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