Mixed Effects Models

Goals of the lecture

The independence assumption of linear regression.
- What about when data are non-independent?
Introducing mixed effects models.
- Fixed vs. random effects.
- Adding random intercepts.
- Adding random slopes.
Mixed models in practice with lme4.
- Model comparison: the logic of likelihood-ratio tests.

Load libraries

### Loading the tidyverse
library(tidyverse)
### Loading lme4
library(lme4)

Part 1: Independence

The independence assumption; why it matters; examples of non-independence.

What is independence?

Independence means that each observation in your dataset is generated by a separate, unrelated causal process.

When you flip a coin, each flip is independent of the others
The outcome of flip #1 doesn’t influence flip #2
No hidden factors connect multiple observations
Each data point provides new information

Why does independence matter?

Standard statistical methods (t-tests, OLS regression, ANOVA) assume independence.

When this assumption is violated:

Standard errors are underestimated
Confidence intervals are too narrow
p-values are too small (inflated Type I error)
Your conclusions may be wrong!

The problem: Treating non-independent observations as independent makes you overconfident in your results.

Example: Independent data

You test whether a new drug increases happiness:

100 people receive the drug
100 people receive a placebo
Each person provides one happiness rating (1-7 scale)
You compare the two groups with a t-test

✓ Independence is reasonable here
Each person contributes exactly one data point.

Another independent example

You test whether people respond differently to metaphor type A vs. B:

200 participants, randomly assigned to condition A or B
Each participant sees one item
Between-subjects design with one item per person
Compare responses with OLS regression

✓ Independence is reasonable here
No repeated measures, no item effects to worry about.

Visualizing independent data

Each observation is independent—no nesting structure.

Analyzing independent data

            Estimate Std. Error    t value      Pr(>|t|)
(Intercept) 723.6329   1.507002 480.180393 1.472707e-305
groupB      -19.4609   2.131223  -9.131332  7.976194e-17

Interpretation: - Intercept (groupA) = 723.63 (mean of group A) - groupB coefficient = -19.46 (difference from A) - Group B is about 20 units lower than Group A

When independence fails

Non-independence occurs when different observations are systematically related, i.e., they were produced by the same generative process.

Common sources of non-independence in behavioral research:

Repeated measures: Same participant tested multiple times
Nested data: Students within classrooms within schools
Stimulus items: Same stimuli shown to multiple participants
Time series: Measurements over time from same units
Clustered designs: Participants recruited from same communities

Example: The sleep study

The sleepstudy dataset tracks reaction times in 18 subjects over subsequent days of sleep deprivation:

  Reaction Days Subject
1 249.5600    0     308
2 258.7047    1     308
3 250.8006    2     308

💭 Check-in

Based on the structure of this study, what might be a source of non-independence?

Naive analysis (wrong!)

Let’s pretend we don’t know about the nested structure:

💭 Check-in

What’s not pictured here?

The real structure

Subjects differ in:

Overall reaction time (baseline speed)
Effect of sleep deprivation (some are more resilient)

Subject-level variability

Some subjects are consistently faster; others are consistently slower.

Two types of nested variance

Random intercepts: Subjects differ in their average reaction time
- Some people are just faster/slower overall
- Even with no sleep deprivation!
Random slopes: Subjects differ in how much sleep deprivation affects them
- Some people are resilient (shallow slope)
- Others deteriorate quickly (steep slope)

Mixed effects models let us account for both types of variance.

Part 2: Introduction to mixed effects models

Fixed vs. random effects; random intercepts; random slopes.

What are mixed effects models?

Mixed effects models combine:

Fixed effects: Variables we care about (e.g., experimental conditions)
- These are your hypotheses!
- Estimated with specific coefficient values
Random effects: Grouping variables that create non-independence
- Sources of variability we need to control for
- Allow intercepts/slopes to vary by group
- Estimated as distributions (variance parameters)

Goal: Get better estimates of fixed effects by accounting for nested structure.

Fixed vs. random effects

Determining which factors to include as fixed vs. random effects is not always straightforward. Here’s a rough guide, but we’ll discuss it again later too:

How do you decide?

Fixed effects:
- Variables you’re theoretically interested in
- Want to estimate specific coefficient values
- Often experimental manipulations
- Examples: treatment condition, time, drug dosage
Random effects:
- Grouping/clustering variables
- Want to account for variance, not estimate specific values
- Usually sampling from larger population
- Examples: participant ID, item ID, school, lab

The mixed model formula

Standard regression:

\[Y = \beta_0 + \beta_1 X + \epsilon\]

Mixed effects model with random intercepts:

\[Y = (\beta_0 + u_0) + \beta_1 X + \epsilon\]

Where:

\(\beta_0\) = overall intercept (fixed effect)
\(u_0\) = subject-specific deviation from overall intercept (random effect)
\(\beta_1\) = fixed effect of X
\(\epsilon\) = residual error

Random intercepts: Intuition

Random intercepts allow each group (e.g., subject) to have its own baseline:

Same slope, different intercepts.

Random intercepts: R syntax

Basic model with random intercepts:

Breaking down the syntax:

Reaction ~ Days: Fixed effect of Days on Reaction
(1 | Subject): Random intercept for each Subject
- 1 = intercept
- | = “grouped by”
- Subject = grouping variable

Extracting random intercepts

After fitting the model, you can extract the random effects:

    (Intercept)
308   40.783710
309  -77.849554
310  -63.108567
330    4.406442

These are deviations from the overall intercept (fixed effect).

Understanding the intercepts

Each subject’s fitted intercept = fixed intercept + random deviation:

(Intercept) 
   251.4051

[1] 40.78371

(Intercept) 
   292.1888

Visualizing random intercepts

Random slopes: Intuition

Random slopes allow the effect of X to vary by group:

Random slopes: R syntax

Model with random intercepts AND slopes:

             Estimate Std. Error   t value
(Intercept) 251.40510   6.824597 36.838090
Days         10.46729   1.545790  6.771481

Breaking down the syntax:

Reaction ~ Days: Fixed effect of Days
(1 + Days | Subject): Random effects for Subject
- 1 = random intercept
- Days = random slope for Days
- | = grouped by
- Subject = grouping variable

Extracting random slopes

After fitting the model, you can extract the random effects:

    (Intercept)      Days
308    2.258551  9.198976
309  -40.398738 -8.619681
310  -38.960409 -5.448856
330   23.690620 -4.814350

Now, we have deviations from the intercept and from the Days slope.

Understanding the slopes

Each subject’s fitted slope = fixed slope + random deviation:

    Days 
10.46729

[1] 9.198976

    Days 
19.66626

Subject 308’s RT increases by ~19.67 ms per day (vs. population average of ~10.47 ms).

Visualizing random slopes

Why random slopes matter

Without random slopes: - Assumes the effect is the same for everyone - Underestimates uncertainty in the fixed effect - Can lead to false positives

With random slopes: - Acknowledges that effects vary across individuals - Provides more conservative (realistic) estimates - Better generalization to new subjects

Rule of thumb: If a variable varies within your grouping variable, include it as a random slope.

Multiple random effects

You can include multiple sources of random effects:

Common in psycholinguistics: - Random effects for subjects (people vary) - Random effects for items (stimuli vary) - Called “crossed random effects”

Conceptual summary

Mixed effects models help when:

You have repeated measures
You have nested/hierarchical structure
You want to generalize beyond your specific sample

Key components: - Fixed effects: Your hypotheses - Random intercepts: Group-level baseline differences
- Random slopes: Group-level differences in effects

Part 3: Using `lme4`

Building and evaluating models; model comparisons; best practices and common issues.

The `lme4` package

Most common R package for mixed models:

Main functions: - lmer(): Linear mixed effects models - glmer(): Generalized linear mixed models (logistic, Poisson, etc.)

Building your first model

Let’s fit a model with random intercepts only:

Note: REML = FALSE uses maximum likelihood estimation, which we need for model comparison.

Viewing model output

Linear mixed model fit by maximum likelihood  ['lmerMod']
Formula: Reaction ~ Days + (1 | Subject)
   Data: sleepstudy

      AIC       BIC    logLik -2*log(L)  df.resid 
   1802.1    1814.9    -897.0    1794.1       176 

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-3.2347 -0.5544  0.0155  0.5257  4.2648 

Random effects:
 Groups   Name        Variance Std.Dev.
 Subject  (Intercept) 1296.9   36.01   
 Residual              954.5   30.90   
Number of obs: 180, groups:  Subject, 18

Fixed effects:
            Estimate Std. Error t value
(Intercept) 251.4051     9.5062   26.45
Days         10.4673     0.8017   13.06

Correlation of Fixed Effects:
     (Intr)
Days -0.380

Adding random slopes

Now let’s add random slopes for the Days effect:

This allows both the intercept AND the slope to vary by subject.

Comparing the models

Linear mixed model fit by maximum likelihood  ['lmerMod']
Formula: Reaction ~ Days + (1 + Days | Subject)
   Data: sleepstudy

      AIC       BIC    logLik -2*log(L)  df.resid 
   1763.9    1783.1    -876.0    1751.9       174 

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-3.9416 -0.4656  0.0289  0.4636  5.1793 

Random effects:
 Groups   Name        Variance Std.Dev. Corr
 Subject  (Intercept) 565.48   23.780       
          Days         32.68    5.717   0.08
 Residual             654.95   25.592       
Number of obs: 180, groups:  Subject, 18

Fixed effects:
            Estimate Std. Error t value
(Intercept)  251.405      6.632  37.907
Days          10.467      1.502   6.968

Correlation of Fixed Effects:
     (Intr)
Days -0.138

Model comparison: The logic

Question: Does adding variable \(X\) improve the model over a model without \(X\)?

Approach: Likelihood ratio test

Compare log-likelihood of two nested models
Model with more parameters should fit better
But: is the improvement “worth it”?
Test statistic: \(\chi^2 = -2(LL_{reduced} - LL_{full})\)
Compare to chi-square distribution

Running model comparison

Data: sleepstudy
Models:
model_reduced: Reaction ~ (1 + Days | Subject)
model_full: Reaction ~ Days + (1 + Days | Subject)
              npar    AIC    BIC  logLik -2*log(L)  Chisq Df Pr(>Chisq)    
model_reduced    5 1785.5 1801.4 -887.74    1775.5                         
model_full       6 1763.9 1783.1 -875.97    1751.9 23.537  1  1.226e-06 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Interpretation:: Adding Days significantly improves fit

Extracting coefficients

Fixed effects (population-level):

(Intercept)        Days 
  251.40510    10.46729

Random effects (subject-specific deviations):

    (Intercept)      Days
308    2.815789  9.075507
309  -40.047855 -8.644152
310  -38.432497 -5.513471
330   22.831765 -4.658665

Visualizing random effects

Conclusion

Mixed models are a valuable tool for modeling non-independent data.
Random intercetps model correlated variance in y; random slopes model correlated variance in y ~ x.
Coming up, we’ll discuss:
- More complex models (e.g., nested variance).
- Deep dive into fitting random effects (e.g., pooling).
- Understanding the outputs (e.g., variance partitioning).
- Best practices (e.g., “keep it maximal”).
- Diagnozing and fixing common issues (e.g., convergence failures).
- Generalizing to generalized linear mixed effects models.

Mixed Effects Models

Goals of the lecture

Load libraries

Part 1: Independence

What is independence?

Why does independence matter?

Example: Independent data

Another independent example

Visualizing independent data

Analyzing independent data

When independence fails

Example: The sleep study

Naive analysis (wrong!)

The real structure

Subject-level variability

Two types of nested variance

Part 2: Introduction to mixed effects models

What are mixed effects models?

Fixed vs. random effects

The mixed model formula

Random intercepts: Intuition

Random intercepts: R syntax

Extracting random intercepts

Understanding the intercepts

Visualizing random intercepts

Random slopes: Intuition

Random slopes: R syntax

Extracting random slopes

Understanding the slopes

Visualizing random slopes

Why random slopes matter

Multiple random effects

Conceptual summary

Part 3: Using lme4

The lme4 package

Building your first model

Viewing model output

Adding random slopes

Comparing the models

Model comparison: The logic

Running model comparison

Extracting coefficients

Visualizing random effects

Conclusion

Part 3: Using `lme4`

The `lme4` package