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
We can fit a logit regression model of the form:
The covariates xtjcan be constant over time or time-varying.
ztj is vector of functions of time (e.g. polynomials or dummy variables) and αTztjis the logit of the baseline hazard function.
Other link functions possible, e.g. clog-log or probit.
Repeated events lead to a two-level hierarchical structure
Level 2: Individuals
Level 1: Episodes
of individual j
are covariates which might be time-varying or defined at
the episode or individual level
random effect representing unobserved characteristics
of individual j – unobserved heterogeneity or frailty
Assume2-level model for recurrent events
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.
where (u1j, u2j) ~ bivariate normal
Note: Generalises to multinomial logit for > 2 states
Start with an episode-based file, e.g.
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.
Convert to discrete-time format:
Eij dummy for Employment, NEij dummy for Non-Employment
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.
Suppose an individual is observed to have an event during
the 17th month, and we wish to group durations into 6-month
- No. deaths for small areas (i) within regions (j) within EC nations (k). Covariates at regional level
Browne, W. J., Steele, F., Golalizadeh, M. & Green, M. (2009). The use of simple reparameterisations in MCMC estimation of multilevel models with applications to discrete-time 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.
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.
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