Multilevel Modeling

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Multilevel Modeling. 1. Overview 2. Application #1: Growth Modeling Break 3. Application # 2: Individuals Nested Within Groups 4. Questions?. Overview. What is multilevel modeling? Examples of multilevel data structures Brief history Current applications

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Multilevel Modeling

1. Overview

2. Application #1: Growth Modeling

Break

3. Application # 2: Individuals Nested Within Groups

4. Questions?

Overview
• What is multilevel modeling?
• Examples of multilevel data structures
• Brief history
• Current applications
• Why multilevel modeling?
• What types of studies use multilevel modeling?
• Computer Programs (HLM 6

SAS Mixed

• Resources

Multilevel Question

• What effects do the following variables have on 3rd grade reading achievement?

School Size

Classroom Climate

Student Gender

Several Types of Nesting
• 1. Individuals Nested Within Groups

Individuals Undivided

Unit of Analysis = Individuals

Individuals Nested Within Groups

Unit of Analysis = Individuals + Classes

… and Further Nested

Unit of Analysis = Individuals + Classes + Schools

Examples of Multilevel Data Structures
• Neighborhoods are nested within communities
• Families are nested within neighborhoods
• Children are nested within families
Examples of Multilevel Data Structures
• Schools are nested within districts
• Classes are nested within schools
• Students are nested within classes
Multilevel Data Structures

Level 4 District (l)

Level 3 School (k)

Level 2 Class (j)

Level 1 Student (i)

2nd Type of Nesting
• Repeated Measures Nested Within Individuals

Focus = Change or Growth

Repeated Measures Nested Within Individuals

Carlos

Day Energy Level

Monday = 0 98

Tuesday = 1 90

Wednes. = 2 85

Thursday = 3 72

Friday = 4 70

3rd Type of Nesting (similar to the 2nd)
• Repeated Measures Nested Within Individuals

Focus is not on change

Focus in on relationships between variables within an individual

Repeated Measures Nested Within Individuals

Carlos

Day Hours of SleepEnergy Level

Monday 9 98

Tuesday 8 90

Wednesday 8 85

Thursday 6 72

Friday 7 70

Repeated Measures Within Persons

Level 2 Student (i)

Level 1 Repeated Measures

Over Time (t)

Nested Data
• Data nested within a group tend to be more alike than data from individuals selected at random.
• Nature of group dynamics will tend to exert an effect on individuals.
Nested Data
• Intraclass correlation (ICC) provides a measure of the clustering and dependence of the data

0 (very independent) to 1.0 (very dependent)

Details discussed later

Brief Historyof Multilevel Modeling

Robinson, W. S. (1950). Ecological correlations and the behavior of individuals. Sociological Review, 15, 351-357.

Burstein, Leigh (1976). The use of data from groups for inferences about individuals in educational research. Doctoral Dissertation, Stanford University.

Table 1

Frequency of HLM application evidenced in Scholarly Journals

Some Current Applications of Multilevel Modeling
• Growth Curve Analysis
• Value Added Modeling of Teacher and School Effects
• Meta-Analysis
Multilevel Modeling Seems New But….

Extension of General Linear Modeling

Simple Linear Regression

Multiple Linear Regression

ANOVA

ANCOVA

Repeated Measures ANOVA

Multilevel Modeling
• Our focus will be on observed variables (not Latent Variables as in Structural Equation Modeling)

• Individual level analysis (ignore group)
• Group level analysis (aggregate data and ignore individuals)
• Individual level analysis (ignore group)

Violation of independence of data assumption leading to misestimated standard errors (standard errors are smaller than they should be).

• Group level analysis

(aggregate data and ignore individuals)

Aggregation bias = the meaning of a variable at Level-1 (e.g., individual level SES) may not be the same as the meaning at Level-2 (e.g., school level SES)

Multilevel Approach
• 2 or more levels can be considered simultaneously
• Can analyze within- and between-group variability
What Types of Studies Use Multilevel Modeling?

Quantitative

Experimental

*Nonexperimental

(Survey, Observational)

How Many Levels Are Usually Examined?

2 or 3 levels very common

15 students x 10 classes x 10 schools

= 1,500

Types of Outcomes
• Continuous Scale (Achievement, Attitudes)
• Binary (pass/fail)
• Categorical with 3 + categories
Software to do Multilevel Modeling

SPSS Users

2 SAV Files: Level 1

Level 2

(Raudenbush, Bryk, Cheong, & Congdon, 2004)

Resources (Sample…see handouts for more complete list)
• Books
• Hierarchical Linear Models: Applications and Data Analysis Methods, 2nd ed. Raudenbush & Bryk, 2002.
• Introducing Multilevel Modeling.

Kreft & DeLeeum, 1998.

• Journals
• Educational and Psychological Measurement
• Journal of Educational and Behavioral Sciences
• Software
• HLM6
• SAS (NLMIXED and PROC MIXED)
• MLwiN
• Journal Articles
• See Handouts for various methodological and applied articles
• Data Sets
• NAEP Data
• NELS:88; High School and Beyond
Self-Check 1
• A teacher with 1 classroom of 24 students used weekly curriculum-based measurements to monitor reading over a 14 week period. The teacher was interested in individual students’ rates of change and differences in change by male and female students.
Self-Check 1
• How would you classify this situation?

(a) not multilevel

(b) 2-level

(c) 3-level

Self-Check 2
• A researcher randomly selected 50 elementary schools and randomly selected 30 teachers within each school. The researcher was interested in the relationships between 2 predictors (school size and teachers’ years experience at their current school) and teachers’ job satisfaction.
Self-Check 2
• How would you classify this situation?

(a) not multilevel

(b) 2-level

(c) 3-level

Self-Check 3
• 60 undergraduates from the research participant pool volunteered for a study that used written vignettes to manipulate the interactional style (warm, not warm) of a professor interacting with a student.  30 randomly assigned students read the vignette depicting warmth and 30 randomly assigned students read the vignette depicting a lack of warmth.  After reading the vignette students used a questionnaire to rate the likeability of the professor.
Self-Check 3
• How would you classify this situation?

(a) not multilevel

(b) 2-level

(c) 3-level

(Select ONLY one)

Growth Curve Modeling
• Studying the growth in reading achievement over a two year period
• Studying changes in student attitudes over the middle school years
Research Questions
• What is the form of change for an individual during the study?
Research Questions
• What is an individual’s initial status on the outcome of interest?
Research Questions
• How much does an individual change during the course of the study?

Rise

Run

Research Questions
• What is the average initial status of the participants?
Research Questions
• What is the average change of the participants?
Research Questions
• To what extent do participants vary in their initial status?
Research Questions
• To what extent do participants vary in their growth?
Research Questions
• To what extent does initial status relate to growth?
Research Questions
• To what extent is initial status related to predictors of interest?
Research Questions
• To what extent is growth related to predictors of interest?
Design Issues
• How many waves a data collection are needed?
• >2
• Depends on complexity of growth curve
Design Issues
• Can there be different numbers of observations for different participants?

Examples

• Missing data
• Planned missingness
Design Issues
• Can the time between observations vary from participant to participant?

Example: Students observed

• 1, 3, 5, & 7 months
• 1, 2, 4, & 8 months
• 2, 4, 6, & 8 months
Design Issues
• How many participants are needed?
• More is better
• Power analyses
• > 30 rule of thumb
Design Issues
• How should participants be sampled?
• What you have learned about sampling still applies
Design Issues
• What is the value of random assignment?
• What you have leaned about random assignment still applies
Design Issues
• How should the outcome be measured?
• What you have learned about measurement still applies
Example
• Context description

A researcher was interested in changes in verbal fluency of 4th grade students, and differences in the changes between boys and girls.

IDGender   Time______

t0    t4    t7

1    0  20    30    30

2     0          40    44    49

3 0          45    40    60

4     0         50    55    59

5     0          42    48    53

6 1          45    52    61

7 1          39    55    63

8 1          46    58    68

9 1          44    49    59

Example
• Level-1 model specification
Example
• Level-2 model specification
Example
• Combined Model
Example
• SAS program

procmixed covtest;

class gender;

model score = time gender time*gender/s;

random intercept / sub=student s;

Example
• SAS output – variance estimates

Covariance Parameter Estimates

Standard Z

Cov Parm Subject Estimate Error Value Pr Z

Intercept Student 62.5125 35.9682 1.74 0.0411

Residual 14.1173 4.9912 2.83 0.0023

Example
• SAS output – fixed effects

Solution for Fixed Effects

Standard

Effect Gender Estimate Error DF t Value Pr > |t|

Intercept 39.8103 3.7975 7 10.48 <.0001

time 1.5077 0.3295 16 4.58 0.0003

Gender F 5.7090 5.6962 16 1.00 0.3311

Gender M 0 . . . .

time*Gender F 1.0692 0.4943 16 2.16 0.0460

time*Gender M 0 . . . .

Example
• Graph – fixed effects
Example
• Conclusions
• Fourth grade girl’s verbal fluency is increasing at a faster rate than boy’s.
Persons Nested in Contexts
• Studying attitudes of teachers who are nested in schools
• Studying achievement for students who are nested in classrooms that are nested in schools
Research Questions
• How much variation occurs within and among groups?
• To what extent do teacher attitudes vary within schools?
• To what extent does the average teacher attitude vary among schools?
Research Questions
• What is the relationship among selected within group factors and an outcome?
• To what extent do teacher attitudes vary within schools as function of years experience?
• To what extent does student achievement vary within schools as a function of SES?
Research Questions
• What is the relationship among selected between group factors and an outcome?
• To what extent do teacher attitudes vary across schools as function of principal leadership style?
• To what extent does student math achievement vary across schools as a function of the school adopted curriculum?
Research Questions
• To what extent is the relationship among selected within group factors and an outcome moderated by a between group factor?
• To what extent does the within schools relationship between student achievement and SES depend on the school adopted curriculum?
Design Issues
• Consider a design where students are nested in schools
• How should schools should be sampled?
• How should students be sampled within schools?
Design Issues
• Consider a design where students are nested in schools
• How many schools should be sampled?
• How many students should be sampled per school?
Design Issues
• What kind of outcomes can be considered?
• Continuous
• Binary
• Count
• Ordinal
Design Issues
• How will level-1 variables be conceptualized and measured?
• SES
• How will level-2 variables be conceptualized and measured?
• SES
Terminology
• Individual growth trajectory – individual growth curve model
• A model describing the change process for an individual
• Intercept
• Predicted value of an individual’s status at some fixed point
• The intercept cold represent the status at the beginning of a study
• Slope
• The average amount of change in the outcome for every 1 unit change in time
HLM
• Hierarchical Linear Model
• The hierarchical or nested structure of the data
• For growth curve models, the repeated measures are nested within each individual
Levels in Multilevel Models
• Level 1 = time-series data nested within an individual
Levels in Multilevel Models
• Level 2 = model that attempts to explain the variation in the level 1 parameters
More terminology
• Fixed coefficient
• A regression coefficient that does not vary across individuals
• Random coefficient
• A regression coefficient that does vary across individuals
More terminology
• Balanced design
• Equal number of observations per unit
• Unbalanced design
• Unequal number of observation per unit
• Unconditional model
• Simplest level 2 model; no predictors of the level 1 parameters (e.g., intercept and slope)
• Conditional model
• Level 2 model contains predictors of level 1 parameters
Estimation Methods
• Empirical Bayes (EB) estimate
• “optimal composite of an estimate based on the data from that individual and an estimate based on data from other similar individuals” (Bryk, Raudenbush, & Condon, 1994, p.4)
Estimation Methods
• Expectation-maximization (EM) algorithm
• An iterative numerical algorithm for producing maximum likelihood estimates of variance covariance components for unbalanced data.