Linear Mixed Models

Why Mixed Models?

When you measure the same people repeatedly, your observations aren't independent. A person who scores high at Time 1 will likely score high at Time 2. Standard regression assumes each data point is independent—violating this assumption inflates your confidence and produces misleading p-values.

Consider a simple example: 200 participants measured at 5 time points each. You have 1,000 observations, but they're not 1,000 independent pieces of information. The 5 observations from Person 1 are more similar to each other than to observations from Person 47.

Linear Mixed Models (LMM) solve this by explicitly modeling the dependency structure. The "mixed" refers to a mixture of:

• Fixed effects: Parameters that apply to everyone (average intercept, average slope)
• Random effects: Parameters that vary across individuals (person-specific deviations)

Quick mental model: Think of LMM as "fit one regression line per person, then summarize those lines." Each individual gets their own intercept and slope, but these estimates are informed by the whole sample—extreme values get pulled toward the group average, and people with sparse data borrow strength from others.

What LMM Provides

LMM offers several capabilities that make it well-suited for longitudinal analysis:

Individual Trajectories Within a Unified Model

Each person gets their own intercept and slope, but these aren't estimated in isolation. The model "borrows strength" across individuals, leading to more stable estimates—especially for people with few observations or extreme values.

Handles Unbalanced Data Gracefully

Real longitudinal data are messy: participants miss waves, measurement timing varies, some people drop out. LMM handles all of this naturally. Unlike repeated measures ANOVA, you don't need complete data from everyone.

Flexible Time Structures

Time can be equally spaced (0, 1, 2, 3, 4), unequally spaced (0, 3, 6, 12, 24 months), or person-specific (actual dates of measurement). Just code time as a continuous variable with appropriate values.

For person-specific timing, set time = 0 at a common within-person reference (e.g., each person's baseline/first visit) so the intercept is interpretable.

Separates Within- and Between-Person Variation

LMM explicitly partitions variance into:

• Between-person: How much do people differ from each other?
• Within-person: How much does each person vary around their own trajectory?

This distinction is fundamental to understanding longitudinal change.

Natural Framework for Predictors

Adding predictors is straightforward:

• Time-invariant predictors (e.g., treatment group, gender): Predict individual differences in trajectories
• Time-varying predictors (e.g., daily stress, current medication): Predict occasion-specific deviations

For time-varying predictors, separate within-person effects (xᵢₜ − x̄ᵢ) from between-person effects (x̄ᵢ) to avoid conflating processes. Report both coefficients explicitly.

When LMM is Appropriate

LMM works well when you have:

Example scenarios where LMM shines:

• Therapy outcome study: 150 participants complete depression inventories at 8 weekly sessions. Some miss sessions. You want to estimate average symptom reduction and identify who improves faster.
• Cognitive aging study: 300 participants complete memory tests every 2 years over a decade. Timing varies slightly. You want to model decline and test whether education predicts slower decline.
• Intensive longitudinal diary study: 80 participants report daily mood for 30 days. You're interested in how daily stress relates to mood within persons, while capturing stable individual differences.

Key Components

The Two-Level Structure

Longitudinal data have a natural hierarchy:

Level 2: Persons (i = 1, 2, ..., N)
    │
    └── Level 1: Observations within persons (t = 1, 2, ..., Tᵢ)

Level 1 describes what happens within each person over time. This is where change lives.

Level 2 describes how people differ from each other. This is where individual differences live.

LMM simultaneously answers two questions: How do people change on average? (fixed effects) and How do individuals differ in that change? (random effects).

Fixed Effects

What they are: Population-average parameters—the intercept and slope that describe the "typical" trajectory.

What they tell you: "On average, people start at X and change by Y per unit time."

Notation: γ₀₀ (average intercept), γ₁₀ (average slope)

Random Effects

What they are: Individual-specific deviations from the fixed effects. Not estimated directly as parameters, but characterized by their variance.

What they tell you: "People vary around the average intercept with SD = X. People vary around the average slope with SD = Y."

Notation: u₀ᵢ, u₁ᵢ (the deviations); τ₀₀, τ₁₁ (their variances)

Key insight: We don't estimate a separate intercept for each person as a "parameter." Instead, we estimate the average intercept (fixed) and the variance of intercepts across people (random). Person-specific estimates are derived quantities (BLUP/EBLUP estimates), not directly estimated coefficients.

The Combined Model

The Level 1 and Level 2 equations combine into:

yᵢₜ = γ₀₀ + γ₁₀(Time) + u₀ᵢ + u₁ᵢ(Time) + εᵢₜ
      \_____________/   \_______________/   \__/
       Fixed effects    Random effects     Residual

This is the "mixed" model: fixed effects (γ) plus random effects (u) plus residual (ε).

When to Treat an Effect as Random

In longitudinal analysis, person is almost always random (you want to generalize beyond your specific participants). Time is typically fixed when modeled as a continuous covariate.

Building Intuition: From Simple to Complex

Let's build intuition by starting simple and adding complexity.

Model 0: No Random Effects (What NOT to Do)

Ignore that observations are nested—one intercept, one slope for everyone:

yᵢₜ = β₀ + β₁(Time) + εᵢₜ

What this assumes: Observations are independent. Everyone follows the exact same trajectory. All variation is random noise.

Problem: Standard errors are wrong because we're treating correlated observations as independent.

Model 1: Random Intercept Only

Allow intercepts to vary, but everyone shares the same slope:

yᵢₜ = [γ₀₀ + u₀ᵢ] + γ₁₀(Time) + εᵢₜ

Each person has their own starting level, but lines are parallel—everyone changes at the same rate.

When this is enough: If you have few time points, or if slope variance is genuinely small. Estimating a random slope typically benefits from ≥4 observations per person; with fewer, consider random-intercept-only.

Model 2: Random Intercept and Slope

Allow both to vary:

yᵢₜ = [γ₀₀ + u₀ᵢ] + [γ₁₀ + u₁ᵢ](Time) + εᵢₜ

Each person has their own intercept AND slope. Lines can have different starting points and different angles—they're non-parallel and can cross.

The correlation: Random intercepts and slopes can be correlated. Negative correlation means high starters tend to have flatter slopes (or steeper declines).

Visualizing the Difference

Random Intercept Only          Random Intercept + Slope
(parallel lines)               (non-parallel lines)

    │    ╱                         │      ╱
    │   ╱                          │   ╱╱
    │  ╱                           │  ╱ ╱
    │ ╱                            │ ╱  ╱
    │╱                             │╱  ╱
    └──────────                    └─────────
       Time                           Time

All same slope,                  Different people,
different intercepts             different slopes

Shrinkage and Partial Pooling

This concept is unique to mixed models and represents one of their key advantages.

The Problem with Extreme Estimates

Suppose Person A has only 2 observations, both very high. If we estimate their trajectory independently (OLS), we'd get an extreme intercept based on just 2 points.

But we have information from 199 other people. Shouldn't that tell us something?

What Shrinkage Does

Mixed models "shrink" extreme individual estimates toward the group mean. The amount of shrinkage depends on:

1. How extreme the person's data are: More extreme = more shrinkage
2. How reliable the person's data are: Fewer observations = more shrinkage
3. How variable the population is: Less population variability = more shrinkage

Three Approaches to Individual Estimation

LMM's partial pooling gives more weight to an individual's data when they have many observations and the data are consistent, and more weight to the group mean when observations are sparse or variable.

Why This Matters

1. Better individual predictions: Shrunken estimates are typically more accurate than raw OLS estimates
2. Handles sparse data: People with few observations still get reasonable estimates
3. Automatic regularization: Protects against overfitting

yᵢₜ = β₀ᵢ + β₁ᵢ(Timeₜ) + εᵢₜ

β₀ᵢ = γ₀₀ + u₀ᵢ
β₁ᵢ = γ₁₀ + u₁ᵢ

[u₀ᵢ]     [0]   [τ₀₀  τ₀₁]
[u₁ᵢ] ~ N([0], [τ₁₀  τ₁₁])

yᵢₜ = γ₀₀ + γ₁₀(Time) + u₀ᵢ + u₁ᵢ(Time) + εᵢₜ

Var(yᵢₜ) = τ₀₀ + 2t·τ₀₁ + t²·τ₁₁ + σ²

Cov(yᵢₜ, yᵢₜ') = τ₀₀ + (t + t')·τ₀₁ + (t × t')·τ₁₁

Interactive Exploration

This tool lets you:

• Adjust fixed effects (average intercept and slope) and see the mean trajectory change
• Modify random effect variances and watch individual trajectories spread out or converge
• Change the intercept-slope correlation to see fan-in or fan-out patterns
• Observe how the ICC changes as you adjust between- and within-person variance

Practical Considerations

Data Format

LMM requires long format: one row per observation.

Wide format (won't work):        Long format (required):

id   y_t1  y_t2  y_t3             id   time   y
1    48    52    56               1    0      48
2    55    54    57               1    1      52
                                  1    2      56
                                  2    0      55
                                  2    1      54
                                  ...

Time Coding

How you code time affects interpretation:

Recommendation: Start with zero at baseline (0, 1, 2, ...). Adjust if your research question focuses on a different time point.

Rescaling time (e.g., months → years) rescales the slope and its variance; choose units that are meaningful for interpretation.

Missing Data

LMM handles missing observations gracefully:

• Uses all available outcome data under ML (observations can differ per person)
• Assumes MAR given the variables in the model; include auxiliary covariates when plausible
• Rows with missing predictors are typically dropped; consider imputation or modeling missingness

Caution: If dropout is related to the outcome trajectory itself (MNAR), estimates may be biased. Consider sensitivity analyses.

Intraclass Correlation (ICC)

The ICC tells you what proportion of total variance is between-person (stable individual differences) versus within-person (change + noise).

An ICC of 0.65 means 65% of variance is between persons—observations within a person are more similar to each other than to other people's observations. This justifies using mixed models: there's meaningful clustering to account for.

With random slopes, the ICC is time-dependent; the single-number ICC interpretation applies to random-intercept-only models.

Common Pitfalls

Before applying LMM, be aware of these frequent mistakes.

Using REML for fixed effects comparison — Comparing models with different fixed effects using REML. Reality: REML likelihoods aren't comparable when fixed effects differ. Use ML (REML = FALSE) for model comparison.

Ignoring singular fit warnings — Dismissing "boundary (singular) fit" messages. Reality: A variance at zero often means the model is over-specified. Check VarCorr() and consider simplifying.

Over-specifying random effects — Including random quadratic with only 4 time points. Reality: You need enough within-person observations to estimate complex random effects. Start simple.

Treating BLUPs as data — Using extracted random effects in secondary analyses as if they were observed data. Reality: BLUPs are estimates with uncertainty. Include predictors in the model, not in post-hoc analyses of BLUPs.

Time coded as factor — Treating time as categorical when you want a continuous slope. Reality: Factor time gives dummy variables, not a growth trajectory. (It's fine if your goal is a nonparametric mean profile; you'll lose a single 'slope' interpretation.)

Misinterpreting ICC — "ICC = 0.65 means the model explains 65%." Reality: ICC is variance partitioning, not variance explained. It tells you how much is between- vs. within-person.

Ignoring residual assumptions — Not checking whether residuals are approximately normal and homoscedastic. Reality: Severe violations can bias standard errors. Always plot residuals vs. fitted. If measurements are closely spaced, consider AR(1) residual correlation and/or time-varying residual variance; inspect residual ACFs and plots.

Confusing marginal and conditional R² — Reporting only marginal R² when random effects matter. Reality: Marginal = fixed effects only; Conditional = fixed + random. Report both.

Centering confusion — Not being clear about what the intercept represents. Reality: The intercept's meaning depends on where time = 0. Be explicit about your centering choice.

Conflating statistical and practical significance — "Slope variance is significant, so individual differences matter." Reality: With large samples, even tiny variances are significant. Interpret effect sizes substantively.

Denominator d.f. and p-values — Mixed-model degrees of freedom depend on the design; use Satterthwaite or Kenward–Roger approximations when reporting tests for fixed effects.

Summary

You now have the conceptual foundation for understanding LMM:

• Nested data require mixed models—ignoring the dependency structure produces misleading standard errors
• Fixed effects capture population-average trajectories; random effects capture individual variation around those averages
• Shrinkage improves individual estimates by borrowing strength across people—especially valuable with sparse data
• Time coding determines what the intercept represents and must match your research question
• ICC tells you how much variance is between- vs. within-person, justifying the mixed model approach

This is enough to understand what LMM does and why. To actually fit one, continue to the worked example.

Next Steps