Discrete-time Event History Analysis

Fiona SteeleCentre for Multilevel Modelling

Institute of Education

2

Discrete-time EHA for …

• Repeated events

• Multiple states

• Competing risks

• Multiple processes

3

Application: Partnership Outcomes and Childbearing in Britain

• Data from National Child Development Study (NCDS) – 1958 birth cohort. Women only.

• Partnership defined as co-resident relationship of 1 month.

• Interested in durations of partnerships and intervals between conceptions (leading to live births) within partnerships.

4

Features of NCDS Data

• Repeated events– Women with > 1 partnership and/or birth

• Multiple states– Marriage and cohabitation

• Competing risks– Outcomes of cohabitation: separation or marriage

• Multiple processes– Partnership durations and conception intervals

5

Discrete-time Data Structure

ti Duration/censoring time of episode i ri Censoring indicator for episode i

(0=censored, 1=uncensored) Convert to sequence of binary responses: yi(t) =0 if no event in interval [t, t+1) =1 if event in interval [t, t+1)

6

Example of Data Structure

ti ri 3 0 2 1

t yi(t) 1 0 2 0 3 0 1 0 2 1

7

Standard Discrete-time Model

H a z a r d f u n c t i o n

)0)1(|1)(Pr()( tytyth iii L o g i t m o d e l

)()()](logit[ txtth iT

i ( t ) f u n c t i o n o f d u r a t i o n , e . g . p o l y n o m i a l x i ( t ) c o v a r i a t e s , p o s s i b l y t i m e - v a r y i n g

8

Model for Repeated EventsH i e r a r c h i c a l s t r u c t u r e : e p i s o d e s ( l e v e l 1 ) w i t h i n i n d i v i d u a l ( l e v e l 2 ) h i j ( t ) i s h a z a r d f o r e p i s o d e i o f i n d i v i d u a l j M u l t i l e v e l ( r a n d o m e f f e c t s ) l o g i t m o d e l

jijT

ij utxtth )()()](logit[

w h e r e ),0(~ 2Nu j

9

Example: Marital Separation

• Duration of marriage episode – time between start of marriage and separation/interview

(t) a cubic polynomial

• Covariates include age at start of marriage, education (time-varying)

10

11

Marital Separation: Selected Results Est. (SE) Age (ref=20-24) <20 0.65 (0.08) 25-29 -0.40 (0.09) 30-34 -0.77 (0.16) 35+ -0.56 (0.30) Post-16 education (ref=0)

1 -0.05 (0.09) 2 -0.27 (0.10) 3-5 -0.36 (0.12) 6+ -0.51 (0.17) 2 0.72 (0.22)

12

Competing Risks

yij(t) is multinomial response, indicating occurrence and type of event Response categories 0, 1, . . ., R. Category 0 is “no event”

)()( thrij is hazard of event type r

13

Discrete-time Competing Risks Model

M u l t i l e v e l m u l t i n o m i a l l o g i t m o d e l R e q u a t i o n s c o n t r a s t i n g e v e n t t y p e r w i t h “ n o e v e n t ”

)()()()()0(

)(

)()()(

)(log r

jr

ijTrr

ij

rij utxt

th

th

w h e r e ),0(~),...,( )()1( RR

jjj MVNuuu

14

Competing Risks: Example

• Outcomes of cohabitation– Separation (r=1)– Marriage to cohabiting partner (r=2)

(r)(t) cubic polynomials

15

16

17

Years to Partnership Transitions: Quartiles

25% 50% 75%

Marriage Separation

13.8 - -

Cohab Separation

3.5 9.1 -

Cohab Marriage

1.3 2.9 10.3

18

Cohabitation Outcomes: Selected Results

S e p a r a t i o n M a r r i a g e E s t . ( S E ) E s t . ( S E ) A g e ( r e f = 2 0 - 2 4 ) < 2 0 0 . 0 9 ( 0 . 1 4 ) - 0 . 0 2 ( 0 . 0 9 ) 2 5 - 2 9 - 0 . 4 2 ( 0 . 1 0 ) - 0 . 0 5 ( 0 . 0 6 ) 3 0 - 3 4 - 0 . 4 1 ( 0 . 1 2 ) - 0 . 2 2 ( 0 . 0 8 ) 3 5 + - 0 . 6 7 ( 0 . 1 6 ) - 0 . 3 5 ( 0 . 1 1 ) P o s t - 1 6 e d u c . 1 0 . 2 1 ( 0 . 1 2 ) - 0 . 0 1 ( 0 . 0 8 ) 2 0 . 1 2 ( 0 . 1 4 ) 0 . 1 0 ( 0 . 0 8 ) 3 - 5 0 . 1 1 ( 0 . 1 3 ) - 0 . 0 4 ( 0 . 0 8 ) 6 + 0 . 1 2 ( 0 . 1 5 ) - 0 . 1 1 ( 0 . 1 0 )

)Var( )( rju 0 . 5 9 ( 0 . 1 4 ) 0 . 2 3 ( 0 . 0 5 )

),Cov( )2()1(jj uu 0 . 0 8 ( 0 . 0 6 )

19

Multiple States

• Estimate equations for marital separation and outcomes of cohabitation jointly.

• State-specific intercepts and covariate effects are fitted by including dummy variables for each state and their interactions with covariates.

• Equations are linked by allowing random effects to correlate across equations.

20

Multiple States: Episode-based File

j i stateij tij rij ageij 1 1 C 3 2 25 1 2 M 2 1 26 r indicates whether episode is censored (r=0) and, if not, type of event (r=1 for separation, r=2 for marriage). Note r is binary for marriage, but multinomial for cohabitation.

21

Multiple States: Discrete-time File

t yij(t) cij mij cij*ageij mij*ageij 1 0 1 0 25 0 2 0 1 0 25 0 3 2 1 0 25 0 1 0 0 1 0 26 2 1 0 1 0 26

22

Multiple States: Estimation

• Include cij, mij, cij*ageij and mij*ageij as explanatory variables.

• Coefficients of mij and mij*ageij are intercept and effect of age on marital separation. Allow coefficient of mij to vary randomly across individuals. cij and cij*ageij will each have two coefficients for r=1 and r=2, and cij will have two random effects.

• Estimation in MLwiN (see Steele et al. 2004), or aML.

23

Multiple States: Random Effects Covariance Matrix

Mar Sep Cohab

Sep Cohab Mar

Mar Sep 1.15*

Cohab Sep 0.46* corr=0.53

0.65*

Cohab Mar 0.12

0.08

0.28*

*95% interval estimate does not contain zero

24

Multiple Processes

• Interested in impact of no. and age of children at time t ,F(t), on hazard of partnership transition

• F(t) are prior outcomes of another, related, dynamic process - fertility

• Partnership and childbearing decisions may be affected by similar unobserved characteristics F(t) may be endogenous

25

Multiprocess Model of Partnership Transitions and Fertility

hP(t): Hazard of partnership transition at time t

hF(t): Hazard of conception at time t

F(t):

Children born

before t

XP(t)

(Observed)

XF(t)

(Observed)

uF

(Unobserved)

uP

(Unobserved)

26

Multiprocess Modelling

• Estimate multistate model for transitions from marriage and cohabitation jointly with model for childbearing within marriage and cohabitation

• Leads to a total of 5 equations, with individual-level random effect in each

• In multiprocess model random effects are correlated across equations, so equations must be estimated simultaneously

27

Selected Random Effect Residual Correlations Across Processes

• Separation from marriage and marital conception

r = -0.28* (*sig. at 5% level)

• Separation from cohabitation and cohabiting conception

r = 0.19• Cohabitation to marriage and cohabiting

conceptionr = 0.59*

28

Example of Interpretation

• Cohabitation to marriage and cohabiting conception, r = 0.59*

• Women with a high propensity to move from cohabitation to marriage tend also to have a high propensity to conceive during cohabitation.

• If this correlation is ignored, hazard of marriage for women who had a child with their partner will be overstated

29

Effects of Fertility Variables on Log-odds of Marrying vs. Staying Cohabiting

Age/Fathera Single Process

Multiprocess

Preschool/Currb 1 -0.15 -0.23* 2+ -0.07 -0.25 Older/Curr 1 -0.36 -0.41* 2+ -0.32 -0.50 Preschool/Prev -0.03 -0.04 Older/Prev -0.04 -0.05 Non-coresid -0.39* -0.42* Corr(uPC(2),uFC) - 0.59* *95% interval estimate does not contain zero aFather is current or previous partner. bReference category for all vars is 0 children.

30

Some References on Discrete-time Event History Analysis

• Competing risks – Steele, Diamond and Wang (1996). Demography, 33: 12-33.

• Multiple states – Goldstein, Pan and Bynner (2004). Understanding Statistics, 3:

85-99.– Steele, Goldstein and Browne (2004). Journal of Statistical

Modelling, 4: 145-159.

• Multiple processes– Upchurch, Lillard and Panis (2002). Demography, 39: 311-329.– Steele, Kallis, Goldstein and Joshi (2004). To appear at

www.mlwin.com/team/mmmpceh