1. What is Event History Analysis?
Fiona Steele
Centre for Multilevel Modelling
University of Bristol

2. What is Event History Analysis?
Methods for analysis of length of time until the occurrence of some event. The dependent variable is the duration until event occurrence.
EHA also known as:
Survival analysis (particularly in biostatistics and when event is not repeatable)
Duration analysis
Hazard modelling

3. Examples of Applications
Education – time to leaving full-time education (from end of compulsory education); time to exit from teaching profession
Economics – duration of an episode of unemployment or employment
Demography – time to first birth (from when?); time to first marriage; time to divorce
Psychology – duration to response to some stimulus

4. Types of Event History Data
Dates of start of exposure period and events, e.g. dates of start and end of an employment spell
Usually collected retrospectively
UK sources include BHPS and cohort studies (partnership, birth, employment, and housing histories)
Current status data from panel study, e.g. current employment status each year
Collected prospectively

5. Special Features of Event History Data
Durations are always positive and their distribution is often skewed
Censoring – there are usually people who have not yet experienced the event when we observe them
Time-varying covariates – the values of some covariates may change over time

6. Censoring
Right-censoring is the most common form of censoring. Durations are right-censored if the event has not occurred by the end of the observation period.
E.g. in a study of divorce, most respondents will still be married when last observed
Excluding right-censored observations leads to bias and may drastically reduce sample size

7. Event Times and Censoring Times

8. Key Quantities in EHA
In EHA, interest is usually focused on the hazard function h(t) and the survivor function S(t)
h(t) is the probability of having at event at time t, given that the event has not occurred before t
S(t) is the probability that an event has not occurred before time t

9. Life Table Estimation of h(t)
Group durations into intervals t=1,2,3,… (often already in this form)
Record no. at risk at start of interval r(t), no. events during interval d(t), and no. censored during interval w(t)
An estimate of the hazard is d(t)/r(t). Sometimes there is a correction for censoring

10. Estimation of S(t)
Estimator of survivor function for interval t is

11. Example: Time to 1st Partnership
Source: Subsample from the National Child Development Study

12. Example of Interpretation
h(16)=0.02 so 2% partnered at age 16
h(20)=0.13 so of those who were unpartnered at their 20th birthday, 13% partnered before age 21
S(20)=0.77 so 77% had not partnered by age 20

13. Hazard of 1st Partnership

14. Survivor Function: Probability of Remaining Unpartnered

15. Introducing Covariates: Event History Modelling
There are many different types of event history model, which
vary according to:
Assumptions about the shape of the hazard function
Whether time is treated as continuous or discrete
Whether the effects of covariates can be assumed constant over time (proportional hazards)

16. The Cox Proportional Hazards Model
The most commonly applied model which:
Makes no assumptions about the shape of the hazard function
Treats time as a continuous or discrete
Assumes that the effects of covariates are constant over time (although this can be modified)

17. The Cox Proportional Hazards Model
hi(t) is hazard for individual i at time t
xi is a covariate with coefficient β
h0(t) is the baseline hazard, i.e. hazard when xi=0
The Cox model can be written
hi(t) = h0(t) exp(βxi)
or sometimes as
log hi(t) = log h0(t) + βxi
Note: x could be time-varying, i.e. xi(t)

18. Cox Model: Interpretation
exp(β) - also written as eβ - is called the relative risk
For each 1-unit increase in x the hazard is multiplied by exp(β)
exp(β)>1 implies a positive effect on hazard, i.e. higher values of x associated with shorter durations
exp(β)<1 implies a negative effect on hazard, i.e. higher values of x associated with longer durations

19. Cox Model: Gender Differences in Age at 1st Partnership
The hazard of partnering at age t is 1.5 times higher for women
than for men.
So women partner at an earlier age than men.
We assume that the gender difference in the hazard is the same for
all ages.

20. Discrete-time Event History Analysis
Event times are often measured in discrete units of time, e.g. months or years, especially when collected retrospectively
Before fitting a discrete-time model we must restructure the data so that we have a record for each time interval

21. Discrete-time Data Structure

22. Discrete-time Model
The response variable for a discrete-time model is the binary indicator of event occurrence yi(t).
The hazard function is the probability that yi(t)=1.
Fit a logistic regression model of the form:

23. Discrete-time Analysis of Age at 1st Partnership
FEMALE Respondent’s sex (1=female, 0=male)
FULLTIME(t) Whether in full-time education at age t
(1=yes, 0=no)
α(t) fitted as quadratic function by including t and t2 as
explanatory variables (after examining plot of hazard)

24. Results Exp(B) are effects on the log-odds of partnering at age t
Women partner more quickly than men.
Enrolment in full-time education is associated with a delay in
partnering.

25. Non-proportional Hazards
So far we have assumed that the effects of x are the same for all values of t
It is straightforward to relax this assumption in a discrete-time model by including interactions between x and t in the model
The following graphs show the predicted log-odds of partnering from 2 different models: 1) the ‘main effects’ model on the previous slide, 2) a model with interactions t*female and t2*female added.

26. Proportional Gender Effects

27. Non-proportional Gender Effects

28. Further Topics Repeated events, e.g. multiple marriages or births
Competing risks, e.g. different reasons for leaving a job (switch to another job, redundancy, sacked)
Multiple states, e.g. may wish to model transitions between unpartnered, marriage and cohabitation states
Multiple processes, e.g. joint modelling of partnership and education histories

29. Some References
Singer, J.D. and Willet, J.B. (1993) “It’s about time: Using discrete-time
survival analysis to study duration and the timing of events.” Journal
of Educational Statistics, 18:
Blossfeld, H.-P. and Rohwer, G. (2002) Techniques of Event History Modeling.
Mahwah (NJ): Lawrence Erlbaum.
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.” Journal of Statistical Modelling, 4: