1. SC968: Panel Data Methods for Sociologists
Introduction to survival/event history models

2. Types of outcome Continuous OLS Linear regression
Binary Binary regression
Logistic or probit regression
Time to event data Survival or event history analysis

3. Examples of time to event data
Time to death
Time to incidence of disease
Unemployed - time till find job
Time to birth of first child
Smokers – time till quit smoking

4. Time to event data
Analyse durations or length of time to reach endpoint
Data are usually censored
Don’t follow sample long enough for everyone to get to the endpoint (e.g. death)

5. 4 key concepts for survival analysis
States
Events
Risk period
Duration

6. States States are categories of the outcome variable of interest
Each person occupies exactly one state at any moment in time
Examples
alive, dead
single, married, divorced, widowed
never smoker, smoker, ex-smoker
Set of possible states called the state space

7. Events A transition from one state to another
From an origin state to a destination state
Possible events depend on the state space
Examples
From smoker to ex-smoker
From married to widowed
Not all transitions can be events
E.g. from smoker to never smoker

8. Risk period
Not all people can experience each state throughout the study period
To be able to have a particular event, one must be in the origin state at some stage
Example
can only experience divorce if married
The period of time that someone is at risk of a particular event is called the risk period
All subjects at risk of an event at a point in time called the risk set

9. Duration
Event history analysis is to do with the analysis of the duration of a nonoccurrence of an event or the length of time during the risk period
Examples
Duration of marriage
Length of life
In practice we model the probability of a transition conditional on being in the risk set

10. Example data ID Entry date Died End date 1 01/01/1991 01/01/2008
1 01/01/ /01/2008
01/01/ /01/ /01/2000
3 01/01/ /01/2005
4 01/01/ /07/ /07/2004

11. Calendar time
Study
follow-up
ended

12. Study time in years
censored
event
censored
event

13. Censoring An observation is censored if it has incomplete information
We will only consider right censoring
That is, the person did not have an event during the time that they were studied
Common reasons for right censoring
the study ends
the person drops-out of the study
the person has to be taken off a drug

14. Data Survival or event history data characterised by 2 variables
Time or duration of risk period
Failure (event)
1 if not survived or event observed
0 if censored or event not yet occurred

15. What is the data structure?
ID Entry date Died End date Duration Event
01/01/ /01/
01/01/ /01/ /01/
01/01/ /01/
01/01/ /07/ /07/
The row is a person
The tricky part is often calculating the duration
Remember we need an indicator for observed events/ censored cases

16. Worked example Random 20% sample from BHPS Waves 1 – 15
One record per person/wave
Outcome: Duration of cohabitation
Conditions on cohabiting in first wave
Survival time: years from entry to the study in 1991 till year living without a partner

17. The data
Duration = 6 years
Event = 1
Ignore data after
event = 1

18. The data (continued)
Note missing waves
before event

19. Preparing the data Select records for respondents who
were cohabiting in 1991
Declare that you want to
set the data to survival time
Important to check that you
have set data as intended

20. Checking the data setup
1 if observation is to be used
and 0 otherwise
time of entry
time of exit
1 if event, 0 if censoring or
event not yet occurred

21. Checking the data setup
How do we know when
this person separated?

22. Trying again!

23. Checking the new data setup
Now censored instead of
an event

24. Summarising time to event data
Individuals followed up for different lengths of time
So can’t use prevalence rates (% people who have an event)
Use rates instead that take account of person years at risk
Incidence rate per year
Death rate per 1000 person years

25. Summarising time to event data
Number of observations
Person-years
<25% of sample had event
by 15 elapsed years
Rate per year

26. List the cumulative hazard function
Default is the survivor function

27. Graphs of survival time
Kaplan-Meier estimate of survival curve
The Kaplan-Meier method estimates the cumulative probability of an individual surviving after baseline to any time, t

29. Kaplan-Meier graphs
Can read off the estimated probability of surviving a relationship at any time point on the graph
E.g. at 5 years 88% are still cohabiting
The survival probability only changes when an event occurs
So the graph is stepped and not a smooth curve

32. Testing equality of survival curves among groups
The log-rank test
A non –parametric test that assesses the null hypothesis that there are no differences in survival times between groups

33. Log-rank test example
Significant difference
between men and women

34. The Cox regression model

35. Event History with Cox regression model Event History with Cox Model
No longer modelling the duration
Modelling the hazard
Hazard: measure of the probability that an event occurs at time t conditional on it not having occurred up until t
Also known as the Cox proportional hazard model

36. Some hazard shapes Increasing Decreasing U-shaped Constant
Onset of Alzheimer's
Decreasing
Survival after surgery
U-shaped
Age specific mortality
Constant
Time till next arrives

37. Cox regression model Regression model for survival analysis
Can model time invariant and time varying explanatory variables
Produces estimated hazard ratios (sometimes called rate ratios or risk ratios)
Regression coefficients are on a log scale
Exponentiate to get hazard ratio
Similar to odds ratios from logistic models

38. Cox regression equation
is the hazard function for individual i
is the baseline hazard function and can take any form
It is estimated from the data (non parametric)
are the covariates
are the regression coefficients estimated from the data
Effect of covariates is constant over time (parameterised)
This is the proportional hazards assumption
Therefore, Cox regression referred to as a semi-parametric model

39. Cox regression in Stata
Will first model a time invariant covariate (sex) on risk of partnership ending
Then will add a time dependent covariate (age) to the model

40. Cox regression in Stata

41. Interpreting output from Cox regression
Cox model has no intercept
It is included in the baseline hazard
In our example, the baseline hazard is when sex=1 (male)
The hazard ratio is the ratio of the hazard for a unit change in the covariate
HR = 1.3 for women vs. men
The risk of partnership breakdown is increased by 30% for women compared with men
Hazard ratio assumed constant over time
At any time point, the hazard of partnership breakdown for a woman is 1.3 times the hazard for a man

42. Interpreting output from Cox regression (ii)
The hazard ratio is equivalent to the odds that a female has a partnership breakdown before a man
The probability of having a partnership breakdown first is = (hazard ratio) / (1 + hazard ratio)
So in our example, a HR of 1.30 corresponds to a
probability of 0.57 that a woman will experience a partnership breakdown first
The probability or risk of partnership breakdown can be different each year but the relative risk is constant
So if we know that the probability of a man having a partnership breakdown in the following year is 1.5% then the probability of a woman having a partnership breakdown in the following year is
0.015*1.30 = 1.95%

45. Time dependent covariates
Examples
Current age group rather than age at baseline
GHQ score may change over time and predict break-ups
Will use age to predict duration of cohabitation
Nonlinear relationship hypothesised
Recode age into 8 equally spaced age groups

46. Cox regression with time dependent covariates

47. Cox regression assumptions
Assumption of proportional hazards
No censoring patterns
True starting time
Plus assumptions for all modelling
Sufficient sample size, proper model specification, independent observations, exogenous covariates, no high multicollinearity, random sampling, and so on

48. Proportional hazards assumption
Cox regression with time-invariant covariates assumes that the ratio of hazards for any two observations is the same across time periods
This can be a false assumption, for example using age at baseline as a covariate
If a covariate fails this assumption
for hazard ratios that increase over time for that covariate, relative risk is overestimated
for ratios that decrease over time, relative risk is underestimated
standard errors are incorrect and significance tests are decreased in power

49. Testing the proportional hazards assumption
Graphical methods
Comparison of Kaplan-Meier observed & predicted curves by group. Observed lines should be close to predicted
Survival probability plots (cumulative survival against time for each group). Lines should not cross
Log minus log plots (minus log cumulative hazard against log survival time). Lines should be parallel

50. Testing the proportional hazards assumption
Formal tests of proportional hazard assumption
Include an interaction between the covariate and a function of time. Log time often used but could be any function. If significant then assumption violated
Test the proportional hazards assumption on the basis of partial residuals. Type of residual known as Schoenfeld residuals.

51. When assumptions are not met
If categorical covariate, include the variable as a strata variable
Allows underlying hazard function to differ between categories and be non proportional
Estimates separate underlying baseline hazard for each stratum

52. When assumptions are not met
If a continuous covariate
Consider splitting the follow-up time. For example, hazard may be proportional within first 5 years, next 5-10 years and so on
Could covariate be included as time dependent covariate?
There are different survival regression methods (e.g. parametric model)

53. Censoring assumptions
Censored cases must be independent of the survival distribution. There should be no pattern to these cases, which instead should be missing at random.
If censoring is not independent, then censoring is said to be informative
You have to judge this for yourself
Usually don’t have any data that can be used to test the assumption
Think carefully about start and end dates
Always check a sample of records

54. True starting time
The ideal model for survival analysis would be where there is a true zero time
If the zero point is arbitrary or ambiguous, the data series will be different depending on starting point. The computed hazard rate coefficients could differ, sometimes markedly
Conduct a sensitivity analysis to see how coefficients may change according to different starting points

55. Other extensions to survival analysis
Discrete (interval-censored) survival times
Repeated events
Multi-state models (more than 1 event type)
Transition from employment to unemployment or leaving labour market
Modelling type of exit from cohabiting relationship- separation/divorce/widowhood

56. Could you use logistic regression instead?
May produce similar results for short or fixed follow-up periods
Examples
everyone followed-up for 7 years
maximum follow-up 5 years
Results may differ if there are varying follow-up times
If dates of entry and dates of events are available then better to use Cox regression

57. Finally….
This is just an introduction to survival/ event history analysis
Only reviewed the Cox regression model
Also parametric survival methods
But Cox regression likely to suit type of analyses of interest to sociologists
Consider an intensive course if you want to use survival analysis in your own work