1. Analysis of Clustered and Longitudinal Data
Module 2
Vocabulary Slam and an Introduction to Linear Mixed Models (LMMs)

2. Agenda Vocabulary Slam Introduction to Linear Mixed Models (LMMs)
We begin this module by defining “clustered” and “longitudinal” data, and address other terms used to identify dependent data structures.
We define and depict clustered and longitudinal data using the multilevel framework.
Introduction to Linear Mixed Models (LMMs)
We give a brief overview of LMMs and introduce the concept of a random effect.
Biostat 512

3. What is Clustered Data? Clustered Data Examples
An outcome is measured once for each subject, and subjects belong to (or are “nested” in) clusters, such as families, schools, or neighborhoods.
The number of subjects in each cluster may vary from cluster to cluster.
Outcomes measured for members of these groups are likely to be correlated.
Examples
Biostat 512

4. What is Longitudinal Data?
An outcome is measured for the same person repeatedly over a period of time.
Different subjects may have different numbers of observations which may be taken at different time points.
Observations made on the same person are likely to be correlated.
Examples
Biostat 512

5. What is Clustered-Longitudinal Data?
An outcome is measured repeatedly for the same subject over time, and subjects are clustered within some unit.
Subjects may have different numbers of measures, and clusters may have differing numbers of subjects.
The outcome values for different time points in the same subject are assumed to be correlated.
Measurements for subjects from within the same cluster are assumed to be correlated.
Examples

6. What is Repeated Measures Data?
Multiple observations are made for the same person over time, space or other dimension.
Each subject need not have all measurements.
Outcomes measured for the same person are likely to be correlated.
Examples

7. What is “Multilevel” Data
Clustered/longitudinal/repeated measures data is more generally known as “multilevel” data.
Levels 1,2,3,…
Level 1 is the lowest or most granular level of the data, and where the outcome variable of interest is measured.
Levels 2,3,… capture higher level information
Cluster-levels for clustered data
Subject-level for longitudinal data
Subject- and cluster-levels for clustered-longitudinal data
We will illustrate the multilevel concept for 2 and 3 level data structures.
Biostat 512

8. Two-Level Clustered Data Example
A research study in education aims to assess the impact of school type (public vs. catholic) as well as student gender and student SES on student-level math achievement scores. Scores are measured once for the students in the school.
Biostat 512

9. Two-Level Clustered Data (Students Nested within Schools)
Level 1 Variables: Student Achievement Score, Gender, Student’s SES….
Level 2 Variables: Public or Catholic School…
Biostat 512

10. Two-Level Longitudinal Data Example
Researchers are studying the effect of a mother’s vocabulary and the child’s gender on the child’s vocabulary growth.
Biostat 512

11. Longitudinal Data (Vocabulary Measured Over Time)
Vocab Measured at Time 2
Vocab Measured at Time n1
Child 1
Vocab Measured at Time 1
Child 2… …
Vocab Measured at Time 2…
Vocab
Measured at
Time 1
Level 2
Level 1
Level 1 Variables (Time-Varying): Child Vocabulary Count, Age at each measurement
Level 2 Variables (Time-Invariant): Mother’s Vocabulary, Child’s Gender
Biostat 512

12. Two-Level Repeated Measures Example
Researchers are studying the effect of two different treatments on nucleotide bonding in three regions of the brain in rats. Measurements are taken from the same three regions of the brain of each rat, after each of the two different treatments.
Biostat 512

13. Repeated Measures Data (Rat Brain Example)
Chemical
Measured in Region 1, Treatment A
Rat 2…
Measured in Region 2, Treatment A
Measured in Region 3, Treatment B
Level 2
Level 1 … …
Level 1 Variables (Varying): Nucleotide bonding measurement, Brain region, Treatment
Level 2 Variables (Invariant): Rat gender
Biostat 512

14. Three-Level Clustered Data Example
A research study in education aims to assess the impact of school, classroom, and student-level variables on student achievement.
Biostat 512

15. Three-Level Clustered Data (Students nested in classrooms nested in schools)
Student n1
Student 1
Student 2… …
Level 1 Variables: Student Achievement Score, Gender, Student’s SES…
Level 2 Variables: Teacher experience, Class size …
Level 3 Variables: School locale (Rural or Urban), School percent low income
Biostat 512

16. Three-Level Clustered-Longitudinal Data Example
Math skills are measured for the same student each year from grades 1 through 6, with students clustered within schools. The goal is to model how student characteristics, such as ethnicity and gender, and school characteristics, such as school size and percent low-income students, affect the math scores of students over time.
Biostat 512

17. Three-Level Clustered-Longitudinal Data (Math scores measured over time for students nested in schools)
Math Score at Grade 2
Math Score at Grade 6
Student 1
Math Score at Grade 1
Student 2… …
Math Score at Grade 3…
Level 3
School 1…
Level 2
Level 1
Level 1 Variables (Time-Varying): Student’s math score, Grade at each measurement
Level 2 Variables (Time-Invariant): Student’s Ethnicity, Student’s Gender
Level 3 Variables (Time-Invariant): School size, Educational Intervention at School Level
Biostat 512

18. What Constitutes a “Level” in Multilevel Data?
In a clustered data set, each “level” represents a factor that can be thought of as a random sample from a larger population.
The students in a two-level clustered data set can be thought of as a random sample of students within each school.
The schools in a two-level clustered data set can be thought of as a random sample from a larger population of schools.
We want to make inferences to the larger population of students and schools, not confine our inference to the particular students and schools included in this study.
In a longitudinal data set, level 1 represents the “occasions” within a subject and Level 2 is the subject.
We think of the subjects as being representative of a larger population of subjects.
Biostat 512

19. What is Not a “Level”
Factors such as Treatment or Gender are not considered to be Levels of data, because they cannot be thought of as a random sample from a larger population.
We wish to make inferences only about the specific values of Treatment or Gender that are included in our study…not to a larger population of treatments or genders.
Biostat 512

20. Why all this talk about “levels”?
Understanding the multilevel nature of your dataset is a critical start to the analysis process.
Is the data “clustered”, “longitudinal” or “clustered-longitudinal”?
How many levels are there? 2,3, more?
What defines each level?
What is the outcome of interest and is it measured at Level 1?
What other variables are of interest at each level?
The answers to these questions will drive the entire analysis.
Biostat 512

21. Models for Multilevel Data
Data are often hierarchical in nature, especially in the social sciences, and we should not ignore this.
Using single-level (OLS, GLM) analysis leads to:
Unit of analysis problem
School or child?
Aggregation bias
School SES vs. child SES?
Incorrectly estimated precision / standard errors
Results in incorrect p-values and incorrect conclusions
Linear Mixed Models can appropriately address these problems.
Biostat 512

22. What are Linear Mixed Models? (LMMs)
LMMs are:
Also known as multilevel models, hierarchical models, random effects models, mixed models
For a continuous outcome variable, Y
Linear in the parameters (β’s)
For multilevel data, where outcomes measured for the same cluster/subject are assumed to be correlated and/or the error variance is not constant. In other words, for situations where the GLM assumption (below) is violated.
Composed of both fixed and random effects, hence, “mixed”
Not the only modeling option for multilevel data with a continuous outcome. Another option is a marginal model, which we will discuss later in the course.

23. Fixed Effects in a LMM Are usually the focus of the analysis
Can be thought of as similar to parameters in an ordinary regression model (the Betas)
Can be taken from any level of the data
Help us to explain the variance in Y at each level of the data
Examples of fixed effects:
Age, sex, treatment, brain region, marital status, teaching experience
Biostat 512

24. Random Effects in a LMM
Are usually not the primary focus of the analysis, but…
Allow us to account for correlation among observations within the same level-2 or higher units (e.g. correlations among observations within the same school)
Allow us to partition the total variance of Y into levels that correspond with the multilevel structure of the data
How much of the variation in student math achievement scores can be attributed to student-level variability (level 1) versus school-level variability (level 2)?
Are summarized by their variance and covariance, if there is more than one random effect in the LMM
Biostat 512

25. Random Effects in a LMM (cont)
Come in two flavors:
Random intercepts
Random slopes
Are explicitly specified in the model. This is in contrast to the random errors, which are never explicitly specified when a model is fit, but always exist and their variance is always estimated.
We will introduce the LMM notation and assumptions in the next module.
Biostat 512

26. In Conclusion
Dependent data structures go by many names – longitudinal, clustered, repeated measures, multilevel.
Understanding the multilevel nature of a dataset is critical to any analysis.
OLS regression is not an appropriate technique for modeling multilevel data.
A Linear Mixed Model is one approach that can be used for dependent data.
Biostat 512