Multilevel or Mixed-effect Models

This tutorials provides an introduction to multilevel models, also known as mixed-effect models or hierarchical linear models. Multilevel models are used to analyze data with a hierarchical or nested structure, where observations are grouped or clustered. These models are particularly useful when traditional regression methods are not appropriate due to the lack of independence among observations. This tutorial covers the key concepts, assumptions, and types of multilevel models, along with examples and applications in R.

  1. Random Intercept Model

  2. Random Slope Model

  3. Generalized Multilevel Model

Introduction to Multilevel Models

Multilevel models, also known as mixed-effect models or hierarchical linear models, are statistical models designed to handle data that is structured hierarchically or grouped. These models are particularly useful when observations are not independent but instead nested within higher-level units. For example:

  • Students nested within schools
  • Patients nested within hospitals
  • Repeated measures on the same individuals

Components of a Mixed-Effect Model

  1. Fixed Effects: These are the population-level effects, which are consistent across all groups (e.g., the average effect of a treatment).

  2. Random Effects: These capture the group-specific deviations from the population-level effects (e.g., variation between schools).

For an observation \(y_{ij}\) (e.g., the score of the \(j\)-th student in the \(i\)-th school):

\[ y_{ij} = X_{ij}\beta + Z_{ij}u_i + \epsilon_{ij} \]

Where:

  • \(y_{ij}\): Dependent variable (outcome).
  • \(X_{ij}\): Row vector of predictors for fixed effects.
  • \(\beta\): Vector of fixed-effect coefficients.
  • \(Z_{ij}\): Row vector of predictors for random effects.
  • \(u_i\): Vector of random effects (specific to group \(i\)).
  • \(\epsilon_{ij}\): Residual error (random noise for observation \(ij\)).

Key Assumptions

  1. Random effects \(u_i\) are normally distributed:

    \[ u_i \sim N(0, \sigma_u^2) \]

    where \(\sigma_u^2\) is the variance of the random effects.

  2. Residual errors \(\epsilon_{ij}\) are normally distributed:

    \[ \epsilon_{ij} \sim N(0, \sigma_\epsilon^2) \]

  3. Independence:

  • \(u_i\) and \(\epsilon_{ij}\) are independent.

Why Use Multilevel Models?

  1. Handle Nested Data:
    • Traditional statistical methods assume independence of observations, which is often not the case with nested data. Multilevel models account for this dependency, leading to more accurate results.
  2. Improve Accuracy:
    • By considering the hierarchical structure, these models provide more precise estimates of the effects of predictors at different levels.
  3. Flexibility:
    • They can handle complex data structures, such as data with multiple levels of nesting or cross-classified data (e.g., students nested within both schools and neighborhoods).
  4. Understand Variability:
    • Multilevel models can partition the variance in the outcome variable into within-group and between-group components, helping to understand how much of the variability is due to differences within groups versus between groups.
  5. Model Growth and Change:
    • They are particularly useful for longitudinal data, allowing researchers to model individual growth trajectories and how these trajectories vary across groups.
  6. Handle Missing Data:
    • These models are more robust to missing data compared to traditional methods, as they can use all available data without requiring complete cases.

Applications

Multilevel models are widely used in fields like education, psychology, public health, and social sciences, where data often has a nested structure. For example, they can be used to study the impact of teaching methods on student performance, accounting for the fact that students are nested within classrooms and schools.

Multilevel Data

Multilevel data, also known as hierarchical data, can come in various forms depending on the structure and context of the data. Here are some common types of multilevel data used in multilevel modeling:

  1. Hierarchical/Nested Data:
    • Example: Students nested within classrooms, which are nested within schools. This type of data has a clear hierarchical structure with multiple levels.
  2. Longitudinal Data (Repeated Measures):
    • Example: Repeated measures of individuals over time, such as tracking a patient’s health metrics across multiple visits. This data type involves multiple observations for each individual over time.
  3. Cross-Classified Data:
    • Example: Students nested within both schools and neighborhoods. Here, the data does not fit neatly into a single hierarchy but rather crosses multiple classifications.
  4. Cross-Nested or Cross-Classified Longitudinal Data:
    • Example: Patients’ recovery tracked over time under different doctors.
  5. Ecological or Spatial Data:
    • Example: Data collected from different geographical locations, such as environmental measurements taken from various regions. This data type often involves spatial hierarchies.
  6. Multivariate Multilevel Data:
    • Example: Multiple outcomes measured for the same individuals, such as test scores in different subjects for students. This involves analyzing several dependent variables simultaneously.

Types of Multilevel Models

Multilevel models (or mixed-effect models) come in various types, depending on the structure of the data and the research question. Here are the common types:

  1. Random Intercept Model
  • Description: Allows groups (e.g., schools, hospitals) to have different intercepts while assuming the slopes for predictors are the same across groups.
  • Use Case: When the outcome depends on both group-level differences and fixed predictors, but you expect only the baseline level (intercept) to vary by group.
  • Example: In a study of students’ test scores nested within schools:
  1. Random Slope Model
  • Description: Allows both the intercept and the slope of a predictor to vary across groups.
  • Use Case: When you expect the relationship between a predictor and the outcome to differ across groups.
  • Example: The effect of study hours on test scores might vary by school (e.g., some schools might provide better resources for studying).
  1. Crossed Random Effects Model
  • Description: Used when data points are nested within multiple, non-hierarchical grouping factors (e.g., students nested in schools and neighborhoods simultaneously).
  • Use Case: When observations are influenced by more than one random factor.
  • Example: Students belong to both a school and a neighborhood, and both factors might influence test scores.
  1. Nested (Hierarchical) Models
  • Description: Data is nested hierarchically (e.g., students within classes, and classes within schools).
  • Use Case: When there are multiple levels of grouping, and each level is nested within the one above it.
  • Example: Students are nested within classrooms, and classrooms are nested within schools.
  1. Growth Curve Models
  • Description: Used for longitudinal data where repeated measurements are taken over time. These models include a random effect for individuals to account for variability in growth trajectories.
  • Use Case: When studying change over time.
  • Example: Tracking students’ reading abilities over multiple years, accounting for individual growth patterns.
  1. Multivariate Multilevel Models
  • Description: Extend multilevel models to handle multiple outcomes simultaneously.
  • Use Case: When studying related outcomes (e.g., math and reading scores) that share hierarchical structures.
  • Example: Math and reading scores of students nested within schools.
  1. Cross-Classified Models
  • Description: A special case of crossed random effects where individuals belong to multiple non-nested groups.
  • Use Case: When individuals belong to categories that aren’t hierarchically nested.
  • Example:Students are influenced by both their primary school and their secondary school, which are not nested.
  1. Three-Level Models
  • Description: Used when data has three levels of hierarchy.
  • Use Case: When observations are nested in intermediate groups, which are further nested within higher-level groups.
  • *Example**:Students nested within classrooms, and classrooms nested within schools.
  1. Generalized Multilevel Models
  • Description: Extend multilevel models to handle non-normal outcomes (e.g., binary, count data).
  • Use Case: When the response variable is not continuous.
  • Example: Predicting whether a student passes or fails (binary outcome) based on study hours and school-level effects.

Summary Table

Type Random Effects Allowed Example R Formula
Random Intercept Model Intercept varies lmer(y ~ x + (1 | group), data)
Random Slope Model Intercept & slope vary lmer(y ~ x + (x | group), data)
Crossed Random Effects Model Multiple random factors (not nested) lmer(y ~ x + (1 | factor1) + (1 | factor2))
Nested Model Hierarchical nesting lmer(y ~ x + (1 | level1/level2), data)
Growth Curve Model Random effects for time lmer(y ~ time + (time | subject), data)
Multivariate Model Multiple outcomes brm(cbind(y1, y2) ~ x + (1 | group), data)
Cross-Classified Model Non-nested groupings lmer(y ~ x + (1 | group1) + (1 | group2))
Three-Level Model Three levels of nesting lmer(y ~ x + (1 | level1/level2/level3), data)
Generalized Model Non-normal outcomes glmer(y ~ x + (1 | group), family = binomial)

Each model type is tailored to specific data structures and research questions, providing flexibility for analyzing complex datasets.

Multilevel Models in R

R offers several powerful packages for multilevel modeling, each tailored to specific needs and levels of complexity. Here’s an overview of commonly used packages:

1. lme4

  • Purpose: Linear and generalized linear mixed-effects models.

  • Key Features:

    • Handles random intercepts and slopes.
    • Supports Gaussian, binomial, Poisson, and other distributions.
    • Efficient for large datasets.

  • Common Functions:

    • lmer(): Linear mixed-effects models.
    • glmer(): Generalized linear mixed-effects models.

  • Example:

    library(lme4)
model <- lmer(score ~ hours_study + (1 | school), data = data)
summary(model)

  • Limitations: Does not compute p-values for fixed effects (requires additional packages like lmerTest).

2. nlme

  • Purpose: Mixed-effects models with more flexible covariance structures than lme4.

  • Key Features:

    • Allows modeling of grouped data with correlation structures.
    • Suitable for longitudinal and repeated-measures data.

  • Common Functions:

    • lme(): Linear mixed-effects models.

  • Example:

    library(nlme)
model <- lme(score ~ hours_study, random = ~ 1 | school, data = data)
summary(model)

3. lmerTest

  • Purpose: Provides p-values and ANOVA tables for models fitted with lme4.

  • Key Features:

    • Easy to compute p-values for fixed effects.
    • Adds hypothesis testing capabilities to lme4.

  • Common Functions:

    • lmer() (enhanced version from lme4).

  • Example:

    library(lmerTest)
model <- lmer(score ~ hours_study + (1 | school), data = data)
summary(model)

4. brms

  • Purpose: Bayesian multilevel models using Stan.

  • Key Features:

    • Handles a wide range of models (linear, non-linear, multivariate).
    • Allows user-defined priors.
    • Supports missing data and complex structures.

  • Common Functions:

    • brm(): Bayesian regression modeling.

  • Example:

    library(brms)
model <- brm(score ~ hours_study + (1 | school), data = data)
summary(model)

5. MCMCglmm

  • Purpose: Bayesian multilevel models using Markov Chain Monte Carlo (MCMC).

  • Key Features:

    • Handles multivariate models.
    • Flexible prior specification.

  • Common Functions:

    • MCMCglmm(): Fits generalized linear mixed models.

  • Example:

    library(MCMCglmm)
model <- MCMCglmm(score ~ hours_study, random = ~ school, data = data)
summary(model)

6. glmmTMB

  • Purpose: Generalized linear mixed models with Tweedie and zero-inflated distributions.

  • Key Features:

    • Suitable for complex data structures (e.g., zero-inflation, dispersion).
    • Faster for certain large datasets compared to lme4.

  • Common Functions:

    • glmmTMB(): Fits generalized linear mixed models.

  • Example:

    library(glmmTMB)
model <- glmmTMB(score ~ hours_study + (1 | school), data = data, family = gaussian())
summary(model)

7. gamm4

  • Purpose: Combines generalized additive models (GAMs) with random effects.

  • Key Features:

    • Extends lme4 to handle smooth terms in GAMs.
    • Useful for modeling non-linear relationships.

  • Common Functions:

    • gamm4(): Fits GAMs with random effects.

  • Example:

    library(gamm4)
model <- gamm4(score ~ s(hours_study) + (1 | school), data = data)
summary(model$mer)

8. tidyLPA

  • Purpose: Latent profile analysis (LPA) with multilevel capabilities.

  • Key Features:

    • Facilitates exploratory analyses of group structures.

  • Common Functions:

    • estimate_profiles(): Estimates latent profiles.

  • Example:

    library(tidyLPA)
results <- estimate_profiles(data, n_profiles = 3)

9. sjPlot (Visualization)

  • Purpose: Visualization of multilevel models.

  • Key Features:

    • Creates plots for fixed effects, random effects, and marginal effects.

  • Common Functions:

    • plot_model(): Visualizes fixed and random effects.

  • Example:

    library(sjPlot)
plot_model(model, type = "re")

10. metafor

  • Purpose: Multilevel meta-analysis.

  • Key Features:

    • Handles meta-analytic data with random effects.

  • Common Functions:

    • rma(): Fits random-effects meta-analysis models.

  • Example:

    library(metafor)
model <- rma(yi, vi, random = ~ 1 | study, data = meta_data)
summary(model)

Comparison Table

Package Type Bayesian P-values Visualization Special Features
lme4 Frequentist No No Moderate Random intercepts/slopes, GLMM
nlme Frequentist No Yes Limited Flexible correlation structures
lmerTest Frequentist No Yes Moderate Hypothesis testing for lme4
brms Bayesian Yes N/A Excellent Wide range of models, flexible priors
MCMCglmm Bayesian Yes N/A Limited Multivariate modeling, MCMC
glmmTMB Frequentist No No Moderate Zero-inflation, dispersion structures
gamm4 GAM + Random Effects No No Limited Smooth terms + mixed models
sjPlot Visualization Tool N/A N/A Excellent For lme4, glmmTMB, etc.
metafor Meta-analysis No Yes Limited Hierarchical meta-analysis

Selecting the Right Package

  • Beginner: Start with lme4 or nlme.
  • Bayesian Models: Use brms or MCMCglmm.
  • Complex Structures: Try glmmTMB or gamm4.
  • Meta-Analysis: Use metafor.

Summary and Conclusions

Multilevel models, also known as mixed-effect models or hierarchical linear models, are powerful tools for analyzing data with a hierarchical structure. These models are particularly useful when observations are not independent but instead nested within higher-level units. By accounting for the dependency among observations, multilevel models provide more accurate estimates of the effects of predictors at different levels. They are widely used in fields like education, psychology, public health, and social sciences to study complex data structures and relationships. R offers several packages for fitting multilevel models, each with its own strengths and capabilities. Whether you’re analyzing nested data, longitudinal data, or multivariate data, there’s a multilevel modeling package in R to suit your needs. By selecting the right package and model type, you can gain valuable insights from your hierarchical data and make more informed decisions based on the results.

References

  1. Gelman, A., & Hill, J. (2006). Data analysis using regression and multilevel/hierarchical models. Cambridge University Press.

  2. Raudenbush, S. W., & Bryk, A. S. (2002). Hierarchical linear models: Applications and data analysis methods. Sage.

  3. Bates, D., Mächler, M., Bolker, B., & Walker, S. (2015). Fitting linear mixed-effects models using lme4. Journal of Statistical Software, 67(1), 1-48.

  4. What are multilevel models and why should I use them?

  5. Multilevel Modeling

  6. Chapter 4 - Generalized Multilevel Model for Examining Intraindividual Covariation