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

2. Types of outcome ContinuousOLS Linear regression BinaryBinary regression Logistic or probit regression Time to event dataSurvival 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 Person IDMarried (Onset of risk) Divorce (event)End of observation period 101/01/1991-01/01/2008 201/01/199101/01/2000 301/01/1995-01/01/2005 401/01/199401/07/2004

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

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 0 3 6 9 12 15 18 censored event censored event

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)= lim Pr(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 Duration (T) CDF Duration (T) CDF PDF S(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. 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 Data structure different: Duration is discrete Duration is continuous Assume: 2 states; 1 transition; no repeated events

21. Data structure-Discrete time IDEntryEnd dateEventX at t0X at t1.... 101/01/199101/01/200801/01/2002 201/01/199101/01/2008- IDDateDuration (t)EventX 101/01/199110 101/01/199220................. 101/01/2002111 201/01/199110....... 201/01/2008170 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 17.0 0 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 Declare that you want to set the data to survival time Important to check that you have set data as intended Generate new duration variable

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

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 Rate per year Less than 50% of the sample has experienced the event by the end of the study

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. Event History with Cox Model Cox regression 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 baseline hazard function and can take any form is the hazard function for individual i 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 h 0 (t) and take logarithms, we obtain: We call h(t) / h 0 (t) the hazard ratio The coefficients b i...b n are estimated by Cox regression, and can be interpreted in a similar manner to that of multiple logistic regression exp(b i ) 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

51. 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

52. Cox regression in Stata

53. Interpreting output from Cox regression Hazard ratios (exp(β)) The effect of the covariate on the hazard rate relative to the baseline hazard Baseline hazard function  not estimated  no intercept 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 = 0.9 for women vs. men The risk of partnership breakdown is lowered by 10% for women compared with men Hazard ratio assumed constant over time At any time point, the hazard of partnership breakdown for a woman is 0.9 times the hazard for a man

54. Interpreting output from Cox regression (ii) The hazard rate may change with time but the hazard ratio 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*0.9 = 1.35% The (instantaneous) hazard rate cannot be derived from Cox estimates alone Need to estimate the baseline hazard function But... can calculate the odds of experiencing the event first = (hazard ratio) / (1 + hazard ratio) So in our example, a HR of 0.9 corresponds to a probability of 0.47 that a woman will experience a partnership breakdown first

55. 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

56. Cox regression with time dependent covariates

57. 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

58. 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.

59. 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

60. 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? Consider another (parametric) model

61. 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

62. 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

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

64. 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

65. 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