Transcript Multi Event History Models with Applications to the Analysis of

Multilevel Event History Models with
Applications to the Analysis of Recurrent
Employment Transitions
Fiona Steele
Outline
• The discrete-time approach
• Multilevel models and examples for:
– Recurrent events
– Multiple states
• Handling large datasets
• Examples of other applications
• Estimation/software
Why use discrete-time methods?
• Events times are often measured in discrete time
units, e.g. months or years.
• Straightforward to allow and test for non-proportional
hazards.
• We can use familiar models for discrete response
data. For more complex data structures and
processes, we can use existing estimation
procedures for multilevel models.
Restructuring data for a discrete-time analysis:
Individual-based file
E.g. records for 2 individuals
INDIVIDUAL (j) DURATION (tj) EVENT
1
5
0
2
3
1
EVENT
AGE (xj)
20
35
=1 if event observed (uncensored)
= 0 if censored.
Restructuring data for a discrete-time analysis:
Person-period file
j
t
ytj
xj
1
1
1
1
1
2
2
2
1
2
3
4
5
1
2
3
0
0
0
0
0
0
0
1
20
20
20
20
20
35
35
35
ytj = 1 if event occurs to individual j in interval t
= 0 if event does not occur in interval t
Discrete-time hazard function
Denote by ptj the probability that individual j has an event during interval
t , given no event before the start of t .
ptj  Pr( ytj  1 | yt 1, j  yt 2, j    y1 j  0)
 Pr( ytj  1)
ptj is a discrete-time approximation to the continuous-time hazard function.
Call ptj the discrete-time hazard.
A simple discrete-time logit model
We can fit a logit regression model of the form:
logit [ ptj ]  αT z tj  βT xtj
The covariates xtj can be constant over time or time-varying.
ztj is vector of functions of time (e.g. polynomials or dummy
variables) and αTztj is the logit of the baseline hazard function.
Other link functions possible, e.g. clog-log or probit.
Recurrent events
• Analyse duration of periods of continuous
exposure (episodes), e.g. employment episodes,
birth intervals, partnerships
• There may be unobserved individual-specific (i.e.
time-invariant) factors which affect the probability
of an event for all of an individual’s episodes
– referred to as unobserved heterogeneity or frailty
Hierarchical data structure
Repeated events lead to a two-level hierarchical structure
Level 2: Individuals
Level 1: Episodes
2-level model for recurrent events
logit [ ptij ]  αT z tij  βT xtij  u j
ptij
is probability of event in time interval t during episode i
of individual j
xtij
are covariates which might be time-varying or defined at
the episode or individual level
uj
random effect representing unobserved characteristics
of individual j – unobserved heterogeneity or frailty
2
u
~
N
(
0
,

Assume j
u)
Example: women’s employment
• Duration of non-employment spells; event is (re)entry into
employment
• Data are subsample from British Household Panel Study: 1401
women, 2290 episodes and 15314 person-year records
• Employment, birth and union histories collected retrospectively at
wave 2. These were linked to subsequent panel data to form
continuous histories
• Focus on effects of duration non-employed and time-varying
indicators of number and age of children, but also adjust for age,
characteristics of previous job (if any)
Unobserved individual heterogeneity
• Estimated standard deviation of woman-level random
effect is 0.65 (se=0.09)
– significant variation between women in log-odds of entering
employment due to unmeasured time-invariant characteristics
• Failure to account for unobserved heterogeneity (UH)
leads to overstatement of negative duration effects and
understatement of positive duration effects
• After accounting for UH, effects of time-varying
covariates (e.g. duration and number/age children) are
subject-specific, i.e. within-woman effects
Duration effects before and after allowing for
unobserved heterogeneity
Duration non-employed (ref is < 1 yr)
Before
After
[1,2) years
-0.788*
-0.648*
[2,3)
-1.133*
-0.927*
[3,4)
-1.489*
-1.225*
[4,5)
-1.372*
-1.077*
[5,6)
-1.256*
-0.930*
[6,7)
-1.353*
-1.025*
[7,8)
-1.575*
-1.248*
[8,9)
-1.688*
-1.350*
9+ years
-2.130*
-1.765*
* p<0.05
Estimates from multilevel logit model of
entry into employment
Child indicator
Est.
(SE)
Imminent birth (within 1 year)
-0.836*
(0.124)
1 child
-0.204*
(0.096)
2
-0.356*
(0.142)
1 child
0.244*
(0.117)
2
0.428*
(0.115)
No. children age <=5 years (ref = 0)
No. children age > 5 years (ref = 0)
* p<0.05
Modelling transitions between multiple states
An individual may pass through various ‘states’, e.g.
employment and non-employment.
Suppose there are 2 states, and denote by pstij the
probability of a transition from state s.
 
logit pstij  αTs z stij  β s x stij  usj ,
T
s  1, 2
where (u1j, u2j) ~ bivariate normal
Note: Generalises to multinomial logit for > 2 states
Multiple states: data structure (1)
Start with an episode-based file, e.g.
j
i
Stateij
tij
EVENT
Ageij
ij
1
1
E
3
1
16
1
2
NE
2
0
19
States are employment (E) and non-employment (NE)
Notes: (i) t in years; (ii) EVENTij =1 if uncensored, 0 if censored;
(iii) age, in years, at start of episode.
Multiple states: data structure (2)
Convert to discrete-time format:
t
ytij
Eij
NEij
1
2
3
1
2
0
0
1
0
0
1
1
1
0
0
0
0
0
1
1
Eij*Ageij NEij*Ag
eij
16
0
16
16
0
0
0
0
19
19
Eij dummy for Employment, NEij dummy for Non-Employment
Example: transitions between
employment and non-employment
• corr(u1j, u2j)=0.58, se=0.13, so large positive
residual correlation between E→NE and NE→E
– Women with high (low) chance of entering E tend to
have a high (low) chance of leaving E
– Positive correlation arises from two sub-groups: short
spells of E and NE, and longer spells of both types
• BUT little impact on estimates for child indicators
on (re)entry into employment
Handling large datasets
• Although flexible, a drawback of the discrete-time
approach is that the analysis file can be very large. This
is a particular problem when we wish to fit complex
models with multiple correlated random effects.
• Two possible approaches:
– Group time intervals
– More efficient algorithms, e.g. reparameterisation in MCMC
estimation (Browne et al. 2009)
Grouped time intervals
Suppose we analyse 6-month rather than monthly intervals.
Need to allow for different lengths of exposure time. In any
6-month interval, some will have the event or be censored
after 1st month while others will be exposed for full 6 months.
Denote by ntij exposure time in grouped interval t.
Estimate binomial logit model with response ytij and
denominator ntij
Note: intervals do not need to be the same width.
Example of grouped time intervals
Suppose an individual is observed to have an event during
the 17th month, and we wish to group durations into 6-month
intervals (t).
j
i
t
ntij
ytij
y*tij
1
1
1
6
0
0
1
1
2
6
0
0
1
1
3
5
1
0.2
Implications of aggregation
• Need to assume that hazard function is constant within
the grouped intervals.
• Need to fix values of time-varying covariates within
intervals, e.g. value at start.
• In practice, aggregation has little impact on estimated
baseline hazard or effects of episode/individual-level
covariates. But impact on coefficients of time-varying
covariates can be substantial.
Examples of other applications
• Hospital admissions: length of stay or duration
between admissions
– Repeated episodes nested within patients if multiple
admissions
– Hospital and GP effects using cross-classified
multilevel model (GPs refer to multiple hospitals, and
hospitals take patients from multiple GPs)
• Area effects on mortality or fertility
– Repeated birth intervals (for fertility) for individuals
nested within areas
Area effects on mortality: alternative
approaches
• As in employment example, set up person-period file with
multiple records per person, e.g. Kravdal (2006)
• Define a single binary response for each person and
include number of years of exposure as offset in a
Poisson regression, e.g. Tarkiainen et al. (2009). Could
also treat as binomial response (as for grouped time
intervals).
• If few, categorical covariates apply Poisson regression to
aggregate data (1 record for each combination of t and
covariate values)
Area effects on mortality: Multilevel Poisson
modelling of aggregate data (1)
• Suppose we want to estimate effect of age, sex and area
characteristics on individual mortality risk
• Suppose we group age into four 5-year age categories.
Then for each area define 8 cells, one for each age-sex
combination
• For area j denote by yij the observed number of deaths for
age-sex cell i
• Denote the total population at risk of mortality in cell i of area
j by nij, or might use expected number of deaths Eij
Area effects on mortality: Multilevel Poisson
modelling of aggregate data (2)
• Analyse (yij, nij) using 2-level Poisson model
• Define age and sex dummies characterising cells and
include these and area-level variables as predictors
• Application to cancer mortality: Langford and Day (2001)
- No. deaths for small areas (i) within regions (j) within EC nations
(k). Covariates at regional level
• Application to teenage conception: Diamond et al. (2002)
– No. conceptions for age-year cell (i) within electoral wards (j).
Deprivation indicators at ward level
Software
• Recurrent events and multiple states. Any
software for multilevel binary responses
• Binomial models for grouped intervals.
GLLAMM, MLwiN, WinBUGS
• Simultaneous equations models for correlated
processes. aML, GLLAMM, MLwiN, Sabre,
WinBUGS. aML is the most general (mixed
response types at different levels)
References
Browne, W. J., Steele, F., Golalizadeh, M. & Green, M. (2009). The use of simple
reparameterisations in MCMC estimation of multilevel models with applications to discretetime survival models. JRSS A, 172, 579-598.
Diamond, I., Clements, S., Stone, N. and Ingham, R. (2002) Spatial variation in teenage
conceptions in south and west England. Journal of the Royal Statistical Society, Series A,
162: 273-289.
Goldstein, H., Pan, H. and Bynner, J. (2004) “A flexible procedure for analysing longitudinal
event histories using a multilevel model.” Understanding Statistics, 3: 85-99.
Kravdal, Ø (2006) Does place matter for cancer survival in Norway? A multilevel analysis of
the importance of hospital affiliation and municipality socio-economic resources. Health and
Place, 12: 527-537.
Langford, I. H. and Day, R.J. (2001) Poisson Regression. In A.H. Leyland and H. Goldstein
(ed) Multilevel Modelling of Health Statistics. London: Wiley. Chapter 4.
References
Steele, F., Goldstein, H. and Browne, W. (2004) “A general multistate competing risks model
for event history data, with an application to a study of contraceptive use dynamics.”
Statistical Modelling, 4: 145-159.
Steele, F. (2011) Multilevel discrete-time event history models with applications to the
analysis of recurrent employment transitions (with discussion). Australian and New Zealand
Journal of Statistics (to appear).
Tarkiainen, L., Martikainen, P., Laaksonen, M. and Leyland, A.H. (2009) Comparing the
effects of neighbourhood characteristics on all-cause mortality using two hierarchical areal
units in the capital region of Helsinki. Health and Place, 16: 409-412.
See also downloadable materials:
http://www.cmm.bris.ac.uk/MLwiN/tech-support/workshops/materials/models.shtml
http://www.cmm.bris.ac.uk/MLwiN/tech-support/workshops/materials/eha.shtml