SC968: Panel Data Methods for Sociologists

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 • Set of a finite, discrete states • Units (individuals, firms, households etc.) –in one state • Transitions between states • Time until a transition takes place

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

6. States • States are categories of the outcome variable of interest • Each unit (ex.: person, household etc.) occupies exactly one state at any moment in time • Examples • alive, dead • single, married, divorced, widowed • never smoker, smoker, ex-smoker • employed, unemployed, inactive • Set of possible states called the state space

7. Events • Event=a transition from one state to another • From an origin state to a destination state • Examples • From smoker to ex-smoker • From married to widowed • Not all transitions may be possible • E.g. from smoker to never smoker

8. Risk period • 2 states: A & B • Event: transition from A B • To be able to undergo this transition, one must be in state A (if in state B already cannot transition) • Not all individuals will be in state A at any given time • 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. Time • Various meanings... • Calendar time • ...but onset of risk usually not simultaneous for all units • Ex: by age 40, some individuals will have smoked for 20+ years, other for 1 year • Duration=time since onset of risk • divorce: time since getting married • finding a job: time since becoming unemployed • death: time since being born

10. Duration • Event history analysis : analyzing the length of duration, i.e. the length of time between the onset of risk and the occurrence of an event • Examples • Duration of marriage • Length of life • In practice we model the probability of a transition conditional on being in the risk set

11. Example data

12. Calendar time Study follow-up ended 1991 1994 1997 2000 2003 2006 2009

13. Censoring • Ideally: observe individual since the onset of risk until event has occurred • ...very demanding in terms of data collection • (ex: risk of death starts when one is born) • Usually– incomplete data  censoring • An observation is censored if it has incomplete information cannot accurately measure duration • Types of censoring • Right censoring • Left censoring

14. Censoring • Right censoring: the person did not experience the event during the time that they were studied • Common reasons for right censoring • the study ends • the person drops-out of the study • We do not know when the person experiences the event but we do know that it is later than a given time T • Left censoring: the person became at risk before we started observing her • We do not know when the person entered the risk set EHA cannot deal with • We know when the person entered the risk set condition on the person having survived long enough to enter the study • Censoring independent of survival processes!!

15. Study time in years censored event censored event 0 3 6 9 12 15 18

16. Why a special set of methods? • duration =continuous variable why not OLS? • Censoring • If excluding  higher probability to throw out longer durations • If treating as complete mis-measurement of duration • Non normality of residuals • Time varying co-variates • Interested in the probability of a transition at any given time rather than in the length of complete spells • Need to simultaneously take into account: • Whether the event has taken place or not • The length of the period at risk before the event occurred

17. Survival function • Length of time (duration) before an event occurs (length of ‘spell’-T)  • probability density function (pdf)- f(t) f(t)= limPr(t<=T<=t+Δt) = δF(t) δt Δt0 Δt • cumulative density function (cdf)- F(t) F(t)= Pr( T<=t) =∫f(t) dt • Survival function: • S(t)=1-F(t)

18. Survival function PDF CDF S(t) Duration (T) CDF Duration (T) Duration (T)

19. Hazard rate • h(t)= f(t)/ S(t) • The exact definition & interpretation of h(t) differs: • duration is continuous • duration is discrete • Conditional on having survived up to t, what is the probability of leaving between t and t+Δt • It is a measure of risk intensity • h(t) >=0 • In principle h(t)= rate; not a probability • There is a 1-1 relationship between h(t), f(t), F(t), S(t) • EHA analysis: • h(t)= g (t, Xs) • g=parametric & semi-parametric specifications

20. 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 Data structure different: Duration is discrete Duration is continuous Assume: 2 states; 1 transition; no repeated events Data

21. Data structure-Discrete time t records (1 for each unit of time the person is at risk)

22. Data structure-Discrete time • The row is a an individual period • An individual has as many rows as the number of periods he is observed to be at risk • No longer at risk when • Experienced event • No longer under observation (censored) • For each period (row)- explanatory variable X  very easy to incorporate time varying co-variates • Stata: reshape long

23. Data structure-continuous time ID Entry Died End date Duration Event X • 1 01/01/1991 01/01/2008 17.00 0 • 2 01/01/1991 01/01/200201/01/2002 11.0 1 0 • 3 01/01/1995 01/01/2000 5.0 0 0 • 3 01/01/2000 01/01/2005 01/01/2005 5.0 1 1

24. Data structure-continuous time • The row is a person • Indicator for observed events/ censored cases • Calculate duration= exit date – entry date • Exit date= • Failure date • Censoring date • If time-varying covariates- • Split the period an individual is under observation by the number of times time-varying Xs change • If many Xs-change often- multiple rows

25. Worked example • Random 20% sample from BHPS • Waves 1 – 15 • One record per person/wave • Outcome: Duration of cohabitation • Condition on cohabiting • Survival time: years from starting cohabitation till year living without a partner

26. The data Duration = 5 years Event = 1 Ignore data after event = 1

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

28. Preparing the data Select records after onset of risk Generate new duration variable Declare that you want to set the data to survival time Important to check that you have set data as intended

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

30. Checking the data setup How do we know when this person divorced?

31. Trying again!

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

33. 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

34. Summarising time to event data Number of observations Person-years Less than 50% of the sample has experienced the event by the end of the study Rate per year

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

36. Graphs of survival time

37. 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

40. 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

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

42. The Cox regression model

43. Cox regression model Event History with Cox Model • Also known as the Cox proportional hazard model • No longer modelling the duration • Modelling the hazard rate • Hazard rate • h(t)= f(t)/ S(t) • conditional on having survived up to t, what is the probability of leaving between t and t+Δt

44. Some hazard shapes • Increasing: • as time elapses, more likely to experience the event • Ex: onset of Alzheimer's • Decreasing • as time elapses, less likely to experience the event • Ex: Survival after surgery • U-shaped • the hazard rate is highest when duration is low/ high • Ex: age specific mortality • Constant • hazard rate does not change with time • Ex: time till next email arrives

45. Hazard rate varying with time

46. 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 • Model is semi-parametric: • Does not estimate how the hazard rate changes with time • Estimates the effect of co-variates in shifting a baseline hazard rate

47. Cox regression equation (i) is the hazard function for individual i is the baseline hazard function and can take any form are the covariates are the regression coefficients estimated from the data • One important assumption!: • the effect of co-variates does not change with time (proportional hazards)

48. Cox regression equation (ii) • If we divide both sides of the equation on the previous slide by h0(t) and take logarithms, we obtain: • We call h(t) / h0(t) the hazard ratio • The coefficients bi...bn are estimated by Cox regression, and can be interpreted in a similar manner to that of multiple logistic regression • exp(bi) is the instantaneous relative risk of an event

49. Cox regression assumptions • Assumption of proportional hazards • No censoring patterns (i.e. censoring process independent of the survival process) • No left censoring • True starting time (on-set of risk unambiguous) • Plus assumptions for all modelling • Sufficient sample size, proper model specification, independent observations, exogenous covariates, no high multicollinearity, random sampling, and so on

50. Proportional hazards assumption • The ratio of the hazard functions of two groups is constant over time • Cox estimates of βs= change in the hazard rate relative to the baseline hazard (=hazard function of the reference group) • 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