Required Packages

Let’s start by loading the necessary packages for our analysis:

# Load required packages
library(lme4)      # For linear mixed-effects models
library(tidyverse) # For data manipulation and visualization
library(haven)     # For reading Stata files

Additional Useful Packages

# Additional packages for analysis
library(lmerTest)  # Adds p-values to lme4 output
library(geepack)   # For GEE models
library(sjPlot)    # For model visualization
library(performance) # For model diagnostics

Introduction to Mixed Effect Models and Linear Effect Model

What are Mixed Effect Models?

• Mixed-effects models incorporate both fixed effects and random effects
• Fixed effects: Parameters associated with entire population or repeatable levels of experimental factors

What are Mixed Effect Models?

• Random effects: Associated with individual experimental units drawn at random from a population
• Also known as hierarchical linear models, multilevel models, or random effects models

Linear Mixed Effect Models

• Extension of linear regression that accounts for:

  • Non-independence of observations
  • Hierarchical data structures
  • Repeated measures data
  • Individual variability across participants/items

Motivations for Linear Mixed Effect Models

Why Use Mixed Effects Models?

1. Account for clustering in data

• Observations within groups are more similar than across groups
• Violates independence assumption of standard regression

2. Handle missing data better than ANOVA

• Uses all available data rather than listwise deletion
• Precision-weighted parameter estimates

Motivations for Linear Mixed Effect Models

Why Use Mixed Effects Models?

3. Model both participant and item variability simultaneously

• Accounts for individual differences
• Generalizable across both subjects and stimuli

4. Flexible with unbalanced designs

• No requirement for equal group sizes
• Handles irregular measurement occasions

Data Suitable for Linear Mixed Effect Models

Types of Data Structures

1. Multilevel/Hierarchical Data

• Students nested within schools
• Patients within hospitals
• Employees within companies

Data Suitable for Linear Mixed Effect Models

Types of Data Structures

2. Longitudinal/Repeated Measures Data

• Multiple measurements per participant over time
• Growth curve analysis
• Panel studies

Data Suitable for Linear Mixed Effect Models

Types of Data Structures

3. Crossed Random Effects

• Participants respond to multiple items
• Multiple raters evaluate multiple subjects

Data Suitable for Linear Mixed Effect Models

Data Format Requirements

• Long format: Each row = one observation
• Variables: Subject ID, Item ID, dependent variable, predictors

Assumptions for Linear Mixed Effect Models

Key Assumptions

1. Linearity

• Linear relationship between predictors and outcome

2. Independence of Residuals

• After accounting for random effects structure

Assumptions for Linear Mixed Effect Models

Key Assumptions

3. Normality of Random Effects

• Random intercepts and slopes normally distributed
• Mean = 0, variance estimated by model

4. Homoscedasticity

• Constant variance of residuals

Assumptions for Linear Mixed Effect Models

Key Assumptions

5. No Perfect Multicollinearity

• Among fixed effects predictors

Less Restrictive Than Traditional ANOVA

• More robust to violations of normality assumption
• Handles unbalanced designs naturally

R Functions and Packages for Linear Mixed Effect Models

Primary Packages

lme4 Package (most common)

```{r}
library(lme4)

# Basic syntax for linear mixed models
lmer(outcome ~ fixed_effects + (random_effects|grouping_factor), 
     data = dataset)

# Example with random intercepts and slopes
lmer(RT ~ condition + (1 + condition|participant) + (1|item), 
     data = mydata)
```

R Functions and Packages for Linear Mixed Effect Models

Primary Packages

nlme Package (alternative with more covariance structures)

```{r}
#| echo: true
#| eval: false

library(nlme)
lme(outcome ~ fixed_effects, random = ~ random_effects|group, 
    data = dataset)
```

R Functions and Packages for Linear Mixed Effect Models

Additional Useful Packages

• lmerTest: Adds p-values to lme4 output
• afex: Simplified ANOVA-type analysis with mixed models
  
• sjPlot: Easy visualization of mixed model results
• performance: Model diagnostics and assumption checking

Loading and Exploring Our Dataset

Let’s read the Stata dataset and explore its structure:

# Read Stata data file
mydata <- read_dta("5.1.dta")

Practical Demonstration with 5.1.dta Dataset

Data Exploration

Loading and Exploring Our Dataset

# Summary of key variables
summary(mydata$score)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
   0.00   19.00   33.00   31.09   45.00   75.00 

Loading and Exploring Our Dataset

# Basic visualization - score by schoolid
mydata %>%
  ggplot(aes(x = as.factor(schoolid), y = score)) +
  geom_boxplot() +
  labs(title = "Distribution of Score by School ID",
       x = "School ID", y = "Score") +
  theme_minimal()

Fitting Linear Mixed Effect Models

Random Intercept Model

• Model 1: Random intercept only with schoolid as grouping factor

model1 <- lmer(score ~ 1 + (1|schoolid), data = mydata)

Fitting Linear Mixed Effect Models

Random Slope Model with cohort90

• Model 2: Random slope model with cohort90 as random slope factor
• This model allows the effect of cohort90 to vary by school

# Fit random slope model with cohort90
model2 <- lmer(score ~ cohort90 + (1 + cohort90|schoolid), data = mydata)

Introduction to Longitudinal Data Analysis

What is Longitudinal Data?

• Longitudinal data: Multiple observations per subject over time
• Also called: panel data, repeated measures data
• Captures individual change trajectories over time

Characteristics

• Within-subject correlation: Observations from same person are correlated
• Time-varying predictors: Can include variables that change over time
• Missing data patterns: Common due to dropout or missed visits
• Individual differences: People may have different starting points and rates of change

Introduction to Longitudinal Data Analysis

Research Questions Addressed

• How does the outcome change over time on average?
• Do individuals differ in their rates of change?
• What predicts individual differences in change?
• Do interventions affect the rate of change?

Different Approaches to Longitudinal Data Analysis

Traditional Approaches

1. Repeated Measures ANOVA

   • Assumes complete data
   • Limited flexibility with time
   • Requires balanced designs

Different Approaches to Longitudinal Data Analysis

Traditional Approaches

2. MANOVA (Multivariate ANOVA)

   • Treats each time point as separate variable
   • Requires complete data
   • Not suitable for many time points

Different Approaches to Longitudinal Data Analysis

Modern Robust Approaches

1. Generalized Estimating Equations (GEE)

   • Population-averaged effects
   • Robust to misspecification of correlation structure

Different Approaches to Longitudinal Data Analysis

Modern Robust Approaches

2. Linear Mixed Effect Models

   • Subject-specific effects
     
   • Flexible with missing data and unbalanced designs
   • Can model complex growth patterns

Generalized Estimating Equations (GEE)

What is GEE?

• Population-averaged approach to longitudinal data
• Estimates marginal effects averaged across the population
• Focuses on mean response as function of covariates
• Treats within-subject correlation as “nuisance”

Generalized Estimating Equations (GEE)

Key Features

Working Correlation Structure:

• Independence: No correlation assumed
• Exchangeable: Constant correlation across all time points
  
• Autoregressive: Correlation decreases with time separation
• Unstructured: Different correlation for each pair of time points

Generalized Estimating Equations (GEE)

R Implementation with sleepstudy Data

• Load sleepstudy data from lme4 package

data(sleepstudy)

• Explore the data structure

Generalized Estimating Equations (GEE)

Fit GEE model using sleepstudy data

library(geepack)

gee_model <- geeglm(Reaction ~ Days, 
                    data = sleepstudy,
                    id = Subject,
                    family = gaussian,
                    corstr = "exchangeable")

Interpretation of GEE Results

• Days coefficient: Population-averaged increase in reaction time per day of sleep deprivation
• Intercept: Population-averaged reaction time at baseline (Day 0)
• Robust standard errors account for within-subject correlation
• Results represent marginal (population-level) effects

Linear Mixed Effect Model for Longitudinal Data

Growth Curve Models: Random Intercept Model

```{r}
# Individual starting points vary, same growth rate
model1 <- lmer(outcome ~ time + (1|subject), data = longdata)
```

Linear Mixed Effect Model for Longitudinal Data

Growth Curve Models: Random Slope Model

```{r}

# Both starting points and growth rates vary
model2 <- lmer(outcome ~ time + (1 + time|subject), data = longdata)
```

Linear Mixed Effect Model for Longitudinal Data

Growth Curve Models: With Treatment Effects

```{r}

# Treatment affects both intercept and slope
model3 <- lmer(outcome ~ time * treatment + (1 + time|subject),
               data = longdata)
```

Linear Mixed Effect Model for Longitudinal Data

R Code Example with sleepstudy Data

# Load required packages
library(lme4)
library(lmerTest)  # for p-values

Linear Mixed Effect Model for Longitudinal Data

Fit random intercept model

model1 <- lmer(Reaction ~ Days + (1|Subject), data = sleepstudy)

Fit random intercept and slope model

model2 <- lmer(Reaction ~ Days + (1 + Days|Subject), data = sleepstudy)

Linear Mixed Effect Model for Longitudinal Data

R Code Example with sleepstudy Data

# Model comparison
anova(model1, model2)

Data: sleepstudy
Models:
model1: Reaction ~ Days + (1 | Subject)
model2: Reaction ~ Days + (1 + Days | Subject)
       npar  AIC  BIC logLik -2*log(L) Chisq Df Pr(>Chisq)    
model1    4 1802 1815   -897      1794                        
model2    6 1764 1783   -876      1752  42.1  2    7.1e-10 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Linear Mixed Effect Model for Longitudinal Data

Interpretation of Linear Mixed Model Results

• Fixed Effects: Population-average effects
  • (Intercept): Average reaction time at Day 0 across all subjects
  • Days: Average increase in reaction time per day of sleep deprivation

Linear Mixed Effect Model for Longitudinal Data

Interpretation of Linear Mixed Model Results

• Random Effects: Individual variability
  • (Intercept): Variability in individual baseline reaction times
  • Days: Variability in individual slopes (how much each person’s reaction time increases per day)
  • Correlation: Relationship between individual baselines and slopes

Linear Mixed Effect Model for Longitudinal Data

Model Diagnostics

# Check model fit
plot(model2)

qqnorm(resid(model2))
qqline(resid(model2))

Linear Mixed Effect Model for Longitudinal Data

Model Comparison and Selection

```{r}
# Compare models using likelihood ratio tests
model_simple <- lmer(outcome ~ time + (1|subject), data = longdata)
model_complex <- lmer(outcome ~ time + treatment + time:treatment +
                      (1 + time|subject), data = longdata)

anova(model_simple, model_complex)

# Check model assumptions
library(performance)
check_model(model_complex)
```

Summary

When to Use Each Approach

GEE:

• Interest in population-averaged effects
• Robust inference about mean trends
• Less concern about individual predictions

Summary

When to Use Each Approach

Linear Mixed Models:

• Interest in individual trajectories
  
• Need to predict individual outcomes
• Want to model complex growth patterns
• Handle irregular measurement occasions

Summary

When to Use Each Approach

Best Practices

1. Explore data graphically before modeling
2. Start with simple models and build complexity
3. Check model assumptions and fit
4. Compare multiple models using information criteria
5. Interpret results in context of research questions

Thank You

Questions and Answer

Resources:

• Pinheiro & Bates (2000). Mixed-Effects Models in S and S-PLUS
• Brown (2021). An Introduction to Linear Mixed-Effects Modeling in R
  
• Twisk (2013). Applied Longitudinal Data Analysis for Epidemiology