Extending Group-Based Trajectory Modeling to Account for Subject Attrition

(Sociological Methods & Research, 2011)

Amelia HavilandBobby Jones

Daniel S. Nagin

Trajectories of Physical Aggression(Child Development, 1999)

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SO (70.9%) LR-D (21.7%) MR-D (5.7%) HR-P (1.6%)

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Trajectories Based on 1979 Dutch Conviction Cohort

The Likelihood Function

.)(N

iYPL

PJ(Yi) = probability of Yi given membership in group j

j= probability of membership in group j

ji

ji

x

x

ij eex

)(

j

ij

iji YPxYP )()()(

Missing Data

• Two Types– Intermittent missing assessments (y1, y2 , . ,y4, . ,y6)– Subject attrition where assessments cease starting

in period τ (y1 , y2 , y3 , . , . , .)

• Both types assumed to be missing at random • Model extension designed to account for

potentially non-random subject attrition• No change in the model for intermittent

missing assessments

Some Notation

τi =period t in which subject i drops out

T=number of assessment periods

jt = Probability of Drop out in group j in period t

Probability of Dropout in Period t

Period Probability of Drop Out 1 0 2 3 4 . . . . . . T

No Drop Out

1 – all the above probabilities

The Dropout Extended Likelihood for Group j

).3()1)(;,,0|(),;,|(1

1

jT

t

jtjiitit

jjii i

i

jagewypjageYP

Specification of

• Binary Logit Model• Predictor Variables

– Fixed characteristics of i, – Prior values of outcome,

• If trajectory group was known within trajectory group j dropout would be “exogenous” or “ignorable conditional on observed covariates”

• Because trajectory group is latent, at population level, dropout is “non-ignorable”

jt

ix,...., 21 itit yy

Simulation Objectives

• Examine effects of differential attrition rate across groups that are not initially well separated

• Examine the effects of using model estimates to make population level projections

Simulation 1: Two Group Model With Different Drop Probabilities and Small Initial Separation

10 10

10 10

No dropoutSlope=.5

Time

E(y) E(y)

E(y) E(y)

Time

Time Time

Group 1 Per Period

Dropout Probability

Expected Group 1

Assessment Periods

Probability of Group 1

Dropout on or before Period 6

Model Without Dropout

Model With Dropout

Group 1 Prob. Est.

(π1)

Percent Bias

Group 1 Prob. Est.

(π1)

Percent Bias

Dropout Prob.Est.

0 6.0 0 .200 0.0 .200 0.0 .000.05 5.3 .226 .171 -14.5 .199 -0.5 .051.10 4.7 .410 .146 -27.0 .199 -0.5 .099.15 4.2 .556 .122 -39.0 .200 0.0 .150.20 3.7 .672 .100 -50.0 .199 -0.5 .199.25 3.3 .762 .079 -60.5 .200 0.0 .250.30 2.9 .832 .061 -69.5 .199 -0.5 .301.35 2.6 .884 .046 -77.0 .199 -0.5 .350.40 2.4 .922 .034 -83.0 .199 -0.5 .398

Simulation Results: Group 1 and Group 2 Initially not Well Separated

An Important Distinction from Zhang and Rubin (2003)

• Dropout due death– Subject exits population of interest-the living– Data said to be “truncated”

• Dropout due termination of study participation– Subject exits the sample but remains in the

population– Data said to be censored

Simulation 2: Projecting to the Population Level from Model Parameter Estimates

Simulation 2 Continued

12.5

10

No Dropout

Dropout=.2 per period

Table 2

Simulation 2: Predicting Population Averages With and Without Adjustments for Dropout

No Dropout Model

Model with dropout

Period Average Y

Predicted Y

1~t Predicted

Y

0 10.5 10.5 .200 10.5 1 10.0 10.1 .166 10.0 2 9.48 9.67 .137 9.48 3 8.95 9.26 .112 8.95 4 8.41 8.84 .092 8.42 5 7.87 8.43 .075 7.87

Chinese Longitudinal Healthy Longevity Survey (CLHLS)

• Random selected counties and cities in 22 provinces

• 4 waves 1998 to 2005• 80 to 105 years old at baseline• 8805 individual at baseline• 68.9% had died by 2005• Analyzed 90-93 years old cohort in 1998

Activities of Daily Living

• On your own and without assistance can you:– Bath – Dress– Toilet– Get up from bed or chair– Eat

• Disability measured by count of items where assistance is required

Table 3

Summary Statistic for the Age 90 to 93 CLHLS Cohort at Baseline

Variable N Average ADL 1998 Count 1078 .84 ADL 2000 Count 580 1.05 ADL 2002 Count 335 1.16 ADL 2005 Count 120 1.26

Female 1078 .52 Life Threatening

Disease 1078 .11

1 2 3 40

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

ADL Trajectory Model Without Dropout

Low (27.1%)Medium (60.0%)High (12.9%)

Wave

ADL Count

1 2 3 40

0.51

1.52

2.53

3.54

4.5

ADL Trajectory Model With Drop Out

Low (20.1% DP=.34)Medium (58.6% DP=.47)High (21.3% DP=.64)

Wave

ADL Count

Table 4

Predict Population Average ADL counts from the Models With and Without Dropout

Model Without Drop Out

Model With Drop Out

Period Average ADL

Count

Predict ADL

Count

% Error

~1

t ~2

t ~3

t Predicted

ADL Count

% Error

1998 .84 .91 8.3 .201 .586 .213 .93 10.7 2000 1.05 1.19 13.3 .254 .600 .146 1.07 1.9 2002 1.16 1.42 22.4 .309 .593 .097 1.17 .9 2005 1.26 1.89 50.0 .366 .571 .063 1.58 25.4

Adding Covariates to Model to Test the Morbidity Compression v. Expansion Hypothesis

• Will increases in longevity compress or expand disability level in the population of the elderly?

• “Had a life threatening disease” at baseline or prior is positively correlated with both ADL counts at baseline and subsequent mortality rate.

• Question: Would a reduction in the incidence of life threatening diseases at baseline increase or decrease the population level ADL count?

Testing Strategy and Results

• Specify group membership probability (πj ) and dropout probability ( ) to be a function of life threatening disease variable

• Both also functions of sex and dropout probability alone of ADL count in prior period

• Life threatening disease significantly related to group membership in expected way but has no relationship with dropout due to death

• Thus, unambiguous support for compression

jt

Projecting the reduction in population average ADL count from a 25% reduction in the incidence of the life threatening disease at

baseline

Year 1998 2000 2002 2005Reduction (%) 3.0 2.2 1.5 .7

Projected % Reduction in Population Average ADL Count

Table 6

Own and Cross Elasticity Estimates (%) for Life Threatening Disease Incidences

Cross Elasticity

Group Own Elasticity

Group 2

Group 3

Total Elasticity

1. Low ()201.1

NA -.033 -.059 -.092

2. Medium ()586.2

.069 NA -.173 -.104

3. High()213.3

.232 -.036 NA .196

Conclusions and Future Research

• Large differences in dropout rates across trajectory groups matter

• Future research– Investigate effects of endogenous selection– Compare results in data sets with more modest

dropout rates– Further research morbidity expansion and

contraction