Linear Mixed Models (LMM)#

If you would use RM-ANOVA but have missing data, unequal time intervals, or nested structures — then linear mixed models (LMM) are your tool. They are not a complicated specialist method but the natural evolution of ANOVA. In fact, RM-ANOVA is a special case of the LMM.

When to Use#

• You have repeated measures but some data are missing — LMM can handle missing values without excluding entire participants
• Your measurements have unequal time intervals (e.g., assessments at 1, 3, and 12 months)
• Your data are nested (e.g., students in classrooms, patients in clinics)
• You need more flexibility than a traditional ANOVA provides — such as different covariance structures
• You want to model individual trajectories (each person can have their own starting point and rate of change)

Fixed and Random Effects — Simply Explained#

Imagine you are studying learning progress of students across different classrooms:

• Fixed effects answer your research question: "Does the new teaching method improve performance?" — these are the effects you want to generalize to the population.
• Random effects model the structure of your data: "Students in the same classroom are more similar to each other than students from different classrooms." — they capture variation between clusters (e.g., classrooms).

Key insight: Fixed effects = what you care about scientifically. Random effects = the nesting in your data.

Assumptions#

• Linearity — the relationship between predictors and outcome is linear
• Normality of residuals (not the raw data!)
• Normality of random effects
• Independence of observations at the highest level (e.g., between classrooms)

Tip: LMM are robust to mild violations of normality, especially with larger samples. Check residuals with a QQ plot. Unlike RM-ANOVA, no sphericity assumption is needed.

Formula#

The core idea is simple — just add a random component:

$Y_{ij} = (\beta_0 + b_{0j}) + (\beta_1 + b_{1j}) \cdot X_{ij} + \varepsilon_{ij}$

Where:

• $\beta_0$ = fixed intercept (overall mean, fixed effect)
• $b_{0j}$ = random deviation of the intercept for group $j$ (random intercept)
• $\beta_1$ = fixed effect of the predictor (e.g., time)
• $b_{1j}$ = random deviation of the slope for group $j$ (random slope)
• $\varepsilon_{ij}$ = residual

In plain language: Each person (or cluster) gets their own starting point and their own rate of change, but all are drawn from a shared distribution.

RM-ANOVA is a special case of the LMM — with a balanced design, complete data, and compound symmetry covariance structure, both yield identical results.

Example#

Effect Size#

For LMM, two $R^2$ measures are reported (following Nakagawa & Schielzeth):

$R^2_{\text{marginal}} = \frac{\sigma^2_f}{\sigma^2_f + \sigma^2_r + \sigma^2_e}$

$R^2_{\text{conditional}} = \frac{\sigma^2_f + \sigma^2_r}{\sigma^2_f + \sigma^2_r + \sigma^2_e}$

• Marginal $R^2$: Variance explained by fixed effects alone
• Conditional $R^2$: Variance explained by fixed + random effects

Tip: When $R^2_m$ and $R^2_c$ diverge strongly, the random effects are important and a simple regression model would be inadequate.