Longitudinal Modelling with Longitudinal Households · 2015. 7. 27. · 22/07/2015 1 Longitudinal Modelling with Longitudinal Households Paul Clarke Workshop Understanding Society

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Longitudinal Modelling with Longitudinal Households

Paul ClarkeWorkshopUnderstanding Society Conference 21 July 2015

Overview

1. Longitudinal households and modelling review

2. Household-level outcomes: Residential mobility example

3. Approaches for modelling person-level outcomes

4. Further issues

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LONGITUDINAL HOUSEHOLDS

AND MODELLING REVIEW

Part 1

3

Longitudinal Households (HHs)

� For British Household Panel Study � Sample of 5500 HHs drawn in 1991 from PAF� HHs followed up annually

� Sample members classified as� Ordinary/Temporary/Permanent

� Families and persons move over time� Especially important for maturing studies (BHPS, PSID)

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Modelling Panel Data

� Types of longitudinal model:� Growth curve� Repeated measures� Survival/Event history

� All based on adapted regression models

� Common themes: trend and autocorrelation

5

A Simple Regression Model

� Linear model for row i of data set (person)

� Variance-covariance structure unspecified� Crucial part of modelling exercise� Often of substantive interest

Fixed part Random part or residual

iii xy εβ +=

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Parameter Estimation: No Clustering

� No clustering, persons independent

� Estimating equations summed over persons

� e.g. GLS for heteroskedastic linear regression( ) )()( ,| 2 ieiiiiii xxxWxyxy σβε =−=

( ) 0;|);(1

=∑ =n

i iiixyxW θεθ

Depends wholly on our model

Can be chosen for efficiency

7

Covariance Structure

=

=

2

2

2

2

1

2

1

00

00

00

varvar

ε

ε

ε

σ

σσ

ε

εε

L

MOM

L

MM

nny

y

y

x

All sample covariate information

Everything homoskedastic from now on!

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The Effects of Clustering

� Persons can be grouped (e.g. households)� Persons a group more alike than general population (> 0)� Less commonly, can also be more unalike (< 0)

� If no clustering then no need to allow for grouping� But clustering is usually present

� Multiple groups: membership can be hierarchical� Focus on most important/known groups

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Autocorrelation

� Repeated measures over time� i now indexes unique wave-person observations (i = t,i)� Clustering: waves within a person

� Almost inevitable: hence autocorrelation� Almost always > 0� Decays over time

� For simplicity, I will ignore it (life is hard enough!)

0),cov( i.e. )( =+ idtti εε10

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Parameter Estimation: Clustering

� Household (HH) clustering; each HH independent

� So estimating equations sum over HHs

� Two different approaches:� Marginal models and conditional models

( ) 0;|,,);(1 1

=∑ =Hn

h hmhhhyyW θεθ xx K

Combined HH covariates

=

mh

h

h

x

x

M

1

x

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Multilevel Models

� Very flexible family of models e.g. Goldstein 2011; Snijders & Bosker 2012

� Two-level linear model for outcomes

� Decompose residual (c.f. variance-components model)� Both residuals normally distributed

� HH-level residual explains HH clustering:

∏==m

i hihihhhmhhuxypuyyp

11),|(),|,,( xK

( )hihihihihih

uex

xy

++=+=

βεβ

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Parameter Estimation

� Specify log-likelihood

� Maximise likelihood � Numerically (quadrature, MCMC)� Iterative least squares (ILS)� Distributional assumptions determine form of “Wε”

)( ),|(log)|,,(log11

udFuxypyyph u

m

i ihihh hmhh ∑ ∫∏∑ ==xK

Normal distn

Integrate over unknown uCond. model plus normality of e

13

Covariance Structure� Allow correlation between household members

� Recall decomposition of model residual:

� Leads to …

0),cov()|,cov( ≠= jhihhjhih yy εεx

hihihih uexy ++= β

),0(~

),0(~2

2

uh

eih

Nu

Ne

σσ

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Residual Covariance Matrix� Example: Three-person HH

++

+=

+++

=

22

222

2222

3

2

1

3

2

1

varvar

ue

uue

uuue

hh

hh

hh

h

h

h

ue

ue

ue

σσσσσσσσσ

εεε

( ) tic)homoskedas (i.e. var where 2eihe σ=

15

Going Longitudinal: Fixed HHs

1

3

2

Wave 2

1

3

2

Wave 1

Household 116

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BHPS/UKHLS-Style Dataset

PID Wave HHID

10017933 1 1001507

10017933 2 2000717

10017968 1 1001507

10017968 2 2000717

10017992 1 1001507

10017992 2 2000717

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Need to convert to …

Dataset for Analysis

PID Wave HHID

1 1 1

1 2 1

2 1 1

2 2 1

3 1 1

3 2 1

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Fixed Households

� 3-level model (wave–person–household)

� To ignore individual autocorrelation

( )hihtihtihtihtihtih

uvex

xy

+++=+=

βεβ

0),cov( )( ==+ ihidtti vee

19

=′=2

22

222

2112

u

uu

uuu

CC

σσσσσσ

Implied Covariance Matrix

=

2221

1211

231

221

211

131

121

111

varCC

CC

εεεεεε

++

+==

22

222

2222

2211

ue

uue

uuue

CC

σσσσσσσσσ

Diagonals: autocorrelation due to HH (between-wave, same person)

Off-diagonal: between-wave covariance, different persons

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HOUSEHOLD-LEVEL

OUTCOMES

Part 2: Multilevel Modelling with Longitudinal Households

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What is a Household?

� Murphy (1996)� “without some additional conditions it is impossible to

use the household as the unit of analysis across time”

� i.e. Must be clear what a HH is to understand what constitutes a change, and the effect this has on the residual correlation structure

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Residential Mobility

� Residential mobility important in HH change� Economically: labour market� Demographically: family dissolution

� HH members decide whether to move or remain� Couple (married or cohabiting)� Singleton

� Dependents of secondary importance

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Some HH-Change Scenarios

� Couple� Both partners move to new address� Separate: one remains, one leaves current address� Separate: both leave for different addresses

� Singleton� Moves to new address (and remains single)� Moves to new address to form couple� Remains at address, joined by new partner

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Transition Models

� Different type of model

� y = HH moved between waves t – 1 and t

� All about covariance structure

� Consider approaches based on multilevel models

L+== βththth xxy )|1Pr(logit

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Review: Part 1

� Pooled cross-sectional, household-level analysis� Clark & Huang (2003)� Collapse person-level data set to household-level� Each HH-wave contribution is considered distinct

βththth xxy == )|1Pr(logit

• ‘Solve’ problem by ignoring it– Cannot model change – Knowingly mis-specified model

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Review: Part 2

� Household-level longitudinal analysis � Pickles & Davies (1985)� Collapse person-level into household-level data set� Each household-wave contribution in separate row� HH-level random effect for autocorrelation

hththth uxxy +== β)|1Pr(logit

• ‘Fudges’ longitudinal HH problem: ‘intact’ HHs only– Ahead of its time … but selection effects?

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Dynamic HHs

� u comprises omitted time-invariant person chars.� e.g. behaviour, personality, health, attitude

� Made up of� HH members’ characteristics � Change with HH composition

� Change in HH membership, change in HH residual

� But all random effectsare independent:� What if two HHs shared individuals: contradiction?

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Review: Part 3� Individual-based approaches

� Davies Withers (1997), Bӧheim & Taylor (2002)� (Ideally) Covariates characterise household members� Person-level residual (B&T 2002)

itititi uxxy +== β)|1Pr(logit

• ‘Double counting’ means SEs under estimated

• Requires extensive HH-level information– But previous applications included little info. on

partner

hyyy ttt HHin both if 21 ==

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Advice from the Study

“Although households were the initial sampling unit, the BHPS does not treat them as longitudinal entities as

households do not remain constant over time. For this reason, longitudinal weights are not calculated for households. To study households longitudinally,

researchers must choose a household member to follow over time and take observations of their household characteristics from wave to wave.”

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Review: Part 4

� Represent HH by a head of household (HoH)� Ioannides & Kan (1996)

� HoH is nominal choice but they chose males

� Limited covariates to represent other partner

� Random effect unchanged even if couple separated

� Improved by Rabe & Taylor (2010) � Included covariates for both partners

� Separate models if household is singleton or couple

� ‘Singleton’ and ‘couple’ random effects independent

itititi uxxy +== β)|1Pr(logit

++

==singleton if~couple if

)|1Pr(logitSiti

Citi

titiux

uxxy

αβ

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Review: Part 5

� Health and place (Chandola et al 2005)� Previous work highlighted area-level effects� But are apparent area effects actually HH effects?� What about changing HHs?

� Used multiple membership multilevel models

� Outcome SF-36 at Wave 9� Weight HH by no. waves� Not truly longitudinal

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Multiple Membership Models

� Due to Browne et al (2001)� Fit by MCMC (Rasbash et al 2009; Leckie & Charlton 2013)

� No HH effects, just weighted sum of person effects

∑ =+n

k ktktiuwx

1β

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New Approaches: Generalisation

� Steele et al (2013)� Extended HoH model� Multiple membership model

� Generalised couple model

=otherwise0

at couplein HoH if1 tcti

Use interactions (no constant term)

)1)(~()()|1Pr(logit )()( titS

itititC

itititi cuxcuxxy −+++== αβ

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Random-Part Specification

� Extended HoH model:

� Multiple membership model:

=

2

2

)(

)(

,0

0~

S

SCC

Si

Ci

tSi

tCi N

u

u

u

u

σσσ

),0(~, where

, ofpartner if 22

2...

)(

)(

u

dii

ji

i

ji

tSi

tCi

Nuu

ij

u

uu

u

u

σ

+=

c.f. consensus HH model (Corfman & Lehmann 1987; Chiappori 1992)

Coupling process independent (given predictors)

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Example: Residential Mobility

3 (Florence)

1 (Gertrude)

2 (Phil)

Wave 2

2

3

1

Wave 1

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Data View

PID Wave HoHID c Name

1 1 1 1 Gert

1 2 1 0 Gert

2 1 1

2 2 2

3 1 2 0 Flo

3 2 2 1 Flo

Phil’s records deleted

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HoH Model: Covariance Matrix

22

1211

C

CC

=

2

2

110

0

S

CCσ

σ

=

SC

SCCσ

σ0

012

=

2

2

220

0

C

SCσ

σ

Florence

Wave 2

Wave 1

Gertrude

Gertrude

Florence

Phil with neither: Correct

Phil with both: Incorrect

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MM Model: Covariance Matrix

22

1211

C

CC

=

2

2

110

02

u

uCσ

σ

=

20

422

22

12

u

uuCσσσ

=

20

02

2

22

u

uCσ

σ

Consensus means couples’ decisions less variable than singletons’

Correct but (over?) structured

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Simulation Study Results

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Ioannides & Kan Rabe & Taylor Extended HoH Multiple membership

Boheim & TaylorNote typo: should equal 1.0402/2 = 0.50201

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Table 5a. Estimates of duration effects and residual variances from alternative models of residential mobility, British Household Panel Study (Steele et al 2013)

Variable HoH-common HoH-joint MM-consensus MM-double

Est. SE Est. SE Est. SE Est. SE

Constant -1.70 0.13 -1.76 0.14 -1.77 0.13 -1.69 0.13

Years since last move(ref ≤ 1)

(1,2] -0.69 0.07 -0.69 0.07 -0.69 0.07 -0.69 0.07

(2,3] -0.94 0.09 -0.93 0.09 -0.93 0.09 -0.94 0.09

(3,4] -0.98 0.10 -0.96 0.10 -0.96 0.10 -0.98 0.10

⁞

(8,9] -1.25 0.18 -1.21 0.19 -1.20 0.19 -1.26 0.18

(9,10] -1.13 0.19 -1.09 0.19 -1.08 0.19 -1.14 0.18

(10,11] -1.40 0.22 -1.37 0.22 -1.35 0.23 -1.41 0.22

>11 -1.43 0.09 -1.39 0.09 -1.38 0.09 -1.44 0.09

Residual variances

Between-individual (Singles) 0.14 0.04 0.22 0.06 0.23 0.05 0.12 0.03

Between-couple 0.14† 0.04† 0.09 0.04 0.12† 0.02† 0.24† 0.06†

Single-couple covariance - - 0.03 0.05 0.12† 0.02† - -

For Singles 41

Table 6a. Estimates of covariate effects from alternative models of residential mobility, British Household Panel Study (Steele et al 2013)

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HoH-common HoH-joint MM-consensus MM-double

Variable Est. SE Est. SE Est. SE Est. SE

Tenure

Owned-mortgage 0 - 0 - 0 - 0 -

Owned-outright 0.30 0.10 0.29 0.10 0.30 0.10 0.29 0.07

Private rented 1.52 0.07 1.49 0.07 1.51 0.06 1.50 0.05

Social rented 0.24 0.08 0.23 0.07 0.23 0.08 0.25 0.06

Living with parents 1.47 0.19 1.45 0.19 1.47 0.19 1.53 0.15

London residence -0.02 0.09 -0.02 0.09 -0.02 0.09 -0.03 0.07

Rooms per person -0.52 0.05 -0.501 0.05 -0.51 0.05 -0.51 0.03

Age, centred at 40 -0.03 0.003 -0.03 0.003 -0.03 0.003 -0.03 0.002

(Age, centred at 40)2 -0.001 0.0002 -0.001 0.0002 -0.001 0.0002 -0.001 0.0002

No. dependent children

0 0 - 0 - 0 - 0 -

1 0.25 0.08 0.25 0.07 0.25 0.08 0.25 0.06

≥ 2 -0.28 0.08 -0.27 0.08 -0.28 0.08 -0.26 0.06

Other covariates (estimates not shown): Age of Youngest Child, Cohabiting, Post-School Education, Employment status

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� On fixed effects (coefficients)� Introduces unnecessary bias

� (See HoH-common, MM-double columns)

� On random-part (variances & covariances)� Can be severely biased/misleading interpretation

� Not big impact in this example� Large sample size (sim and data example) limit impact: worse if

sample size smaller

Summary: Impact of Mis-specification

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PERSON-LEVEL OUTCOMES

Part 3: Multilevel Modelling with Longitudinal Households

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Handling Longitudinal HHs

� Back to linear models

� Drop household subscriptand ignoring autocorrelation …

� How do we specify ui(HH(t))?

hihtihtihtih uvexy +++= β

45

))((0 tHHitititi uexy +++= β

Approach 1: HH = House

� u comprises omitted household characteristics� e.g., draughty, damp, noisy, small

� Random effect unaffected by� No. HH members� Characteristics of HH members

� u1 and u2 are� a) Distinct: HHs 1 and 2 have distinct physical locations� b) Independent: person 3 chose new HH ‘randomly’

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Longitudinal HHs: Simple Example

1

3

2

Wave 2

1

3

2

Wave 1

HH 1

HH ?

HH ??

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Approach 1: HH Definitions

� Persons 1 and 2 still in same household�Keep same random effect (u1) at wave 2

� Person 3 left to join all-new household�Different random effect (u2) at wave 2

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Approach 1: Data View

PID Wave HHID

1 1 1

1 2 1

2 1 1

2 2 1

3 1 1

3 2 2

49

Approach 1: Model

iii euxy 1111 ++= β

2,1for 2122 =++= ieuxy iii β

2322323 euxy ++= β

Wave 1

Wave 2

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Approach 1: Covariance Matrix

2221

1211

CC

CC

++

+=

22

222

2222

11

ue

uue

uuue

C

σσσσσσσσσ

=0

0

02

22

12 u

uu

C σσσ

++

+=

22

22

222

22 0

0

ue

ue

uue

C

σσσσ

σσσ

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Approach 2: HoH-Type Models

� No need to worry about double counting� Rabe & Taylor (2010) HoH model: HoH = HH

� Wave 1: All in HH 1 (u1)

� Wave 2: Persons 1 and 2 in ‘different’ HH (u2)

� Wave 2: Person 3 left in another HH (u3)

� Steele et al (2013) extended-HOH model� Dummy variables for size of original HH

� Random effects uHH1, uHH2, uHH3 etc. with covariances

� Leavers HH effects independent of ‘core’ HH

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Approach 2: Data View

PID Wave HHID

1 1 1

1 2 2

2 1 1

2 2 2

3 1 1

3 2 3

53

Approach 2: Rabe&Taylor-type Model

iii uxy 1111 εβ ++=

2,1for 2222 =++= iuxy iii εβ

2332323 εβ ++= uxy

Wave 1

Wave 2

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Approach 2: Covariance Matrix

2221

1211

CC

CC

++

+=

22

222

2222

11

ue

uue

uuue

C

σσσσσσσσσ

=0

00

000

12C

++

+=

22

22

222

22 0

0

ue

ue

uue

C

σσσσ

σσσ

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Approach 3: Super-Household

� Super-household (SHH)� A set of HHs which have shared at least one individual

� Include SHH random effect (v)� 3-level model: Persons within HHs within SHHs� allows between-wave correlation

� v is very hard to interpret� Earlier wave outcomes can depend on people who have yet to move in� Or on people who did not move in but were co-resident with someone who

did

� Covariance structure too crude

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Approach 3: SHH Definition

� Combine with distinct HHs� SHHs can be determined graphically:

� Wave 1:� Each node represents person at Wave 1� Insert edge between person pairs in same HH

� Wave 2: Leaving existing edges intact:� Inset new node for new sample members� Insert edge between pairs in same HH

� Repeat for Waves 3, …, T� SHH: Set of persons connected by paths

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Approach 3: Data View

PID Wave HHID SHHID

1 1 1 1

1 2 2 1

2 1 1 1

2 2 2 1

3 1 1 1

3 2 3 1

N.B. Distinct HHs

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Approach 3: Model

iii vuxy 11111 εβ +++=

2,1for 21222 =+++= ivuxy iii εβ

23132323 εβ +++= vuxy

Wave 1

Wave 2

59

Approach 3: Covariance Matrix

+++++++++

=222

22222

2222222

11

vue

vuvue

vuvuvue

C

σσσσσσσσσσσσσσσ

=2

22

222

12

v

vv

vvv

C

σσσσσσ

++++

+++=

222

2222

222222

22

vue

vvue

vvuvue

C

σσσσσσσσσσσσσ

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Approach 4: Multiple Membership

� Pros: Flexible enough to handle larger HHs and person-level outcomes

� Cons: Independence assumptions about who forms HHs with whom become more tenuous

61

Approach 4: Data View

PID Wave HHID

1 1 1

1 2 2

2 1 1

2 2 2

3 1 1

3 2 3

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Approach 4: Model

113211111 3

1

3

1

3

1iii uuuxy εβ ++++=

2,1for 2

1

2

122212222 =+++= iuuxy iii εβ

23333233233 εβ +++= uuxy

Wave 1

Wave 2

63

Approach 4: Covariance Matrix

++

+=

3

33

333

22

222

2222

11

ue

uue

uuue

C

σσσσσσσσσ

=3

33

333

2

22

222

12

u

uu

uuu

C

σσσσσσ

++

+=

22

22

222

22 02

022

ue

ue

uue

C

σσσσ

σσσ

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FURTHER ISSUES

Part 4

65

Selection

� Usual ‘random effects’ assumption� Covariates and person/HH effects uncorrelated� Unlikely if person-HH selection non-random� Estimates are purely descriptive: not causal effects

� On-going further work� Modelling residential move process incorporating

‘push’ and ‘pull’ factors (Steele et al 2015, under review)

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Sample Design

� Stratified multistage sampling design� Design weights

� Self-weighting if extended samples excluded� Clustering handled by multilevel model (more or less)

� Post-stratification and non-response weights� Depends on how correlated these things are with your

outcome (given included covariates)� Sensitivity analysis (with and without/SEs wrong)

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Missing Data

� Multiple imputation� Only for standard hierarchical structures

� Using MCMC:� Fills in missing outcomes using data augmentation� MAR assumption� More difficult for missing covariates

� Drop incomplete wave contributions� c.f. noninformative censoring in survival analysis� Different assumption to MAR but not unrealistic

� Informative drop-out (Washbrook et al 2014)

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References� Böheim, R., M. Taylor. 2002. "Tied down or room to move? Investigating the relationships between housing tenure,

employment status and residential mobility in Britain." Scottish Journal of Political Economy 49:369-92.� Browne, W.J., H. Goldstein, J. Rasbash. 2001. "Multiple membership multiple classification (MMMC) models."

Statistical Modelling 1:103-24.� Chandola, T., P. Clarke, R. Wiggins, M. Bartley. 2005. "Who you live with and where you live: setting the context for

health using multiple membership multilevel models." Journal of Epidemiology and Community Health 59(2):170-75.� Chiappori, P.A. 1992. "Collective labor supply and welfare." Journal of Political Economy 100:437-67.� Clark, W.A.V., Y. Huang. 2003. "The life course and residential mobility in British housing markets." Environment and

Planning Series A 35:323-39.� Corfman, K.P., D.R. Lehmann. 1987. "Models of cooperative group decision-making and relative influence - an

experimental investigation of family purchase decisions." Journal of Consumer Research 14:1-13.� Davies Withers, S. 1997. "Methodological considerations in the analysis of residential mobility: A test of duration, state

dependence, and associated events." Geographical Analysis 29:354-72.� Goldstein, H. 2011. Multilevel Statistical Models (4th ed.). Wiley: Chichester.� Goldstein, H., J. Rasbash, W.J. Browne, G. Woodhouse, M. Poulain. 2000. "Multilevel models in the study of dynamic

household structures." European Journal of Population 16:373-87.� Ioannides, Y.M., K. Kan. 1996. "Structural estimation of residential mobility and housing tenure choice." Journal of

Regional Science 36:335-63.� Leckie, G.B., C. Charlton. 2013. "runmlwin - a program to run the MLwiN multilevel modelling software from within

Stata." Journal of Statistical Software 52:11.� Murphy, M.J. 1996. "The dynamic household as a logical concept and its use in demography." European Journal of

Population 12:363-81.� Pickles, A., R.B. Davies. 1985. "The longitudinal analysis of housing careers." Journal of Regional Science 25:85-101.� Rabe, B., M. Taylor. 2010. "Residential mobility, quality of neighbourhood and life course events." Journal of the Royal

Statistical Society Series A (Statistics in Society) 173:531-55.� Rasbash, J., C. Charlton, W.J. Browne, M.J.R. Healy, B. Cameron. 2009. MLwiN version 2.1. University of Bristol:

Centre for Multilevel Modelling.� Snijders, T.A.B., R. Bosker. 2012. Multilevel Analysis: An Introduction to Basic and Advanced Multilevel Modeling (2nd

ed.). Sage: London.� Steele, F., P. Clarke, E.Washbrook. 2013. “Modeling household decisions using longitudinal data from household

panel surveys, with applications to residential mobility.” Sociological Methodology 43: 220-271. � Washbrook, E.V., P. Clarke, F. Steele. 2014. “Investigating non-ignorable dropout in panel studies of residential

mobility.” Journal of the Royal Statistical Society Series C (Applied Statistics) 63: 239-266.

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