Interpret linear mixed model diagnostics and summary statistics to evaluate the linear mixed model structure and variable selection.

A claims-modeling team fits a random-intercept GLMM with policy-year as a grouping factor. The fixed-effect t-statistics look healthy, but the QQ plot of conditional residuals has heavy tails and the random-intercept variance is suspiciously close to zero. Do you trust the coefficients, drop the random effect, or refit with a different covariance structure? LMM diagnostics tell you which.

Why diagnostics matter for LMMs. An LMM has more moving parts than ordinary least squares (OLS): fixed-effect coefficients, variance components, and a covariance structure. Each piece can be wrong independently. Diagnostics partition the question "is this model adequate" into "is the mean structure right," "is the random-effect structure right," and "are the residuals well-behaved."

The summary output gives each ${\hat{\beta }}_j$, a standard error, a t-value (or z for GLMM), and often a degrees-of-freedom approximation (Satterthwaite or Kenward-Roger). Three habits:

• Compare SEs to an OLS fit. If the LMM SEs are much larger, the random effect is absorbing real between-group variation that OLS was wrongly attributing to residuals.
• Watch for inflated df. Naive df equal to total $n$ overstates precision when observations cluster. Satterthwaite df closer to the number of groups is the honest answer for between-group covariates.

Common mistakes

• Comparing REML log-likelihoods across models with different fixed effects. REML conditions on $X$; the comparison is meaningless when $X$ changes. Refit by ML before reading $\Delta$AIC. A wrong answer of $\Delta$AIC = 4.6 based on REML fits is a common trap when the true ML answer differs.
• Using a plain ${\chi }_1^2$ for the random-variance LRT. The boundary makes the reference a 50:50 mixture, which halves the p-value. Reporting a non-significant 0.30 instead of the correct 0.15 is the textbook error.
• Reading an ICC of 0.20 as small. In actuarial data, ICC above roughly 0.05 already inflates OLS Type I error meaningfully. Treat anything north of 0.10 as a structural feature, not a nuisance.

Bottom line

• Two residual layers. Marginal residuals $y-X\hat{\beta }$ check the fixed part. Conditional residuals $y-X\hat{\beta }-Z\hat{b}$ check the within-group fit.
• Random effects need their own QQ plot. Predicted BLUPs ${\hat{b}}_i$ should look Normal; banana, step, or two-cloud patterns flag the wrong grouping or an omitted covariate.
• AIC, BIC, and $-2\log L$ rank models, but compare REML fits only when fixed effects are identical, and ML fits when fixed effects differ.
• LRT for variance components is one-sided. Testing $H_0:{\sigma }_b^2=0$ uses a 50:50 mixture of ${\chi }_0^2$ and ${\chi }_1^2$, not a plain ${\chi }_1^2$.

Exam shortcut

If the problem changes only the covariance structure, score with REML AIC and use the mixture LRT for nested variance components. If the problem changes the fixed-effect set, refit by ML before computing $\Delta$AIC.

The full lesson (about 2,207 words, 15 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• B4

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