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MIXED EFFECTS MODELING
USING SPSS®
Instructor: Manolo ROMERO ESCOBAR
http://mixedreviews.wordpress.com
12 January 2011
At the Research and Outcome Measurement Branch.
Ministry of Children and Youth Services
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"Two facts - that children's growth is
typically the object of inquiry, and that
such growth occurs in organizational
settings - correspond to two of the most
troublesome and persistent methodological
problems in the social sciences: the
measurement of change and the
assessment of multi-level effects“
(Bryk & Raudenbush, 1989, p. 160).
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AGENDA
Review of general linear model
Mixed Effects Model for Clustered/Grouped
Data
◦ Lab 1
Lunch Break
Longitudinal Outcome Analysis with Mixed
Effects Modeling
◦ Lab 2 and 3
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“Nested” data
Common in behavioural and social sciences
Students-classes-schools-boards-etc…
Repeated observations
Violation of assumptions (e.g. students in
same classroom might share characteristics,
non-independent, non-random)
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Different names: Same idea
Multilevel linear models (cf. Goldstein, 1995)
Multilevel analysis (cf. Hox, 1995)
Hierarchical Linear Models (cf. Raudenbush &
Bryk, 2002)
Mixed effects models
Slopes as Outcomes
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General Linear Model (GLM)
A continuous outcome (dependent) variable
Do not confuse with “generalized linear
model” in which DV is not continuous (e.g.
Logistic regression)
Predictors can be continuous (multiple
regression) or categorical (ANOVA) or a
combination of both (ANCOVA).
Estimation via Ordinary Least Squares
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GLM formulation
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),0(~
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Nrwhere
rxy
i
iii
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A GLM example
Simulated data
DV: Self Esteem
Factors: Psycho-therapy treatment and
Gender
Covariates: Intelligence, Resilience, and
Socio-economic Status.
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Observed Data
Predicted Data
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Unconditional Means Model12/0
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),0(~
,
2
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Nrwhere
ry
ij
ijjij
),0(~
,
000
0000
Nuwhere
u
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ijjij ruY 000Fixed Effect Random Effects
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Level 2 (Group) Predictors
),0(~),0(~ 000
2
020210100
0202101000
0
NuandNr
ruzzY
uzz
ry
jij
ijjij
jj
ijjij
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Level 1 (individual) variables
ijjjij rxy 110
jjjj uu 11010000 ;
ijjjij rxuuxy 11011000
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Level 1 and Level 2
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1110
0100
1
0
2
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111111010100
1111101
0101000
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,0
0~
),0(~
Nu
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Nrwhere
rxuu
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uz
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Grouped data predictors
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Level Type/Field Fixed Effect
Interaction
Random Effect
Level 1 (Within
groups)
Continuous/
Covariate
With Level 2 predictor
by default (can be taken
off if n.s.)
With Level 1 predictor if
it is of research interest.
Yes, but test if it
is necessary
Level 1 (Within
groups)
Categorical/ Factors
Level 2 (Between
groups)
Continuous/
Covariate
With Level 1 predictor
by default (can be taken
off if n.s.)
With Level 2 predictor if
it is of research interest.
No.
Level 2 (Between
groups)
Categorical/ Factors
Grouping Variable Continuous/Subjects
on first dialog
window
No. Unless it is also a
level 1 predictor
As grouping
variable
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A clustered data example
High school and beyond
Example of HLM par excellence
Students within schools
Schools by sector: Catholic-Public
DV: Math achievement
Predictors: SES, MEANSES, school size,
minority, sex, others.
From Bryk & Raudenbush via nlme package
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Unconditional means model
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Adding level 2 predictor
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Adding level 1 predictor
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Level 1 and 2 predictors
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Lab 1: Clustered Data
File: BrykRauden_HighSchool&Beyond.sav
inside data folder
Open on SPSS
Fit Unconditional means model:
Find the parameters and estimate the intra-
class correlation given by this formula
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Lab 1: continuation
Add a level 2 and a level 1 predictor one at
a time
Why is Sector not a grouping variable?
What would happen if there were more
than two levels of Sector?
Try other predictors, one or two at a time
and note the results
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LONGITUDINAL OUTCOME ANALYSIS
Part II
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Time variable as level 1 predictor
Time variable is continuous
Adds a random effect
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ijjjij rtimey 10
jjjj uu 11010000 ;
ijjjij rtimeuutimey 101000
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Adding time varying predictor
Level 1 predictor
Complex random effects covariance
structure:
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ijjjjij rxutimeuuxtimey 12101201000
222120
121110
020100
2
1
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2
,
0
0
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~
),0(~
N
u
u
u
and
Nrwhere
j
j
j
ij
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Adding a time constant predictor
Level 2 predictor
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ijjj
jjij
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Time relationship
(level)
Type/Field Fixed Effect
Interaction
Random Effect
Time variant
(Level 1- within
subjects)
Continuous/ Covariate With Level 2 predictor
by default (can be taken
off if n.s.)
With other Level 1
predictors if it is of
research interest. It is
unlikely an interaction
with time would be of
interest
Yes, but test if it
is necessary
Time variant
(Level 1- within
subjects)
Categorical/ Factors
Time constant(
Level 2 – between
subjects)
Continuous/ Covariate With time predictor by
default.
With Level 2 predictor
if it is of research
interest.
No.
Time
constant(Level 2 -
Between subjects)
Categorical/ Factors
Grouping Variable Continuous/Subjects
on first dialog window
No. Unless it is also a
level 1 predictor
As grouping
variable
Time variable Continuous/Covariate With level 2 predictors Yes, but test if it
is necessary
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Covariance Structures
AR(1)
Compound symmetry
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Wide vs Long format
Open the wide.sav file inside the data folder
◦ There are columns for each time point for each
time varying variable, including the DV (DV is
wm)
Open the long.sav file inside the data folder
◦ There is one column per predictor and a “wave”
variable to indicate time. Notice that Centered
Age could also be taken as a time variable
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Re-structuring datasets in SPSS
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Lab 2: Longitudinal data 3 time points
Simulated data: Development of Working
Memory from 10 to 12 years of age
Somewhat accelerated longitudinal design
Variables in the long.sav file ID, a time
constant variable (const), Age in months, a
time varying covariate (cov), WM.
Also: Age in years (aiy), Age in years rounded
(aiy_r), Age in years centered (aiy_c) and
Age in months centered (aim_c)
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Lab continuation
Try different time variables and different
covariance structures
Also add the time constant predictor and
the time varying predictor accordingly
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Lab 3: Unbalanced data
Simulated data inspired by consulting case
Longitudinal study on depression and
treatment techniques
Up to five timepoints : baseline, one week,
one month, three months, and six months
follow ups
Not all participants with observations at all
points, at least two timepoints
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Variables in dep.sav
id: Subject ID;time: in months 0, .25, 1,3,6,
centered at baseline
cond: treatment condition
Changes over time:
◦ LDep3: Depression scale from 0 to 5
◦ selfest: Self-esteem scale from 1 to 7
Time constant: Age, anxiety, avoidance,
extraversion openness, Frequency(number
of observations for the subject)
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If random effect of time not needed…
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Adding treatment condition
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Adding time constant predictor
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Adding time varying predictor
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Test for different covariance structures
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Lab 3: Fit different models
Try adding different predictors
Change DV from depression to self-esteem
Center the time variable at different
timepoints:
◦ At 1 month assuming that was the length of the
treatment
◦ At 6 months to test for differences among
conditions in maintaining results
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APPENDIX: Re-structuring using syntax
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