Multi-level Modeling (MLM)
Refresher
Jessaca Spybrook
Western Michigan University
MLM Refresher
Goal of session
 Brief review of multilevel models
 Establish common language
 Establish common notation
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Hierarchical Data
 Individuals grouped into larger units
 Examples
 Students in schools
 Citizens in communities
 Focus Example
 Africa Program for Education Impact Evaluation in the
Gambia
Students in schools
Teachers in schools
Classrooms in schools
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Hierarchical Data
Methods for analyzing data
 Put everything at one level
 Aggregate data up to level 2
 Model both levels together
Model both levels together
 Improved estimation of individual effects
 Questions related to cross-level effects
 Partitioning of variance among levels
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Hierarchical Data
Modeling both levels together
 Hierarchical linear models, multilevel models,
mixed effects models, random effects models,
random coefficient models
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Hierarchical Data
Scenario




The Gambia data (2008)
Students nested in schools
2,657 ->2,008 students (pupils data)
271 ->204 schools (head teacher data)
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Hierarchical Data
Variables
 DV:
Number of words read correctly in 60 seconds
(reading fluency) [S2Q3_PP]
 IVs:
Treatment (L2) [trmt –WSC and Grant]
Age (L1) [age_PP]
Mean school age (L2) [age_PP_m]
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Guiding Questions (A)
Guiding Questions (A)
 What is the mean reading fluency for all
students?
 How much variation in reading fluency is
between schools? Within schools?
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The Model
Level 1 (students):
Yij   0 j  rij
Yij is reading fluency for student i in school j
 0 j is the mean reading fluency for school j
rij is the random error associated with student i in school j, var( rij )   2
Level 2 (schools):
 0 j   00  u0 j
 00 is the average school mean reading fluency across the schools
u0j is the random error associated with school means, var( u0 j )   00
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The Model
Combined Model:
Yij   00  u0 j  rij
- Demo in HLM
- Fill in Table as we go
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ICC
Intraclass Correlation:
 00
85.33



0
.
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2
 00   85.33  381.22
 00 is the between school variance
 is the within school variance
2
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Guiding Question (B)
Guiding Question (B)
 Is there a difference in reading fluency at
baseline for those in the treatment condition
compared to those in the control condition?
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The Model
Level 1 (students):
Yij   0 j  rij
Yij is reading fluency for student i in school j
 0 j is the mean reading fluency for school j
rij is the random error associated with student i in school j, var( rij )   2
Level 2 (schools):
 0 j   00   01W j  u0 j
 00 is the average school mean reading fluency for the control schools
Wj is the indicator for condition (1=treatment including WSC and Grant, 0=control)
 01 is the main effect of treatment, average difference in mean reading fluency for
treatment and control schools
u 0 j is the random error associated with control school means, now a conditional
variance, var( u0 j )   00|W
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The Model
Combined Model:
Yij   00   01W j  u0 j  rij
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Guiding Questions (C)
Guiding Questions (C)
 What is the relationship between students age
and reading fluency?
Consider 5 options
•
•
•
•
1 - Age is group mean centered at L1
2 - Age is uncentered at L1
3 - Age is grand mean centered at L1
4 - Age is group mean centered at L1, grand mean
centered at L2
• 5 - Age is grand mean centered at L1, grand mean
centered at L2
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The Model-Option 1
Level 1 (students):
Yij   0 j  1 j (age)  rij
Yij is reading fluency for student i in school j
 0 j is the average unadjusted mean reading fluency for school j
1 j is the average change in reading fluency for a 1 unit increase in student age in school j
(within school age-reading fluency slope)
rij is the random error associated with student i in school j, var( rij )   2
Level 2 (schools):  0 j   00  u0 j
1 j   10
 00 is the average school mean reading fluency across the schools
 10 is the average age-reading fluency slope within schools
u0j is the random error associated with school means, var( u0 j )   00
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The Model
Combined Model:
Yij   00   10 (age)  u0 j  rij
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Option 2
What if we left age uncentered?
 Same models, age uncentered
 Intercept is now average school mean reading
fluency for schools when age = 0
 Slope is now composite of within school agereading fluency relationship and betweenschool age reading fluency relationship
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Option 3
What if we grand mean centered age?
 Same models, age grand mean centered
 Intercept is now average adjusted school
mean reading fluency for schools
 Slope is now composite of within school agereading fluency relationship and betweenschool age reading fluency relationship
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Option 4
What if we group mean centered age at L1
and grand mean centered age at L2?
 Need new model
 Aggregate version of age for each school at
L2
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The Model – Option 4
Level 1 (students):
Yij   0 j  1 j (age)  rij
Yij is reading fluency for student i in school j
 0 j is the average unadjusted mean reading fluency for school j
1 j is the average change in reading fluency for a 1 unit increase in student age in school j
(within school age-reading fluency slope)
rij is the random error associated with student i in school j, var( rij )   2
Level 2 (schools):
 0 j   00   10 ( sm _ age)  u0 j
1 j   10
 00 is the average school mean reading fluency across the schools adjusted for school
mean age
 10 is the average change in school mean reading fluency for a 1 unit increase in school
mean age across schools (between school mean age-reading fluency relationship)
 10 is the average age-reading fluency slope within schools
var( u0 j )   00|W
u0j is the random error associated with adjusted school means, now a conditional,
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Option 5
What is we grand mean centered age at
L1 and grand mean centered age at L2?
 Same model, age grand mean centered at L1
 Intercept is same but adjusted mean
  10 is same
  01 is the compositional effect of age, weighted
composite of the within and between slopes,
difference between 2 students with same age
value but who attend schools that differ by
one unit of school mean age
 Note:  c  b   w
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Next Steps
Practice session in lab
Questions/comments via video session
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