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 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 • 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 • ...intensity may not be the same • EX: one smoker may smoke 5 cigarettes a day, another 20 • 1 unit of time -same for all individuals

10. 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

11. Example data ID Entry date Died End date 1 01/01/1991 01/01/2008 • 01/01/1991 01/01/2000 01/01/2000 3 01/01/1995 01/01/2005 4 01/01/1994 01/07/2004 01/07/2004

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 • 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 ocurred

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. 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

19. 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

20. Data structure-Discrete time

21. 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

22. 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

23. 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

24. 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

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

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

27. 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

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

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

30. Trying again!

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

32. 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

33. Summarising time to event data Number of observations Person-years <25% of sample had event by 15 elapsed years Rate per year stvary-check whether a variable varies within individuals and over time

34. Descriptive analysis • To recap…. • pdf= probability that a spell has a length of exactly T f(t)= limPr(t<=T<=t+Δt) = δF(t) δt Δt0 Δt • cdf=probability that a spell has a length<=T • F(t)= Pr( T<=t) =∫f(t) dt • Survival function • S(t)=1-F(t)

35. Kaplan-Meier estimates of survival time • The Kaplan-Meier  cumulative probability of an individual surviving to any time, t • Analysis can be made by subgroup • Nonparametric method • First period: S1=1-d1/n1 exit rate • After t periods: St=(1-d1/n1)*(1-d2/n2)*……*(1-dt/nt) • Survival function  estimated only at times where you observe exits!!! • Last t that can be estimated highest non-censored time observed

36. Survival/ failure function • Describing the survival/ failure function

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 graph not smooth but (irregular) stepwise • sts graph, survival

41. 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

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

43. More elaborate models… • Modeling the hazard rate not survival time directly • h(t)=transitioning at time t, having survived up to t • Time: • Continuous- parametric • Exponential • Weibull • Log-logistic • Continuous-semi-parametric • Cox • Discrete • Logistic • Complementary log-log

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

45. Proportional-hazards (PH) models • h(t) is separable into h0(t) and the effects of Xs • h0(t)=‘baseline’ hazard that depends on t but not on individual characteristics • h(t)=h0(t)exp(βX) • Absolute differences in X proportional differences in h(t) ~scaling of h0(t)

46. The Cox regression model

47. 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

48. Cox regression equation (i) 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 PH assumption needed Estimate βs without estimating h0(t)  semi parametric model

49. 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

50. 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