Multilevel Modeling: Why, When and How?

Frank Dong1-9-2013

Outline

• Why do we need the Multilevel Modeling• When do we need Multilevel Modeling• How can we conduct Multilevel Modeling

analysis (live demo)

Background

• Everyone knows about ordinary least squares regression, aka, linear regression

• The formula is

• We typically assume the error term has a normal distribution N(0, )

• Everyone knows how to do it in SPSS

Problems

• Ordinary least squares analysis does not solve everything

• There are often times where data present certain hierarchy

• For example, the performance of students on the test score may depends on the students themselves, but also may depends on schools

• School effects are often ignored

Purpose of this presentation

• To introduce the idea of multilevel modeling• Not everything can be done with the linear

regression• Live demonstration of how to conduct

multilevel analysis in SPSS.

An example

• This example is from a book called Multilevel Statistical Models, 4th Edition by Harvey Goldstein

• Have data on 728 elementary students• N=50 schools• Interested in the following question: Does the

student’s 8-year math score predict the 11-year math score?

• Y= 11-year math score• X=8-year math score

Some data points

11-year Math Score

8-year Math Score School ID

Gender: Boy=1Girl=0

Social class: Manual=1Non-manual=0

39 36 1 1 0

11 19 1 0 1

32 31 1 0 1

27 23 1 0 0

36 39 1 0 0

Inappropriate Analysis

• For each school, • The overall model becomes

• We have 50 pairs of to estimate, one for each school

• We also have a variance term, to estimate

Issues

• Too many unknown (N=2*50+1) parameters• Unable to compare school performance if we

desires to do so• Some schools have fewer students than other

schools

Solutions

• Multilevel Modeling• Instead of estimating N=2*50+1 unknown

parameters, we will simplify the model• -----Original model• More importantly, and are also treated as

random variable• They are assumed to have a normal

distribution with certain M and SD

Final Solution

• The final model becomes

• The unknown parameters are , variance of , and , and covariance between

• We reduced the number of parameters from 101 to 6

ResultsParameter Multilevel Modeling

Estimate (s.e.)OLS Estimate (s.e.)

FixedIntercept 13.9 13.88-year Math Score 0.65 (0.025) 0.65 (0.026)Random EffectBetween School Variance

3.28

Between Students Variance

19.8 23.34

Variance Partition Coefficient

0.14

Research Question 2

• We also have the gender (1=boy, 2=girl), and social class (1=manual, 0=non-manual), would those two variables affect the performance of the 11-year math grade?

• Is gender significant?• Is social class significant?

Parameters Multilevel Modeling Estimate (s.e.)

OLS Modeling Estimate (s.e.)

Fixed EffectsIntercept 14.88 14.798-year Math Score 0.638 (0.025) 0.638 (0.026)Gender (boy vs girl) -0.357 (0.340) -0.363 (0.358)Social Class (manual vs non-manual)

-0.720 (0.387) -0.697 (0.397)

Random EffectBetween School Variance

3.312

Between Students Variance

19.728 49.36

Variance Partition Coefficient

0.144

How to conduct a Multilevel Modeling

• You do not need to do it by yourself• You are required to be aware of the existence

of multilevel modeling• The benefit is to improve the estimate

accuracy• Here is how to do it in SPSS (live demo)

Summary

• Ordinary least squares regression is not almighty

• When there is a clear structure of hierarchy, multilevel modeling will be useful

• Multilevel modeling can also be used to compare the performance of hospitals