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Multilevel Modeling

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1.Overview

2.Application #1: Growth Modeling

Break

3.Application # 2: Individuals Nested Within Groups

4.Questions?

- 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

Nested Data Structures

- 1.Individuals Nested Within Groups

Individuals Undivided

Unit of Analysis = Individuals

Unit of Analysis = Individuals + Classes

Unit of Analysis = Individuals + Classes + Schools

- Neighborhoods are nested within communities
- Families are nested within neighborhoods
- Children are nested within families

- Schools are nested within districts
- Classes are nested within schools
- Students are nested within classes

Level 4 District (l)

Level 3 School (k)

Level 2 Class (j)

Level 1 Student (i)

- Repeated Measures Nested Within Individuals
Focus = Change or Growth

Carlos

DayEnergy Level

Monday = 098

Tuesday = 190

Wednes. = 285

Thursday = 372

Friday= 470

- Repeated Measures Nested Within Individuals
Focus is not on change

Focus in on relationships between variables within an individual

Carlos

DayHours of SleepEnergy Level

Monday998

Tuesday890

Wednesday885

Thursday672

Friday770

Level 2 Student (i)

Level 1 Repeated Measures

Over Time (t)

- 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.

- 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

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

- Growth Curve Analysis
- Value Added Modeling of Teacher and School Effects
- Meta-Analysis

Extension of General Linear Modeling

Simple Linear Regression

Multiple Linear Regression

ANOVA

ANCOVA

Repeated Measures ANOVA

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

Traditional Approaches – 1-Level

- 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)

- 2 or more levels can be considered simultaneously
- Can analyze within- and between-group variability

Quantitative

Experimental

*Nonexperimental

(Survey, Observational)

2 or 3 levels very common

15 students x 10 classes x 10 schools

= 1,500

- Continuous Scale (Achievement, Attitudes)
- Binary (pass/fail)
- Categorical with 3 + categories

SPSS Users

2 SAV Files: Level 1

Level 2

HLM 6 (Menu Driven)

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

SAS Users

Proc Mixed

- 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

- 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

- 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.

- How would you classify this situation?
(a) not multilevel

(b) 2-level

(c) 3-level

- 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.

- How would you classify this situation?
(a) not multilevel

(b) 2-level

(c) 3-level

- 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.

- How would you classify this situation?
(a) not multilevel

(b) 2-level

(c) 3-level

(Select ONLY one)

- Studying the growth in reading achievement over a two year period
- Studying changes in student attitudes over the middle school years

- What is the form of change for an individual during the study?

- What is an individual’s initial status on the outcome of interest?

- How much does an individual change during the course of the study?

Rise

Run

- What is the average initial status of the participants?

- What is the average change of the participants?

- To what extent do participants vary in their initial status?

- To what extent do participants vary in their growth?

- To what extent does initial status relate to growth?

- To what extent is initial status related to predictors of interest?

- To what extent is growth related to predictors of interest?

- How many waves a data collection are needed?
- >2
- Depends on complexity of growth curve

- Can there be different numbers of observations for different participants?
Examples

- Missing data
- Planned missingness

- 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

- How many participants are needed?
- More is better
- Power analyses
- > 30 rule of thumb

- How should participants be sampled?
- What you have learned about sampling still applies

- What is the value of random assignment?
- What you have leaned about random assignment still applies

- How should the outcome be measured?
- What you have learned about measurement still applies

- 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

30 45 40 60

4 0 50 55 59

5 0 42 48 53

61 45 52 61

71 39 55 63

81 46 58 68

91 44 49 59

- Level-1 model specification

- Level-2 model specification

- Combined Model

- SAS program
procmixed covtest;

class gender;

model score = time gender time*gender/s;

random intercept / sub=student s;

- 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

- 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 . . . .

- Graph – fixed effects

- Conclusions
- Fourth grade girl’s verbal fluency is increasing at a faster rate than boy’s.

- Studying attitudes of teachers who are nested in schools
- Studying achievement for students who are nested in classrooms that are nested in schools

- 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?

- 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?

- 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?

- 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?

- Consider a design where students are nested in schools
- How should schools should be sampled?
- How should students be sampled within schools?

- Consider a design where students are nested in schools
- How many schools should be sampled?
- How many students should be sampled per school?

- What kind of outcomes can be considered?
- Continuous
- Binary
- Count
- Ordinal

- How will level-1 variables be conceptualized and measured?
- SES

- How will level-2 variables be conceptualized and measured?
- SES

- 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

intercept

- Hierarchical Linear Model
- The hierarchical or nested structure of the data
- For growth curve models, the repeated measures are nested within each individual

- Level 1 = time-series data nested within an individual

- Level 2 = model that attempts to explain the variation in the level 1 parameters

- Fixed coefficient
- A regression coefficient that does not vary across individuals

- Random coefficient
- A regression coefficient that does vary across individuals

- 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

- 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)

- Expectation-maximization (EM) algorithm
- An iterative numerical algorithm for producing maximum likelihood estimates of variance covariance components for unbalanced data.