LMM Assumptions & Write-up

Info sheet · Statistics for Psychology & Neuroscience

Author

Andrew Bell

Published

August 13, 2026

Info sheet 0.2 (draft) · Prerequisites: building & evaluating an LMM; regression assumptions · Give feedback ↗

Working notes for the author — not shown to students once collapsed; remove before publishing.

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What you’ll get from this sheet

The assumptions of a linear mixed model — and how to write one up. By the end you should be able to:

  1. Say which GLM assumptions an LMM keeps and which it relaxes.
  2. Report an LMM analysis clearly.

An LMM keeps three GLM assumptions — normal residuals, linearity, and no excessive multicollinearity — but gets to drop two: independence (it models the dependency via random effects) and homogeneity of variance (it doesn’t partition variance like ANOVA). When reporting, err on the side of more detail — and share your data and code.

Which assumptions carry over?

Recall the five general‑linear‑model assumptions: independence of observations, normal residuals, homogeneity of variance, linearity between predictors and outcome, and no excessive multicollinearity. As if you needed another reason to love mixed models — you get to ditch a couple. Toggle between them:

Switch to the LMM and two rows go green. Independence is dropped because we explicitly model the dependency — in the cosmetic‑surgery example, patients at the same clinic share variance in quality of life, and including clinic as a random effect captures exactly that (a fixed‑effects model just ignored it). Homogeneity of variance is dropped because an LMM doesn’t partition variance the way ANOVA does. The remaining three — normal residuals, linearity, no excessive multicollinearity — still apply, and you test them the usual way: a Q–Q plot (or residuals‑vs‑fitted, or Shapiro–Wilk) for normality, a scatter for linearity, and a correlation matrix / VIF for collinearity. (With a single continuous predictor, collinearity can’t even arise.)

Writing it up

Because LMMs are relatively new, there’s no settled convention yet for running, interpreting, or reporting them — so the safe move is to include as much information as you can. A minimal write‑up:

“I fitted a linear mixed model using the lme4 package (Bates et al.). Quality of life was predicted by surgery and baseline QoL as fixed effects, with a random intercept and slope for clinic. I compared this full model against a null model omitting surgery using a likelihood‑ratio χ² test. Residual normality was assessed via a Q–Q plot; [assumptions]. The comparison was [non]significant, χ²(1) = …, p = …”

Naming the package matters (different packages handle things differently), as does specifying the random‑effects structure and the model comparison. Go further where you can — a figure, the coefficient estimates or their confidence intervals — but above all, share your data and your code. That’s the ultimate “cover‑your‑back” move: if anyone questions the analysis, they can rerun it themselves.

See it in code

library(lme4)
m <- lmer(QoL ~ surgery + baseline + (surgery | clinic), data = cosmo, REML = FALSE)

# check the assumptions that still apply:
qqnorm(resid(m)); qqline(resid(m))     # normality of residuals
plot(m)                                # residuals vs fitted (linearity / outliers)
shapiro.test(resid(m))                 # (formal normality test)
# car::vif(m)                          # multicollinearity — only with 2+ predictors

Your LMM includes clinic as a random effect. Do you still need to worry about the independence assumption?

No — that’s the point of the random effect. Observations from the same clinic aren’t independent (patients there share unmeasured clinic‑level influences), and the random effect for clinic models that dependency explicitly. So the independence assumption that a plain regression would violate is handled by design. You do still check the surviving assumptions: normal residuals, linearity, and (with 2+ predictors) multicollinearity.

“Relaxed” isn’t “ignored”: an LMM only handles the dependency you actually model — if there’s clustering you didn’t put in the random‑effects structure, it’s still lurking. Check residual normality (LMMs are sensitive at the extremes / to outliers) and don’t over‑trust a single fixed‑effect p‑value — judge terms by model comparison. And because reporting norms are still unsettled, under‑reporting is the real risk: name your package, specify the random structure, and share code and data.

Where this shows up next

This closes the mixed‑models thread. For the fitting and evaluation it builds on, see Building & Evaluating a Linear Mixed Model; for the philosophy, the Bayesian sheets offer another way to weigh evidence.