Mixed Effects Models - Level-2 Variables

Week 5.1: Level-2 Variables
! Notation
! Multiple Random Effects
! Combining Datasets in R
! Modeling
! Level-2 Variables
! Including Level-2 Variables in R
! Modeling Consequences
! Cross-Level Interactions
! Lab

Recap
• Last week, we created a model of middle
schoolers’ math performance that included a
random intercept for Classroom
• model1 <- lmer(FinalMathScore ~ 1 + TOI +
(1|Classroom), data=math)
Fixed effect of
naive theory
of intelligence
Average
intercept
(averaged all
classrooms)
Variance in
that intercept
from one class
to the next
Residual
(unexplained)
variance at
the child level

Notation
• What is this model doing mathematically?
• Let’s go back to our model of individual students
(now slightly different):
Student
Error
Ei(j)
=
End-of-year math
exam score
+ +
Baseline
Yi(j) B0j
Fixed mindset
γ10x1i(j)

Notation
Student
Error
Ei(j)
=
End-of-year math
exam score
+ +
Baseline
Yi(j) B0j
Fixed mindset
γ10x1i(j)
What now determines the baseline that
we should expect for students with
fixed mindset=0?

Notation
• Baseline (intercept) for a student in classroom j
now depends on two things:
Student
Error
Ei(j)
=
End-of-year math
exam score
+ +
Baseline
Yi(j) B0j
Fixed mindset
γ10x1i(j)
U0j
=
Intercept
+
Overall intercept
across everyone
B0j γ00
Teacher effect for this
classroom (Error)

Notation
• Essentially, we have two regression models
• Hierarchical linear model
• Model of classroom j:
• Model of student i:
Student
Error
Ei(j)
=
End-of-year math
exam score
+ +
Baseline
Yi(j) B0j
Fixed mindset
γ10x1i(j)
U0j
=
Intercept
+
B0j γ00
classroom (Error)
LEVEL-1
MODEL
(Student)
LEVEL-2
MODEL
(Classroom)
Overall intercept
across everyone

Hierarchical Linear Model
Student
1
Student
2
Student
3
Student
4
Level-1 model:
Sampled STUDENTS
Mr.
Wagner’s
Class
Ms.
Fulton’s
Class
Ms.
Green’s
Class
Ms.
Cornell’s
Class
Level-2 model:
Sampled
CLASSROOMS
• Level-2 model is for the superordinate level here,
Level-1 model is for the subordinate level
Variance of classroom intercept is
the error variance at Level 2
Residual is the error variance at
Level 1

Notation
• Two models seems confusing. But we can simplify
with some algebra…
• Model of classroom j:
• Model of student i:
Student
Error
Ei(j)
=
End-of-year math
exam score
+ +
Baseline
Yi(j) B0j
Fixed mindset
γ10x1i(j)
U0j
=
Intercept
+
B0j γ00
classroom (Error)
LEVEL-1
MODEL
(Student)
LEVEL-2
MODEL
(Classroom)
Overall intercept
across everyone

Notation
• Substitution gives us a single model that combines
level-1 and level-2
• Mixed effects model
• Combined model:
Student
Error
Ei(j)
=
End-of-year math
exam score
+ +
Yi(j)
Fixed mindset
γ10x1i(j)
U0j
+
Overall
intercept
γ00
classroom (Error)

Notation
• Just two slightly different ways of writing the same
thing. Notation difference, not statistical!
• Mixed effects model:
• Hierarchical linear model:
Ei(j)
= + +
Yi(j)
γ10x1i(j)
U0j
+
γ00
Ei(j)
=
Yi(j) B0j
γ10x1i(j)
U0j
= +
B0j γ00
+ +

Notation
• lme4 always uses the mixed-effects model notation
• lmer(
FinalMathScore ~ 1 + TOI + (1|Classroom)
)
• (Level-1 error is always implied, don’t have to
include)
Student
Error
Ei(j)
=
End-of-year math
exam score
+ +
Yi(j)
Fixed mindset
γ10x1i(j) U0j
+
Overall
intercept
γ00
Teacher
effect
for this
class (Error)

! We’re continuing our study of naïve theories of
intelligence & math performance
! We’ve now collected data at three different
schools
! math1.csv from Jefferson Middle School
! math2.csv from Highland Middle School
! math3.csv from Hoover Middle School
Combining Datasets in R

! Look at the math1, math2, math3 dataframes
! How are they similar? How are they different?
! TOI and final math score for each student

Columns not always in same order

Only Hoover has GPA
reported

! Overall, this is similar information, so let’s
combine it all
! Paste together the rows from two (or more)
dataframes to create a new one:
! bind_rows(math1, math2, math3) -> math
! Useful when observations are spread across files
! Or, to create a dataframe that combines 2 filtered
dataframes
math1
math2
math3
math

bind_rows(): Results
! Resulting dataframe:
! nrow(math) is 720 – all three combined
math1
math2
math3
math

! bind_rows() is smart!
! Not a problem that column order varies across
dataframes
! Looks at the column names
! Not a problem that GPA column only existed in one of
the original dataframes
! NA (missing data) for the students at the other schools

! You can also add the optional .id argument
! bind_rows(math1, math2, math3,
.id='OriginalDataframe’) -> math
! Adds another column that tracks which of the original
dataframes (by number) each observation came from

Other, Similar Functions
! bind_rows() pastes together every row from
every dataframe, even if there are duplicates
! If you want to skip duplicates, use union()
! Same syntax as bind_rows(), just different function name
! Other related functions:
! intersect(): Keep only the rows that appear in all of
the source dataframes
! setdiff(): Keep only the rows that appear in a single
source dataframe—if duplicates, delete both copies

Multiple Random Effects
• Schools could differ in math achievement—let’s add
School to the model to control for that
• Is SCHOOL a fixed effect or a random effect?
• These schools are just a sample of possible schools of
interest " Random effect.
School
1
School
2
Sampled SCHOOLS
Sampled
CLASSROOMS
Sampled STUDENTS
LEVEL 3
LEVEL 2
LEVEL 1
Student
1
Student
2
Student
3
Student
4
Mr.
Wagner’s
Class
Ms.
Fulton’s
Class
Ms.
Green’s
Class
Ms.
Cornell’s
Class

• No problem to have more than 1 random effect in
the model! Let’s a random intercept for School.
School
1
School
2
Sampled SCHOOLS
Sampled
CLASSROOMS
Sampled STUDENTS
LEVEL 3
LEVEL 2
LEVEL 1
Student
1
Student
2
Student
3
Student
4
Mr.
Wagner’s
Class
Ms.
Fulton’s
Class
Ms.
Green’s
Class
Ms.
Cornell’s
Class

• model2 <- lmer(FinalMathScore ~ 1 + TOI
+ (1|Classroom) + (1|School), data=math)
School
1
School
2
Sampled SCHOOLS
Sampled
CLASSROOMS
Sampled STUDENTS
LEVEL 3
LEVEL 2
LEVEL 1
Student
1
Student
2
Student
3
Student
4
Mr.
Wagner’s
Class
Ms.
Fulton’s
Class
Ms.
Green’s
Class
Ms.
Cornell’s
Class

+ (1|Classroom) + (1|School), data=math)
• Less variability across schools than classrooms in a school

• This is an example of nested random effects.
• Each classroom is always in the same school.
• We’ll look at crossed random effects next week
School
1
School
2
Sampled SCHOOLS
Sampled
CLASSROOMS
Sampled STUDENTS
LEVEL 3
LEVEL 2
LEVEL 1
Student
1
Student
2
Student
3
Student
4
Mr.
Wagner’s
Class
Ms.
Fulton’s
Class
Ms.
Green’s
Class
Ms.
Cornell’s
Class

Level-2 Variables
• So far, all our model says about classrooms is
that they’re different
• Some classrooms have a large intercept
• Some classrooms have a small intercept
• But, we might also have some interesting
variables that characterize classrooms
• They might even be our main research interest!
• How about teacher theories of intelligence?
• Might affect how they interact with & teach students

Level-2 Variables
Student
1
Student
2
Student
3
Student
4
Sampled STUDENTS
Mr.
Wagner’s
Class
Ms.
Fulton’s
Class
Ms.
Green’s
Class
Ms.
Cornell’s
Class
Sampled
CLASSROOMS
• TeacherTheory characterizes Level 2
• All students in the same classroom will experience
the same TeacherTheory
LEVEL 2
LEVEL 1
TeacherTheory
TOI

Level-2 Variables
• Is TeacherTheory a fixed effect or random
effect?
• Teacher mindset is a fixed-effect variable
• We ARE interested in the effects of teacher mindset
on student math achievement … a research
question, not just something to control for
• Even if we ran this with a new random sample of 30
teachers, we WOULD hope to replicate whatever
regression slope for teacher mindset we observe
(whereas we wouldn’t get the same 30 teachers
back)

Level-2 Variables
• This becomes another variable in the level-2
model of classroom differences
• Tells us what we can expect this classroom to be like
Student
Error
Ei(j)
=
End-of-year math
exam score
+ +
Baseline
Yi(j) B0j
Growth mindset
γ10x1i(j)
U0j
=
Intercept
+
Overall
intercept
B0j
γ00
classroom (Error)
LEVEL-1
MODEL
(Student)
LEVEL-2
MODEL
(Classroom)
Teacher
mindset
+
γ20x20j

Level-2 Variables
• Since R uses mixed effects notation, we don’t
have to do anything special to add a level-2
variable to the model
+ TeacherTheory + (1|Classroom) +
(1|School), data=math)
• R automatically figures out TeacherTheory is a
level-2 variable because it’s invariant for each
classroom
• We keep the random intercept for Classroom
because we don’t expect TeacherTheory will
explain all of the classroom differences. Intercept
captures residual differences.

What Changes? What Doesn’t?
• Random classroom & school variance is reduced.
• Teacher theories of intelligence accounts for some of the variance
among classrooms (and among the schools those classrooms are in).
• TeacherTheory explains some of the “Class j” effect we’re substituting
into the level 1 equation. No longer just a random intercept.
WITHOUT TEACHERTHEORY WITH TEACHERTHEORY

• Residual error at level 1 essentially unchanged.
• Describes how students vary from the class average
• Divergence from the class average cannot be explained by teacher
• Regardless of what explains the “Class j” effect, you’re still substituting it
into the same Lv 1 model

• Similarly, our level-1 fixed effect is essentially unchanged
• Explaining where level-2 variation comes from does not change our
level-1 model
• Note that average student TOI and TeacherTheory are very slightly
correlated (due to random chance); otherwise, there’d be no change.

Cross-Level Interactions
• Because R uses mixed effects notation, it’s also
very easy to add interactions between level-1
and level-2 variables
+ TeacherTheory + TOI:TeacherTheory +
(1|Classroom) + (1|School), data=math)
• Does the effect of a student’s theory of intelligence
depend on what the teacher’s theory is?
• e.g., maybe matching theories is beneficial

Cross-Level Interactions
• Because R uses mixed effects notation, it’s also
very easy to add interactions between level-1
and level-2 variables
• In this case, the interaction is not significant

Mixed Effects Models - Level-2 Variables

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