Transcript Slide 1
Hierarchical Linear
Modeling (HLM): A
Conceptual Introduction
Jessaca Spybrook
Educational Leadership, Research, and
Technology
Overview
What is hierarchical data?
Why is it a problem for analysis?
Example
Modeling the hierarchical structure
Example
1
student level predictor
1 student level predictor, 2 school level predictors
Questions
Slide 2
What is hierarchical (nested) data?
Examples
Kids
in classrooms
Kids in classrooms in schools
Kids in classrooms in schools in districts
Workers in firms
Patients in doctors offices
Repeated measures on individuals
Other examples?
Slide 3
Why is it problematic?
What is the relationship between SES and
math achievement?
Dependent
variable: Math achievement
Independent variable: Student SES
Case 1: 1 School (school A)
School
A
Yi 0 1 ( SESi ) ri
ri ~ N (0, 2 )
^
Mean achievement: 0 9.73
^
SES achievement slope: 1 2.51
Slide 4
Why is it problematic?
Case 2: 1 school (School B)
School
BY
i
0 1 ( SESi ) ri
ri ~ N (0, 2 )
^
Mean achievement: 0 13.51
^
SES-achievement slope: 1 3.26
Case 3: 160 schools
160
means, mean varies
160 SES-achievement slope parameters, slope
varies
Within school variation
Slide 5
Why is it problematic?
Case 3: 160 schools
Option
A: Ignore nesting
Violate assumptions for traditional linear model
Standard errors too small
Option
Lose information
Option
B: Aggregate to school level
C: Model the hierarchical structure
Hierarchical linear models, multilevel models, mixed
effects models, random effects models, random
coefficient models
Slide 6
Modeling the hierarchical structure
Advantages
Improved
estimation of individual (school effects)
Test hypotheses for cross level effects
Partition variance and covariance among levels
Level 1 : Yij 0 j 1 j ( SESij ) rij
rij ~ N (0, 2 )
Level 2 : 0 j 00 u0 j
1 j 10 u1 j
u0 j
~ MVN 00 01
u
ij
10 11
Combined : Yij 00 10 ( SESij ) u0 j u1 j ( SESij ) rij
Slide 7
Example
Results – what do they mean?
Fixed Effect
Coefficient
Standard
Error
t-ratio
p-value
Overall mean
achievement 00
12.64
0.24
51.84
<0.001
Mean SES-ach
slope 10
2.19
0.13
17.16
<0.001
Random Effects
Variance
Df
Chi-square
p-value
School means,u0j
8.68
159
1770.86
<0.001
SES-ach slope, u1j
0.68
159
213.44
0.003
Within school, rij
36.70
Slide 8
Example
School-level predictors
Do
Catholic schools differ from public schools in
terms of mean achievement (controlling for school
mean ses)?
Do Catholic schools differ from public schools in
terms of strength of association between student
SES and achievement (controlling for school mean
ses)?
Slide 9
Example
School level predictors
Level 1 : Yij 0 j 1 j ( SESij ) rij
Level 2 : 0 j 00 01 (Catholic j ) 02 ( MeanSES j ) u0 j
1 j 10 11 (Catholic j ) 12 ( MeanSES j ) u1 j
Combined : Yij 00 01 (Catholic j ) 02 ( MeanSES j ) 10 ( SESij )
11 (Catholic j )( SESij ) 12 ( MeanSES j )( SESij ) u0 j u1 j ( SESij ) rij
Slide 10
Example
Results – what do they mean?
Fixed Effect
Coefficient
Standard
Error
t-ratio
p-value
Model for school
means
Intercept 00
12.09
0.17
69.64
<0.001
0.31
3.98
<0.001
0.33
15.94
<0.001
0.15
19.90
<0.001
0.24
-6.91
<0.001
0.33
3.11
0.002
Catholic 01
1.23
MEAN SES 02 5.33
Model for SES-ach
slope
Intercept 10
2.94
Catholic 11
-1.64
MEAN SES 12 1.03
Slide 11
Example
Visual Look
22.79
SECTOR = 0
SECTOR = 1
17.40
MATHACH
12.01
6.61
1.22
-3.76
-2.41
-1.05
0.30
1.65
SES
Slide 12