Introduction to Multilevel Modeling

Stephen R. Porter

Associate Professor

Dept. of Educational Leadership and Policy Studies

Iowa State University

Lagomarcino Hall

Ames, IA 50011

Email: srporter@iastate.edu

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Goals of the workshop

Understand why multilevel modeling is important and understand basic 2-level models.

Become informed consumer of multilevel research. Know how to estimate some simple models using the

software package HLM. Have a thorough grounding in the basics so you can

learn more complicated multi-level techniques (3-level, SEM, etc.) on your own.

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Schedule

1st day Review and discuss multilevel terminology and

theory Begin reviewing choices in model building

2nd day Estimate simple 2-level models using student

version of HLM Discuss in detail model building.

Introduction

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Why multilevel modeling?

Nested data are very common in higher education. Analysis of nested data poses unit of analysis

problem – should we analyze the individual or the group? Unfortunately, we often can’t choose one over the other.

Traditional linear models offer a simple view of a complex world – generally assume same effects across groups.

If effects do differ across groups, we can explain these differences with multilevel modeling.

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Unit of analysis problem: individual, group or both?

Example: studying what affects student retention (1000 students per college) in a group of colleges (n=50). Total dataset N=50,000.

We can assign college-level variables to each individual, but … We end up estimating the standard errors for

college-level variables using N=50,000. Yet we only have 50 different college

observations, so N really equals 50.

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Unit of analysis problem: individual, group or both?

Alternatively, we can average student data for each college so that we have 1 observation per college (N=50). Now we have reduced variance on our student-

level variables. We also have variables which measure both

individual student characteristics (SAT score=aptitude/preparation) and college environment (average SAT score=selectivity).

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What are nested data?

Simply put, sub-units are grouped (or “nested”) within larger units.

Often the data are observations of individuals nested within groups. Key: individuals within groups are more similar to

one another than to individuals in other groups. We can empirically verify this.

Sometimes data are multiple observations nested within an individual.

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Students/faculty nested within departments/disciplines

Paul

M aria Jerry Vicky

History

Bill Anthon y

Basketw eaving

Jose

Steve Claire

Physics

Liz John

G overnm ent

Note that this could be one institution, or individuals from several different institutions.

Examples: student satisfaction, gains in skills; faculty salaries, research productivity.

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Students/faculty nested within institutions

Paul

M aria Jerry Vicky

FloridaState

University

Bill Anthony

University ofM aryland,

CollegePark

Jose

Steve Claire

W esleyanUniversity

Liz John

PrinceGeorge's

Com m unityCollege

Examples: student satisfaction, retention

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Time periods nested within students

Fall1998

Spring1999

Fall1999

Steve

Spring1999

Fall1999

Claire

Fall1998

Fall1998

Fall1999

Vicky

Spring1999

Fall1999

Paul

Example: grade-point average

Terms and theory

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Terminology: HLM

HLM stands for hierarchical linear models. It is both a statistical technique and a software

package. People also use the term multilevel models. Economists often refer to these models as random-

coefficient regression models

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Terminology: levels

Level-1 variables: These are the variables that are nested within

groups. Typically these are individual-level variables.

Level-2 variables Typically these are unit-level variables.

Note that growth models have time periods at level-1, and individuals at level-2.

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Terminology: variance

Numbers represent people, each number is a person’s question response on a 5-point Likert scale; 6 groups

Variance between groups only:

1111 2222 5555 4444 2222 3333

Variance within groups only:

1235 1235 1235 1235 1235 1235

Variance both between and within groups:

1112 2233 2333 3344 3444 4455

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Terminology: random and fixed

Fixed effects are variable coefficients that are constant across groups, they do not vary. Typical OLS coefficients.

Random effects are coefficients that can vary across groups. This means the coefficient can take a different value

for each group. E.g., if we allow an intercept for each group, then the intercept is said to be random.

It is random because we assume it is stochastic. Yet we can also explain some of this variance with

other variables.

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One way to think about multilevel models: “slopes-as-outcomes”

Suppose we estimate 1 regression equation for each group, e.g., for the 1,000 students in school A, the 1,000 students in school B, etc. The result is 50 regression equations.

We then take the slope coefficients for each school, as well as information about each school such as private/public status, and make a new dataset.

We run a regression model on these 50 observations using the slope coefficients (or intercepts) as the dependent variable and public/private status as the independent variable.

The result is a single set of coefficients for the school dataset.

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Now for some algebra!

You must learn some of the basic mathematical notation used in multilevel modeling. As we will see, the program HLM uses this

notation to express the models that you estimate. Understanding these basic symbols and

expressions will allow you to tackle more complex analyses, and understand other researchers’ more complex analyses.

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A level-1 model: multiple students in one school (familiar OLS equation)

Student i is viewed as having average achievement in the school, plus a positive deviation due to SES, plus a positive or negative deviation due to the unique circumstances of the student.

term) (error i student for effect unique is r

variable) nt(independe SES edstandardiz si' student is X

(slope) tachievemen on SES of effect average is β

)(intercept school withintachievemen average is β

score tachievemen math si' student is Y

rXββY

i

i

1

0

i

ii10i

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A level-1 model: multiple students in multiple schools

Now we’re estimating the equation from before for each school. Each school can have a different average achievement (or intercept), and a different impact of SES on achievement (or slope).

j school in i student for effect unique is r

j school in i student of SES edstandardiz is X

j school for tachievemen on SES of effect average is β

j school withintachievemen average is β

j number school in tachievemen si' student is Y

rXββY

ij

ij

1j

0j

ij

ijij1j0jij

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Need to make some additional assumptions about the coefficients, because they vary

Student-level errors are normally distributed. Gamma’s: we expect the average achievement for school j

to equal the average school mean for all j schools, and the slope of SES for school j to equal the average of the slopes for all j schools.

Tau’s: these are the variances of the intercepts and slopes, and the covariance between them.

011j,0j

111j101j

000j000j

2ij

)βCov(β

)(β Var,)E(β

)β Var(,)E(β

),0(N~r

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Level-2 model: explaining the Level-1 coefficients

Since our intercepts and slopes vary by school, we can now model why they vary.

Suppose we hypothesize that levels of achievement and impact of SES are related to whether a school is public or Catholic.

We need equations for the intercept and slope to describe our hypothesis:

j school for tachievemen on SES of effect average is β

j school withintachievemen average is β

t)coefficien (slope uWβ

)(intercept uWβ

1j

0j

1jj11101j

0jj01000j

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Level-2 model: continued

SES of effect on j school of effect unique is u

tachievemen mean on j school of effect unique is u

SES of effect the on type school of impact is

schools in tachievemen mean on type school of impact is

schools across SES of impact average is

schools across tachievemen mean is

public if 0 Catholic, is j school if 1 variable,dummy a is W

uWβ

uWβ

j1

j0

11

01

10

00

j

1jj11101j

0jj01000j

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So math achievement of an individual student in school j is explained by …

ijij1j0j

ijj11ij10

j0100ij

rXuu

XWX

WY

mean achievement in public schools,plus impact of a school being Catholicon mean achievement (if j is Catholic)

the effect of SES on achievement, plus the impact of a school being Catholic on how SES affects achievement (again, ifj is Catholic)

student- and school-specific error terms

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Summary

0jj01000j

1jj11101j

ijij1j0jij

uWβ

uWβ

rXββY

Level-1

Level-2

Level-2

Explain dependent variable

Explain slopes

Explain intercepts

Some practical aspects of multilevel modeling

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Questions to answer

Can you use multilevel techniques to study your dependent variable?

Should you use multilevel techniques to study your dependent variable?

How will you center your level-1 and level-2 predictors?

Which of the level-1 coefficients will be explained at level-2? I.e., are they fixed or random?

How does my model perform?

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Can I use HLM?

HLM requires a large amount of data. Minimum:

number of groups: 30, but most recommend 50+ number of individuals within groups: 5-10, but can

have low as 1. average group size: 10, obviously more is better.

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Should I be using HLM?

How much of the variance in your dependent variable is explained by group membership?

Intraclass correlation coefficient (ICC) = var between groups (var between groups+var within groups)

)/( 20000

variance level-student the is and means, school

the or ,intercepts the of variance the is Remember,2

00

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Centering variables

Whether and how you center is a very important decision: interpretation of results depends on your choice.

Important because the intercept at level-1 is also a dependent variable.

Centering Refers to subtracting a mean from your

independent variables. The transformed value for an individual measures

how much they deviate (+/-) from the mean.

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Centering variables

Suppose we center verbal SAT scores around a student mean of 500.

How would we interpret a regression coefficient if all variables were similarly transformed?

Actual score

Centered score

Steve 800 300Claire 750 250Bill 500 0Paul 200 -300

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Centering variables

Why would we want to center? Variable may lack a natural zero point, such as

SAT score. Stability of estimates at level-1 affected by location

of variables. Location at level-2 is less important.

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Centering variables

Generally two types of centering. For a specific variable: Grand mean centering – subtract the mean for the

entire sample from each observation in the sample.

Group mean centering – subtract the mean for each group from each member of the group.

To fully understand the implications of centering, see the discussion in Bryk and Raudenbush (2002) pp. 134-149.

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Fixed or random?

It would be nice to have everything random; that is, a different set of coefficients for each group.

But due to HLM demands on data, usually only the intercept and a few variables can be random.

Important: if you randomize gender and you have a group without females, that group will be dropped.

Generally you should run parallel models for intercept and slopes, as in our theory example.

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Model statistics

Goodness of fit: Proportion of variance explained at level-1

Variance explained at level-2

)(

)()(2

22

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fullnull

)(

)()(

null

fullnull

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Some thoughts about building your models

Before using HLM, run OLS regressions for sample and for each group.

Building the null model: This is should be your first step. Calculate the ICC

Building the level-1 models: Should be theory driven Step-up approach Be cautious about what you leave as random – it’s

often difficult to leave more than the intercept and one variable as random

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Some thoughts about building your models

Building the level-2 models Rule of thumb: 10 observations/variable Parallel models

Many scholars drop insignificant variables at both levels. (I disagree with this.)