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

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

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

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

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

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

Event Times and Censoring Times

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

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

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

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

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

Hazard of 1st Partnership

Survivor Function: Probability of Remaining Unpartnered

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)

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)

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)

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

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.

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

Discrete-time Data Structure

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:

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)

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.

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.

Proportional Gender Effects

Non-proportional Gender Effects

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

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: 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.