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Multilevel Modeling PowerPoint PPT Presentation


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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

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Multilevel modeling l.jpg

Multilevel Modeling

1.Overview

2.Application #1: Growth Modeling

Break

3.Application # 2: Individuals Nested Within Groups

4.Questions?


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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


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Multilevel Question

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

    School Size

    Classroom Climate

    Student Gender


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What is Multilevel or Hierarchical Linear Modeling?

Nested Data Structures


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Several Types of Nesting

  • 1.Individuals Nested Within Groups


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Individuals Undivided

Unit of Analysis = Individuals


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Individuals Nested Within Groups

Unit of Analysis = Individuals + Classes


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… and Further Nested

Unit of Analysis = Individuals + Classes + Schools


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Examples of Multilevel Data Structures

  • Neighborhoods are nested within communities

  • Families are nested within neighborhoods

  • Children are nested within families


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Examples of Multilevel Data Structures

  • Schools are nested within districts

  • Classes are nested within schools

  • Students are nested within classes


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Multilevel Data Structures

Level 4 District (l)

Level 3 School (k)

Level 2 Class (j)

Level 1 Student (i)


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2nd Type of Nesting

  • Repeated Measures Nested Within Individuals

    Focus = Change or Growth


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Time Points Nested Within Individuals


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Repeated Measures Nested Within Individuals

Carlos

DayEnergy Level

Monday = 098

Tuesday = 190

Wednes. = 285

Thursday = 372

Friday= 470


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Repeated Measures Nested Within Individuals


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Repeated Measures Nested Within Individuals


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Changes for 5 Individuals


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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


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Repeated Measures Nested Within Individuals

Carlos

DayHours of SleepEnergy Level

Monday998

Tuesday890

Wednesday885

Thursday672

Friday770


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Repeated Measures Nested Within Individuals (Not Change)


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Repeated Measures Nested Within Individuals (Not Change)


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Repeated Measures Nested Within Individuals


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Repeated Measures Within Persons

Level 2 Student (i)

Level 1 Repeated Measures

Over Time (t)


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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.


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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


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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.


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Table 1

Frequency of HLM application evidenced in Scholarly Journals


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Multilevel Articles


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Some Current Applications of Multilevel Modeling

  • Growth Curve Analysis

  • Value Added Modeling of Teacher and School Effects

  • Meta-Analysis


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Multilevel Modeling Seems New But….

Extension of General Linear Modeling

Simple Linear Regression

Multiple Linear Regression

ANOVA

ANCOVA

Repeated Measures ANOVA


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Multilevel Modeling

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


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Why Multilevel Modelingvs. Traditional Approaches?

Traditional Approaches – 1-Level

  • Individual level analysis (ignore group)

  • Group level analysis (aggregate data and ignore individuals)


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Problems withTraditional Approaches

  • Individual level analysis (ignore group)

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


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Problems withTraditional Approaches

  • 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)


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Multilevel Approach

  • 2 or more levels can be considered simultaneously

  • Can analyze within- and between-group variability


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What Types of Studies Use Multilevel Modeling?

Quantitative

Experimental

*Nonexperimental

(Survey, Observational)


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How Many Levels Are Usually Examined?

2 or 3 levels very common

15 students x 10 classes x 10 schools

= 1,500


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Types of Outcomes

  • Continuous Scale (Achievement, Attitudes)

  • Binary (pass/fail)

  • Categorical with 3 + categories


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Software to do Multilevel Modeling

SPSS Users

2 SAV Files: Level 1

Level 2

HLM 6 (Menu Driven)

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


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HLM 6


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Software to do Multilevel Modeling

SAS Users

Proc Mixed


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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


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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


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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.


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Self-Check 1

  • How would you classify this situation?

    (a) not multilevel

    (b) 2-level

    (c) 3-level


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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.


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Self-Check 2

  • How would you classify this situation?

    (a) not multilevel

    (b) 2-level

    (c) 3-level


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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.


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Self-Check 3

  • How would you classify this situation?

    (a) not multilevel

    (b) 2-level

    (c) 3-level

(Select ONLY one)


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Growth Curve Modeling

  • Studying the growth in reading achievement over a two year period

  • Studying changes in student attitudes over the middle school years


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Research Questions

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


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Research Questions

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


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Research Questions

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

Rise

Run


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Research Questions

  • What is the average initial status of the participants?


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Research Questions

  • What is the average change of the participants?


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Research Questions

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


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Research Questions

  • To what extent do participants vary in their growth?


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Research Questions

  • To what extent does initial status relate to growth?


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Research Questions

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


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Research Questions

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


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Design Issues

  • How many waves a data collection are needed?

    • >2

    • Depends on complexity of growth curve


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Design Issues

  • Can there be different numbers of observations for different participants?

    Examples

    • Missing data

    • Planned missingness


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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


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Design Issues

  • How many participants are needed?

    • More is better

    • Power analyses

    • > 30 rule of thumb


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Design Issues

  • How should participants be sampled?

    • What you have learned about sampling still applies


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Design Issues

  • What is the value of random assignment?

    • What you have leaned about random assignment still applies


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Design Issues

  • How should the outcome be measured?

    • What you have learned about measurement still applies


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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.


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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


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Example

  • Level-1 model specification


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Example

  • Level-2 model specification


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Example

  • Combined Model


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Example

  • SAS program

    procmixed covtest;

    class gender;

    model score = time gender time*gender/s;

    random intercept / sub=student s;


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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


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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 . . . .


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Example

  • Graph – fixed effects


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Example

  • Conclusions

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


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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


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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?


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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?


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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?


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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?


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Design Issues

  • Consider a design where students are nested in schools

    • How should schools should be sampled?

    • How should students be sampled within schools?


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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?


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Design Issues

  • What kind of outcomes can be considered?

    • Continuous

    • Binary

    • Count

    • Ordinal


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Design Issues

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

    • SES

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

    • SES


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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


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intercept


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HLM

  • Hierarchical Linear Model

    • The hierarchical or nested structure of the data

    • For growth curve models, the repeated measures are nested within each individual


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Levels in Multilevel Models

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


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Levels in Multilevel Models

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


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More terminology

  • Fixed coefficient

    • A regression coefficient that does not vary across individuals

  • Random coefficient

    • A regression coefficient that does vary across individuals


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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


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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)


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Estimation Methods

  • Expectation-maximization (EM) algorithm

    • An iterative numerical algorithm for producing maximum likelihood estimates of variance covariance components for unbalanced data.


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