.
• BLUPs shrink each group's raw residual mean toward zero by credibility factor $n_i\sigma_u^2/(n_i\sigma_u^2+\sigma_\varepsilon^2)$; shrinkage shrinks as $n_i$ grows.
• REML estimates variance components without bias from estimating $\beta$; use ML when comparing models with different fixed effects via likelihood ratio.

Exam shortcut

If the question asks for a variance ratio or "proportion of variation between groups," go straight to $\rho = \sigma_u^2/(\sigma_u^2+\sigma_\varepsilon^2)$; the answer is rarely more than two arithmetic steps. For any LRT on a random-effect variance, compute the naive $\chi^2$ $p$-value and then halve it.

The full lesson (about 2,905 words, 19 min read) adds 8 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• B3

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