Multilevel Modeling 1.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 structures 3. Brief history 4. Current applications 5. Why multilevel modeling? 6. What types of studies use multilevel modeling? 7. Computer Programs (HLM 6 SAS Mixed 8. Resources

Multilevel Question What effects do the following variables have on 3 rd grade reading achievement? School Size Classroom 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)

2 nd Type of Nesting Repeated Measures Nested Within Individuals Focus = Change or Growth

Time Points Nested Within Individuals

Repeated Measures Nested Within Individuals Carlos DayEnergy Level Monday = 098 Tuesday = 190 Wednes. = 285 Thursday = 372 Friday= 470

Repeated Measures Nested Within Individuals

Changes for 5 Individuals

3 rd Type of Nesting (similar to the 2 nd ) 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 DayHours of SleepEnergy Level Monday998 Tuesday890 Wednesday885 Thursday672 Friday770

Repeated Measures Nested Within Individuals (Not Change)

Repeated Measures Nested Within 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 History of Multilevel Modeling Robinson, W. S. (1950). Ecological correlations and the behavior of individuals. Sociological Review, 15, 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 Total by journal American Educational Research Journal 3543?~15 Child Development Cognition and Instruction Contemporary Educational Psychology Developmental Psychology Educational Evaluation and Policy Analysis Educational Technology, Research and Development Journal of Applied Psychology Journal of Counseling Psychology Journal of Educational Computing Research Journal of Educational Psychology Journal of Educational Research Journal of Experimental Child Psychology Journal of Experimental Education Journal of Personality and Social Psychology Journal of Reading Behavior/Literacy Research Journal of Research in Mathematics Education Reading Research Quarterly Sociology of Education Total by Year ~168

Multilevel Articles

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)

Why Multilevel Modeling vs. Traditional Approaches? Traditional Approaches – 1-Level 1. Individual level analysis (ignore group) 2. Group level analysis (aggregate data and ignore individuals)

Problems with Traditional 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 with Traditional 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 Users 2 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, 2 nd ed. Raudenbush & Bryk, Introducing Multilevel Modeling. Kreft & DeLeeum, 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

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 4 th grade students, and differences in the changes between boys and girls.

ID Gender Time______ t0 t4 t

Example Level-1 model specification

Example Level-2 model specification

Example Combined Model

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 Z Cov Parm Subject Estimate Error Value Pr Z Intercept Student Residual

Example SAS output – fixed effects Solution for Fixed Effects Standard Effect Gender Estimate Error DF t Value Pr > |t| Intercept <.0001 time Gender F Gender M time*Gender F 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

intercept

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.