Transcript Groups change too - Daniel J. Bauer, Ph.D.

Groups Change Too: Analyzing Repeated Measures
on Individuals Embedded Within Dynamic Groups
Daniel J. Bauer
Outline of Talk

Goal:


To offer a more realistic model for repeated measures data when
individuals are clustered within groups that undergo structural or
functional change over time
Roadmap:
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Causes and consequences of clustered data
Multilevel modeling
Analyzing change over time
Stable versus dynamic groups
Two applications of dynamic group models
Clustering is a Natural Feature of Data
Humans exist within a social ecology including both
natural and constructed groups (e.g., family and school)
Clustering Usually Implies Correlation

Observations on individuals from the same group tend to be
correlated
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Peer groups subject to selection effects (homophily) and socialization
effects (group norms)
Schools include students drawn from similar sociodemographic
backgrounds, and students are exposed to common teachers and
curricula
Family members have common genes, environmental exposures, and
social influences
Yet most statistical models assume independence of observations
(more specifically, independent residuals)
Consequences of Ignoring Dependence

What happens if we erroneously analyze the data as if they were
independent?
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Standard errors, test statistics, degrees of freedom, p-values, and
confidence intervals are all incorrect
Tests tend to be too liberal, inflating Type I errors
Most importantly, we neglect important processes in the data
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How similar are individuals within groups?
How strong are group effects on individuals?
What predictors account for within- versus between-group differences?
Appropriately Analyzing Clustered Data
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There are several possible ways to analyze cluster-correlated data
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Fixed-effects approaches
Generalized estimating equations
Multilevel models with random effects
Multilevel models offer unique insights into both individual and
group-level processes
A Classic Example
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Science achievement scores for student from different schools:
A Simple Multilevel Model

A basic two-level model for clustered data:
yij   0  v j  rij
Overall average
(fixed effect)
Group-level
influences
(random effect)
Individual-level
influences that are
independent of group
(random effect)
rij ~ N  0,  r2 
v j ~ N  0,  v2 
 v2
ICC  2
 v   r2
Variance component
associated with each
random effect
Correlation between
individuals’ scores
The Variance Components

Here we see how each component of variability maps onto our
plot of the data
0
vj
rij
Extending the Model

Normally, our next step would be to incorporate predictors at
the individual and group level to explain each source of variability
in the data

Suppose, however, we didn’t just measure our outcome once, but
multiply over time, with the goal of capturing individual
trajectories of change
Modeling Individual Trajectories
y
Person i=1 in Group j=1
0
1
2
Time
3
Modeling Individual Trajectories
Person 1, Group 1
y
Person 2, Group 1
Mean
Person 3, Group 2
Person 4, Group 2
ytij   0  1Timetij  u0ij  u1ijTimeti  v j  rti
0 Mean
Trajectory
1
2
Individual
Time
Differences
3Group
Effect
Time-Specific
Residual
Taking a Closer Look

This is a typical three-level model for capturing individual change
when individuals are clustered within groups
ytij   0  1Timetij  u0ij  u1ijTimeti  v j  rti
Mean
Trajectory

Individual
Differences
Group
Effect
Time-Specific
Residual
Note that the group effect, vj, is constant over time
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Is this consistent with theory?
Dynamic Groups
Just an individuals change, so too does the social ecology
Chronosystem
Dynamic Groups

We refer to dynamic groups as those that undergo structural
and/or functional change over time yet maintain their core
integrity as units
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Examples:
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Rockbridge and Hickman High Schools both experience turnover in
students, teachers, administrators, and curricula, yet continue to be
characterized by distinctive school cultures
The Jones Family undergoes structural changes as a consequence of
divorce, remarriage, and child birth, and undergoes functional changes
as a consequence of parental addiction and unemployment, yet the
Jones Family remains distinct from the Smith Family
Rewriting the Model
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With dynamic groups, we would expect group effects to be
correlated over time, but not necessarily constant
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A more realistic model might thus be
ytij   0  1Timetij  u0ij  u1ijTimeti  vtj  rti
Mean
Trajectory
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Individual
Differences
Group
Effect
The group effect is now time-varying
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Key is then to discern its temporal structure
Time-Specific
Residual
Temporal Structure of Group Effects
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Say we have 4 time points
How should we structure the covariance matrix of the group
effects over time?
v1 j
v1 j
v2 j
v3 j
v4 j
17
a
a

a

a
v2 j
a
a
a
v3 j
?a
a
v4 j





a 
Temporal Structure of Group Effects
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
Say we have 4 time points
How should we structure the covariance matrix of the group
effects over time?
Traditional “stable
groups” model
v1 j
v2 j
v3 j
v4 j
v1 j
v2 j
v3 j
v4 j
1
1

1

1
1
1
1
1
1
1
1
1
1
1
1
1
Correlated 1.0
over time


 2
 v


Temporal Structure of Group Effects
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
Say we have 4 time points
How should we structure the covariance matrix of the group
effects over time?
Toeplitz model
v1 j
v2 j
v3 j
v4 j
v1 j
v2 j
v3 j
v4 j
1
a

b

c
a
1
a
b
b
a
1
a
c
b  2
v
a

1
Banded
Covariance Matrix
Temporal Structure of Group Effects
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
Say we have 4 time points
How should we structure the covariance matrix of the group
effects over time?
Stabilization
model
v1 j
v2 j
v3 j
v4 j
v1 j
v2 j
v3 j
v4 j
1
a

b

b
a
1
a
b
b
a
1
a
b
b  2
v
a

1
Stabilizing Banded
Covariance Matrix
Temporal Structure of Group Effects
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
Say we have 4 time points
How should we structure the covariance matrix of the group
effects over time?
Compound
symmetric
model
v1 j
v2 j
v3 j
v4 j
v1 j
v2 j
v3 j
v4 j
1
a

a

a
a
1
a
a
a
a
1
a
a
a  2
v
a

1
Equal covariances
Temporal Structure of Group Effects
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Say we have 4 time points
How should we structure the covariance matrix of the group
effects over time?
v1 j
AR(1) model
v1 j
v2 j
v3 j
v4 j
1


 2
 3
 
v2 j
v3 j
v4 j

2

3 
2
  2
v

1

2
1


1 
Exponential Decay
Temporal Structure of Group Effects
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Say we have 4 time points
How should we structure the covariance matrix of the group
effects over time?
v1 j
ARMA(1,1)
model
v2 j
v3 j
v4 j
v1 j  1
   2 

v2 j  
1 
  2

v
v3 j   
1
 
 2

v4 j   
1 
Rapid Decay
Temporal Structure of Group Effects
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
Say we have 4 time points
How should we structure the covariance matrix of the group
effects over time?
v1 j
Unstructured
model
v1 j
v2 j
v3 j
v4 j
v2 j
v3 j
v4 j
 12v a
b
c 


2
e 
 a  2v d
b
d  32v
f 

2 
e
f  4v 
 c
No Structure
Imposed
Why It Matters
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Specifying a poor temporal structure for the group effects risks
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Incorrect tests of regression coefficients for predictors
Biased estimates of variance components at each level of the model
Occluding important findings regarding the nature and stability of group
effects over time
Goal is thus to identify a theoretically plausible structure that fits
the data well
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Some structures are nested and can be compared using LRT
Others are non-nested and can be compared by BIC, AIC, etc.
Example: Attitudes About Science
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Data drawn from the Longitudinal Study of American Youth (LSAY)
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Cohort 1 (N=2091): 1987, 1988, 1989
Cohort 2 (N=1407): 1990, 1991, 1992
51 schools, 3498 students, 7756 observations from grades 10-12
Outcome is an IRT developmental scale score of science ability:
26
Goals for analysis
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Evaluate the relationship between religious attitudes towards
science and science achievement
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“Science undermines morality”
“We need less science, more faith”
“The theory of evolution is true” (R)
“The bible is God’s word”
Separate within-school and between-school effects, while
controlling for SES
Obtain accurate tests of these effects by appropriately accounting
for school effects
Determine the temporal structure of school effects
Fitted Model
sci
science
cience
encetitijj   0  1 gradetij   2 cohortiijj  3ccohort
ohortijij  grade
gradetijttiijj 
 4 studentSESij  5 studentattiijj   6 schoolSES j   7 schoolatt
schoolatttt j 
u0ij  u1ij gradetij  vttjj  rtitijj
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Captures average trajectory for each cohort
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Captures within- and between- school effects of SES and
attitudes
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Captures individual differences in change over time, time-varying
school effects, and residuals from the individual trajectories
Selecting a Temporal Structure
Structure
Traditional
models
Dynamic
group
models
Parameters
AIC
BIC
Intercept
1
52055.0
52064.6
Intercept + Slope
3
51981.6
51995.1
Toeplitz
6
51903.1
51922.5
Stabilizing LagLag
Stabilizing
4 4
5
51901.8
51901.8
51919.2
51919.2
Stabilizing Lag 3
4
51920.9
51936.4
Stabilizing Lag 2
3
51919.0
51932.5
CS
2
51929.2
51940.7
AR(1)
2
51909.1
51920.7
ARMA(1,1)
3
51911.1
51924.6
Fixed Effects
Stable Group Model
Dynamic Group Model
Estimate
95% CI
Estimate
95% CI
Intercept
60.51*
(59.61,61.42)
60.54*
(59.55,61.54)
Grade
2.58*
(2.43,2.73)
2.49*
(2.17,2.81)
Cohort
1.44*
(.74,2.14)
1.32*
(.37,2.27)
Grade*Cohort
-.78*
(-1.04,-.53)
-.61*
(-1.11,-.11)
Student
Attitudes
-2.67*
(-3.37,-1.98)
-2.68*
(-3.37,-1.98)
School
Attitudes
-8.20*
(-16.01,-.38)
-8.83*
(-16.47,-.29)
Student SES
.13*
(.10,.15)
.13*
(.10,.15)
School SES
.12
(-.09,.32)
.12
(-.09,.34)
* p<.05
School Effects Over Time
87
87
88
89
90
91
92
1.00
1.00

1.00

1.00
1.00

1.00
88
1.00
1.00
1.00
1.00
1.00
89
90
91
92
1.00
1.00
1.00
1.00





1.00


1.00 1.00

1.00 1.00 1.00 
Dynamic group model shows
diminishing correlation of school effect
over time
** Superior model fit
Traditional model with random
intercept for school assumes constant
school effect over time
87
88
89
90
91
92
87 1.00



88  .90 1.00


89  .82 .90 1.00


90  .76 .82 .90 1.00


91  .62 .76 .82 .90 1.00


92  .62 .62 .76 .82 .90 1.00 
Summary
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Though the regression coefficient estimates are similar, the
dynamic groups model fits the data better and likely provides
more accurate tests of these coefficients
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Suggests both within and between-school effects of fundamentalist
religious attitudes on science achievement
The dynamic groups model also provides insights into the
stability and change of school effects over time
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School effects highly stable from one year to the next
But the correlation decays to .62 over a period of 4 years, indicating
some drift in nature of school effects over time
Example: Family Effects on Psychopathology
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Data drawn from the Michigan Longitudinal Study (PI: Zucker)
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280 families, 588 children, 2468 repeated measures
Repeated measures included from age 11-17 and span 12 calendar years
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Outcomes are IRT scores of self-reported externalizing and
depression
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Primary goal is to examine temporal stability of family effects on
psychopathology
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Ancillary goals are to estimate trajectories of externalizing and
depression for boys and girls, and to evaluate added risk due to
parental impairment (alcoholism, depression, ASP)
Fitted Model
yttijij   0  1agetij   2 agetij2 
3maleiijj   4 agetij  maleijij  5 agetij2  mal
maleeij 
 6 pAlc j   7 pDep j  8 pASPj 
u0ij  u1ij agetij  u2ijij age
agetij2  vtj  rttijtiij

Captures differences in average trajectories of girls and boys
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Captures effects of parental impairment
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Captures individual differences in change over time, timevarying family effects, and residuals from the individual
trajectories
Selecting a Temporal Structure
Externalizing
Depression
Structure
Parameters
AIC
BIC
AIC
BIC
Intercept
1
4769.5
4798.6
6251.9
6280.9
Intercept + Slope
3
4768.9
4805.2
6252.7
6289.0
CS
2
4770.0
4802.7
6252.3
6284.9
AR(1)
AR(1)
2
4754.5
4754.5
4787.2
4787.2
6245.9
6245.9
6278.6
6278.6

For both outcomes, the AR(1) dynamic group model fits best
Fixed Effects
Externalizing
Internalizing
Estimate
95% CI
Estimate
95% CI
Intercept
-.133
(-.272,.006)
-1.307*
(-1.495,-1.120)
Age
.055*
(.027,.084)
.070*
(.031,.110)
Age2
-.034*
(-.047,-.021)
.010
(-.000,.020)
Male
.217*
(.096,.338)
-.273*
(-.410,-.135)
Age × Male
-.070*
(-.104,-.036)
-.078*
(-.124,-.031)
Age2 × Male
.028*
(.012,.043)
Parent Alc
.415*
(.291,.540)
.207*
(.024,.390)
Parent Dep
.098
(-.027,.223)
.179
(-.004,.363)
Parent ASP
.207*
(.049,.364)
.257*
(.028,.486)
* p<.05
Family Effects Over Time
Summary
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Gender differences consistent with other literature
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Parental history of alcoholism elevates risk of both depression
and externalizing, and this is compounded by history of ASP
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History of parental depression does not have a significant effect
Family effects are highly fluid
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A family that is troubled in one year is likely to continue to function
poorly in the next year or two, but may right itself over the longer term
Conversely, a family functioning well at one point in time is not immune
from later difficulties
Family effects less stable for externalizing than depression
Conclusions

Standard multilevel models fail to account for the fact that groups
undergo change over time
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The effect of the group on its members is unlikely to be constant

We propose the use of dynamic group models to obtain new
insights on the temporal structure of group effects
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
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At a time lag of five years, school effects on science achievement were
correlated .62
In contrast, family effects were correlated .53 for depression and only
.25 for externalizing behavior
This difference in stability is perhaps not surprising. Schools are large
institutions with a great deal of inertia, whereas families are small groups
that are potentially more vulnerable to stochastic events
Acknowledgements and Disclaimer
The project described was supported by supported by National Institutes
of Health grants R01 DA 025198 (PI: Antonio Morgan-Lopez), R37 AA
07065 (PI: Robert Zucker), and R01 DA 015398 (PIs: Andrea Hussong and
Patrick Curran). The content is solely the responsibility of the author and
does not represent the official views of the National Institute on Drug
Abuse, National Institute on Alcohol Abuse and Alcoholism, or the National
Institutes of Health.
Nisha Gottfredson
Danielle Dean
Robert Zucker
Antonio Morgan-Lopez
Andrea Hussong