groups change too: analyzing repeated measures on individuals embedded within dynamic groups daniel...

Groups Change Too: Analyzing Repeated Measures on Individuals Embedded Within

Dynamic Groups

Daniel J. Bauer

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: Causes and consequences of clustered data Multilevel modeling Analyzing change over time Stable versus dynamic groups Two applications of dynamic group models

Outline of Talk

Clustering is a Natural Feature of DataHumans exist within a social ecology including both natural and constructed groups (e.g., family and school)

Observations on individuals from the same group tend to be correlated 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)

Clustering Usually Implies Correlation

Consequences of Ignoring Dependence

What happens if we erroneously analyze the data as if they were independent? 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 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

There are several possible ways to analyze cluster-correlated data 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

Science achievement scores for student from different schools:

A Simple Multilevel Model

A basic two-level model for clustered data:

0ij j ijy v r

2~ 0,j vv N

2~ 0,ij rr N

2

2 2ICC v

v r

Overall average (fixed effect)

Group-level influences (random effect)

Individual-level influences that are independent of group (random effect)

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

b0vj

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

0 1 2 3

Person i=1 in Group j=1

Time

y

Modeling Individual Trajectories

Person 1, Group 1

Person 2, Group 1

Person 4, Group 2

Time

y

0 1 2

Person 3, Group 2

Mean

3

0 1 0 1tij tij ij ij ti j tiy Time u u Time v r

MeanTrajector

y

Individual Difference

s

Time-Specific Residual

Group Effect

Taking a Closer Look

This is a typical three-level model for capturing individual change when individuals are clustered within groups

Note that the group effect, vj, is constant over time Is this consistent with theory?

0 1 0 1tij tij ij ij ti j tiy Time u u Time v r

MeanTrajector

y


s


Group Effect

Chronosystem

Dynamic Groups

Just an individuals change, so too does the social ecology

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

Examples: 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

With dynamic groups, we would expect group effects to be correlated over time, but not necessarily constant

A more realistic model might thus be

The group effect is now time-varying Key is then to discern its temporal structure

0 1 0 1tij tij ij ij ti tj tiy Time u u Time v r

MeanTrajector

y


s


Group Effect

17

Temporal Structure of Group Effects

Say we have 4 time points How should we structure the covariance matrix of

the group effects over time?

1

2

3

4

j

j

j

j

v

v

v

v

1 2 3 4j j j jv v v v

a

a a

a a a

a a a a

?


Traditional “stable groups” model

1

2

3

4

j

j

j

j

v

v

v

v

1 2 3 4j j j jv v v v

2

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

v

Correlated 1.0 over time






Toeplitz model

1

2

3

4

j

j

j

j

v

v

v

v

1 2 3 4j j j jv v v v

2

1

1

1

1

v

a b c

a a b

b a a

c b a

Banded Covariance

Matrix




Stabilization model

1

2

3

4

j

j

j

j

v

v

v

v

1 2 3 4j j j jv v v v

2

1

1

1

1

v

a b b

a a b

b a a

b b a

Stabilizing Banded

Covariance Matrix




Compound symmetric model

1

2

3

4

j

j

j

j

v

v

v

v

1 2 3 4j j j jv v v v

2

1

1

1

1

v

a a a

a a a

a a a

a a a

Equal covariances




AR(1) model1

2

3

4

j

j

j

j

v

v

v

v

1 2 3 4j j j jv v v v

2 3

22

2

3 2

1

1

1

1

v

Exponential Decay

ARMA(1,1) model

1

2

3

4

j

j

j

j

v

v

v

v

1 2 3 4j j j jv v v v

2

2

2

1

1

1

1

v




Rapid Decay


Unstructured model

1

2

3

4

j

j

j

j

v

v

v

v

1 2 3 4j j j jv v v v

21

22

23

24

v

v

v

v

a b c

a d e

b d f

c e f



No Structure Imposed

Why It Matters

Specifying a poor temporal structure for the group effects risks 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 Some structures are nested and can be compared using LRT Others are non-nested and can be compared by BIC, AIC,

etc.

26

Example: Attitudes About Science Data drawn from the Longitudinal Study of American

Youth (LSAY) 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:

Goals for analysis

Evaluate the relationship between religious attitudes towards science and science achievement “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

2 3

4

0

5 6 7

1

0 1

tij tij ij ij

ij ij j j

ij ij tij tj ti

ti

j

jgrade

studentSES studentatt schoolSES schoola

science grade

tt

cohort cohort

u u v rgrade

Fitted Model

4 5

1

6 7

0

0 1

2 3

ij i

tij tij i t

j j j

ij ij tij tj ti

iji

j

j j


s

t

cience grade cohort c

t

u u

ohor

grade

gt

r

rad

v

e

4 5

1

6 7

0 2 3

0 1

ij i

tij ij ij tij

ij ij tij tj

j

j

j

j

j

ti

ti


s

t

grade cohort cohort grade

u u grade v

ci ce

r

t

en

Captures average trajectory for each cohort

Captures within- and between- school effects of SES and attitudes

0 1 2 3

4 5 6 7

0 1

tij ij ij tij

ij

ij ij tij t

ti

j tij

j

j

ij j

grade cohort cohort grade

studentSES studentatt schoolSES schoolatt

u u grade v r

science

Captures individual differences in change over time, time-varying school effects, and residuals from the individual trajectories

Structure Parameters

AIC BIC

Intercept 1 52055.0 52064.6

Intercept + Slope

3 51981.6 51995.1

Selecting a Temporal Structure


AIC BIC

Intercept 1 52055.0 52064.6

Intercept + Slope

3 51981.6 51995.1

Toeplitz 6 51903.1 51922.5

Stabilizing Lag 4

5 51901.8 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


AIC BIC

Intercept 1 52055.0 52064.6

Intercept + Slope

3 51981.6 51995.1

Toeplitz 6 51903.1 51922.5

Stabilizing Lag 4

5 51901.8 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

Traditional models

Dynamic group models

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

Dynamic group model shows diminishing correlation of school effect over time

** Superior model fit

School Effects Over Time

1.00

1.00 1.00

1.00 1.00 1.00

1.00 1.00 1.00 1.00

1.00 1.00 1.00 1.00 1.00

1.00 1.00 1.00 1.00 1.00 1.00

Traditional model with random intercept for school assumes constant school effect over time

87 88 89 90 91 92

8

7

8

8

8

9

9

0

9

1

9

2

1.00

.90 1.00

.82 .90 1.00

.76 .82 .90 1.00

.62 .76 .82 .90 1.00

.62 .62 .76 .82 .90 1.00

87 88 89 90 91 92

8

7

8

8

8

9

9

0

9

1

9

2

Summary

Though the regression coefficient estimates are similar, the dynamic groups model fits the data better and likely provides more accurate tests of these coefficients 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 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

Data drawn from the Michigan Longitudinal Study (PI: Zucker) 280 families, 588 children, 2468 repeated measures Repeated measures included from age 11-17 and span 12

calendar years

Outcomes are IRT scores of self-reported externalizing and depression

Primary goal is to examine temporal stability of family effects on psychopathology

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

0 12

2

23 4 5

6 7 8

21 20

tij

ij tij ij tij ij

j j j

ij i

tij tij

ij tij j tij tj tij

y age age

male age male age male

pAlc pDep pASP

u age u agu e v r

0 1

6 7

22

23 4

8

20 1 2

5

tij

ij

j j j

ij ij tij i

tij i

j tij tj ti

tij tij

j tij

j

ij

y age age

male age male age mal

pAlc pDep pASP

u u age u ag

e

e v r

Captures differences in average trajectories of girls and boys

6 7

20 1 2

23 4

20 1 2

8

5

tij tij

ij tij ij tij ij

ij ij tij ij tij tj tij

tij

j j j

age age

male age male age male

u u age u age v

y

pAlc pDep pASP

r

Captures effects of parental impairment

Captures individual differences in change over time, time-varying family effects, and residuals from the individual trajectories

20 1 2

23 4 5

6 7

2

8

0 1 2

tij tij

i

ij ij tij ij tij tj t

j tij i

t

j

ij

i

tij ij

j

j

j j

age age

male age male age mal

y

u u age u age v

e

pAlc pDep pASP

r

Selecting a Temporal Structure

For both outcomes, the AR(1) dynamic group model fits best

Externalizing Depression


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) 2 4754.5 4787.2 6245.9 6278.6

Externalizing Depression


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) 2 4754.5 4787.2 6245.9 6278.6

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

Gender differences consistent with other literature

Parental history of alcoholism elevates risk of both depression and externalizing, and this is compounded by history of ASP History of parental depression does not have a significant

effect

Family effects are highly fluid 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

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 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

groups change too: analyzing repeated measures on individuals embedded within dynamic groups daniel...

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