What is Event History Analysis?

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 t1,2,3, (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 ß h0(t) is the
baseline hazard, i.e. hazard when xi0 The Cox
model can be written hi(t) h0(t) exp(ßxi) or
sometimes as log hi(t) log h0(t) ßxi Note
x could be time-varying, i.e. xi(t)
18
Cox Model Interpretation

• exp(ß) - also written as eß - is called the
  relative risk
• For each 1-unit increase in x the hazard is
  multiplied by exp(ß)
• exp(ß)gt1 implies a positive effect on hazard,
  i.e. higher values of x associated with shorter
  durations
• exp(ß)lt1 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 Respondents sex (1female,
0male) FULLTIME(t) Whether in full-time
education at age t (1yes, 0no) a(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 tfemale and t2female
  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) Its about
time Using discrete-time survival analysis to
study duration and the timing of events. Journal
of Educational Statistics, 18 155-195.
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,
4 145-159.