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Multilevel Modeling1. Overview
2. Application #1: Growth Modeling
Break
3. Application # 2: Individuals Nested Within Groups
4. Questions?
Overview
1. What is multilevel modeling?2. Examples of multilevel data structures3. Brief history4. Current applications5. Why multilevel modeling?6. What types of studies use multilevel
modeling?7. Computer Programs (HLM 6
SAS Mixed8. Resources
Multilevel Question
What effects do the following variables have on 3rd grade reading achievement?
School SizeClassroom Climate
Student Gender
What is Multilevel or Hierarchical Linear Modeling?
Nested Data Structures
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
Time Points Nested Within Individuals
Repeated Measures Nested Within Individuals
CarlosDay Energy LevelMonday = 0 98Tuesday = 1 90Wednes. = 2 85Thursday = 3 72Friday= 4 70
Repeated Measures Nested Within Individuals
DAY
543210
EN
ER
GY
100
90
80
70
60
Repeated Measures Nested Within Individuals
DAY
543210
EN
ER
GY
100
90
80
70
60 Rsq = 0.9641
Changes for 5 Individuals
0 1.00 2.00 3.00 4.000
25.00
50.00
75.00
100.00
Time
En
erg
y L
evel
Changes in Energy Level Over the Week
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
CarlosDay Hours of Sleep Energy
LevelMonday 9 98Tuesday 8 90Wednesday 8 85Thursday 6 72Friday 7 70
Repeated Measures Nested Within Individuals (Not Change)
HOURS
9.59.08.58.07.57.06.56.05.5
EN
ER
GY
100
90
80
70
60
Repeated Measures Nested Within Individuals (Not Change)
HOURS
9.59.08.58.07.57.06.56.05.5
EN
ER
GY
100
90
80
70
60
Repeated Measures Nested Within Individuals
2.00 4.50 7.00 9.50 12.000
25.00
50.00
75.00
100.00
Hours of Sleep
En
erg
y L
evel
Repeated Measures Nested Within Individuals (3 Individuals)
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
Journal 1999 2000 2001 2002 2003 Total by journal
American Educational Research Journal 3 5 4 3 ? ~15
Child Development 3 2 6 5 13 29
Cognition and Instruction 1 0 0 0 0 1
Contemporary Educational Psychology 0 0 0 0 0 0
Developmental Psychology 2 1 2 5 7 17
Educational Evaluation and Policy Analysis 2 1 5 2 2 12
Educational Technology, Research and Development 0 0 0 0 0 0
Journal of Applied Psychology 1 1 5 7 6 20
Journal of Counseling Psychology 0 2 1 0 0 3
Journal of Educational Computing Research 0 0 0 0 0 0
Journal of Educational Psychology 1 2 3 6 1 13
Journal of Educational Research 2 0 3 3 5 13
Journal of Experimental Child Psychology 0 0 0 0 0 0
Journal of Experimental Education 0 0 0 0 1 1
Journal of Personality and Social Psychology 4 4 6 5 13 32
Journal of Reading Behavior/Literacy Research 0 0 0 0 0 0
Journal of Research in Mathematics Education 0 0 0 0 0 0
Reading Research Quarterly 0 0 0 1 0 1
Sociology of Education 1 2 5 2 1 11
Total by Year 20 20 40 39 49 ~168
Multilevel ArticlesFrequency of Studies Employing HLM in Education or Related Journals
0
25
50
1999 2000 2001 2002 2003
Year
Fre
qu
ency
Total for 19 Journals Reviewed
Journal of Personality and Social Psychology
Child Development
Journal of Educational Research
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 RegressionMultiple Linear Regression
ANOVAANCOVA
Repeated Measures ANOVA
Multilevel Modeling
Our focus will be on observed variables (not Latent Variables as in Structural Equation Modeling)
Why Multilevel Modelingvs.
Traditional Approaches?
Traditional Approaches – 1-Level
1. Individual level analysis (ignore group)
2. Group level analysis (aggregate data and ignore individuals)
Problems withTraditional Approaches
1. Individual level analysis (ignore group)
Violation of independence of data assumption leading to misestimated standard errors (standard errors are smaller than they should be).
Problems withTraditional Approaches
1. 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 Users2 SAV Files: Level 1
Level 2
HLM 6 (Menu Driven) (Raudenbush, Bryk, Cheong, &
Congdon, 2004)
HLM 6
Software to do Multilevel Modeling
SAS Users
Proc Mixed
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 Multilevel Modeling Newsletter
Resources (cont)(Sample…see handouts for more complete list)
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?
Run
Research Questions
How much does an individual change during the course of the study?
Rise Riseb
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.
ID Gender Time______ t0 t4 t7
1 0 20 30 302 0 40 44 493 0 45 40 604 0 50 55 595 0 42 48 536 1 45 52 617 1 39 55 638 1 46 58 689 1 44 49 59
Example
Level-1 model specification
0 1 1*( )fluencyY Time error
Example
Level-2 model specification
0 00 01 2
1 10 11
*( )
*( )
G G Gender error
G G Gender
Example
Combined Model
00 01 10
11 2 1
*( ) *( )
*( ) *( )fluencyY G G Gender G Time
G Gender Time error error
Example
SAS program
proc mixed covtest;
class gender;
model score = time gender time*gender/s;
random intercept / sub=student s;
Example
SAS output – variance estimates
Covariance Parameter Estimates Standard ZCov Parm Subject Estimate Error Value Pr Z Intercept Student 62.5125 35.9682 1.74 0.0411Residual 14.1173 4.9912 2.83 0.0023
Example SAS output – fixed effects
Solution for Fixed Effects StandardEffect Gender Estimate Error DF t Value Pr > |t| Intercept 39.8103 3.7975 7 10.48 <.0001time 1.5077 0.3295 16 4.58 0.0003Gender F 5.7090 5.6962 16 1.00 0.3311Gender M 0 . . . .time*Gender F 1.0692 0.4943 16 2.16 0.0460time*Gender M 0 . . . .
Example Graph – fixed effects
0
25.00
50.00
75.00
100.00
SC
OR
E
0 2.50 5.00 7.50 10.00
TIME
GENDER = 0
GENDER = 1
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
I ntercept & Slope I llustration
0
5
10
15
20
25
0 1 2 3 4 5 6 7 8 9 10
Time
Sco
re
RiseRise
bRun
Run
intercept
Curvature =Acceleration=Quadratic Component
0
5
10
15
20
25
30
35
0 1 2 3 4
Time
Sco
re
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
0 1 *( )Y Time error
Levels in Multilevel Models
Level 2 = model that attempts to explain the variation in the level 1 parameters
0 00 01
1 10 11
*( )
*( )
G G Sessions error
G G Sessions error
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.