SC968: Panel Data Methods for Sociologists

Introduction to survival/event history models

Types of outcome

Continuous OLS

Linear regression

Binary Binary regression

Logistic or probit regression

Time to event data Survival or event history analysis

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

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)

4 key concepts for survival analysis

States Events Risk period Duration

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

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

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

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

Example data

ID Entry date Died End date

1 01/01/1991 01/01/2008

2 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

Calendar time

1991 1994 1997 2000 2003 2006 2009

Study follow-up ended

Study time in years

0 3 6 9 12 15 18

censored

event

censored

event

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

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

What is the data structure?

ID Entry date Died End date Duration Event

1 01/01/1991 01/01/2008 17.0 0

2 01/01/1991 01/01/2000 01/01/2000 9.0 1

3 01/01/1995 01/01/2005 10.0 0

4 01/01/1994 01/07/2004 01/07/2004 10.5 1

The row is a personThe tricky part is often calculating the durationRemember we need an indicator for observed events/ censored cases

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

The data

+----------------------------+ | pid wave mastat | |----------------------------| | 10081798 1 married | | 10081798 2 married | | 10081798 3 married | | 10081798 4 married | | 10081798 5 married | | 10081798 6 married | | 10081798 7 widowed | | 10081798 8 widowed | | 10081798 9 widowed | | 10081798 10 widowed | | 10081798 11 widowed | | 10081798 12 widowed | | 10081798 13 widowed | | 10081798 14 widowed | | 10081798 15 widowed | |----------------------------|

Duration = 6 years

Event = 1

Ignore data after event = 1

The data (continued)

+----------------------------+ | pid wave mastat | |----------------------------| | 10162747 1 living a | | 10162747 2 living a | | 10162747 3 living a | | 10162747 4 living a | | 10162747 5 living a | | 10162747 6 living a | | 10162747 10 separate | | 10162747 11 . | | 10162747 12 . | | 10162747 13 . | | 10162747 14 never ma | | 10162747 15 never ma | +----------------------------+

Note missing waves before event

Preparing the data

. sort pid wave . generate skey=1 if wave==1&(mastat==1|mastat==2) . by pid: replace skey=skey[_n-1] if wave~=1 . keep if skey==1 . drop skey . . stset wave,id(pid) failure(mastat==3/6) id: pid failure event: mastat == 3 4 5 6 obs. time interval: (wave[_n-1], wave] exit on or before: failure ------------------------------------------------------------------------------ 15058 total obs. 1628 obs. begin on or after (first) failure ------------------------------------------------------------------------------ 13430 obs. remaining, representing 1357 subjects 270 failures in single failure-per-subject data 13612 total analysis time at risk, at risk from t = 0 earliest observed entry t = 0 last observed exit t = 15

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

Checking the data setup

. list pid wave mastat _st _d _t _t0 if pid==10081798,sepby(pid) noobs +-------------------------------------------------+ | pid wave mastat _st _d _t _t0 | |-------------------------------------------------| | 10081798 1 married 1 0 1 0 | | 10081798 2 married 1 0 2 1 | | 10081798 3 married 1 0 3 2 | | 10081798 4 married 1 0 4 3 | | 10081798 5 married 1 0 5 4 | | 10081798 6 married 1 0 6 5 | | 10081798 7 widowed 1 1 7 6 | | 10081798 8 widowed 0 . . . | | 10081798 9 widowed 0 . . . | | 10081798 10 widowed 0 . . . | | 10081798 11 widowed 0 . . . | | 10081798 12 widowed 0 . . . | | 10081798 13 widowed 0 . . . | | 10081798 14 widowed 0 . . . | | 10081798 15 widowed 0 . . . | +-------------------------------------------------+ 1 if observation is to be used

and 0 otherwise

1 if event, 0 if censoring orevent not yet occurred

time of exit

time of entry

Checking the data setup

. list pid wave mastat _st _d _t _t0 if pid==10162747,sepby(pid) noobs +--------------------------------------------------+ | pid wave mastat _st _d _t _t0 | |--------------------------------------------------| | 10162747 1 living a 1 0 1 0 | | 10162747 2 living a 1 0 2 1 | | 10162747 3 living a 1 0 3 2 | | 10162747 4 living a 1 0 4 3 | | 10162747 5 living a 1 0 5 4 | | 10162747 6 living a 1 0 6 5 | | 10162747 10 separate 1 1 10 6 | | 10162747 11 . 0 . . . | | 10162747 12 . 0 . . . | | 10162747 13 . 0 . . . | | 10162747 14 never ma 0 . . . | | 10162747 15 never ma 0 . . . | +--------------------------------------------------+ How do we know when

this person separated?

Trying again!

. fillin pid wave . stset wave,id(pid) failure(mastat==3/6) exit(mastat==3/6 .) id: pid failure event: mastat == 3 4 5 6 obs. time interval: (wave[_n-1], wave] exit on or before: mastat==3 4 5 6 . ------------------------------------------------------------------------------ 20355 total obs. 7524 obs. begin on or after exit ------------------------------------------------------------------------------ 12831 obs. remaining, representing 1357 subjects 234 failures in single failure-per-subject data 12831 total analysis time at risk, at risk from t = 0 earliest observed entry t = 0 last observed exit t = 15

. list pid wave mastat _st _d _t _t0 if pid==10162747,sepby(pid) noobs +--------------------------------------------------+ | pid wave mastat _st _d _t _t0 | |--------------------------------------------------| | 10162747 1 living a 1 0 1 0 | | 10162747 2 living a 1 0 2 1 | | 10162747 3 living a 1 0 3 2 | | 10162747 4 living a 1 0 4 3 | | 10162747 5 living a 1 0 5 4 | | 10162747 6 living a 1 0 6 5 | | 10162747 7 . 1 0 7 6 | | 10162747 8 . 0 . . . | | 10162747 9 . 0 . . . | | 10162747 10 separate 0 . . . | | 10162747 11 . 0 . . . | | 10162747 12 . 0 . . . | | 10162747 13 . 0 . . . | | 10162747 14 never ma 0 . . . | | 10162747 15 never ma 0 . . . | +--------------------------------------------------+

Checking the new data setup

Now censored instead of an event

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

Summarising time to event data

Number of observationsPerson-years Rate per year

<25% of sample had event by 15 elapsed years

. stsum failure _d: mastat == 3 4 5 6 analysis time _t: wave exit on or before: mastat==3 4 5 6 . id: pid | incidence no. of |------ Survival time -----| | time at risk rate subjects 25% 50% 75% ---------+--------------------------------------------------------------------- total | 12831 .0182371 1357 . . .

List the cumulative hazard function

Default is the survivor function. sts list, failure failure _d: mastat == 3 4 5 6 analysis time _t: wave exit on or before: mastat==3 4 5 6 . id: pid Beg. Net Failure Std. Time Total Fail Lost Function Error [95% Conf. Int.] ------------------------------------------------------------------------------- 2 1357 29 162 0.0214 0.0039 0.0149 0.0306 3 1166 33 89 0.0491 0.0061 0.0384 0.0625 4 1044 16 64 0.0636 0.0070 0.0513 0.0789 5 964 35 58 0.0976 0.0088 0.0818 0.1164 6 871 12 34 0.1101 0.0094 0.0931 0.1300 7 825 20 24 0.1316 0.0103 0.1128 0.1534 8 781 14 17 0.1472 0.0109 0.1271 0.1701 9 750 12 30 0.1609 0.0115 0.1398 0.1848 10 708 15 23 0.1786 0.0121 0.1563 0.2038 11 670 9 32 0.1897 0.0125 0.1666 0.2155 12 629 8 16 0.2000 0.0128 0.1762 0.2266 13 605 13 24 0.2172 0.0134 0.1922 0.2449 14 568 8 24 0.2282 0.0138 0.2025 0.2566 15 536 10 526 0.2426 0.0143 0.2160 0.2719 -------------------------------------------------------------------------------

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

0.00

0.25

0.50

0.75

1.00

0 5 10 15analysis time

Kaplan-Meier survival estimate

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

0.0

00

.25

0.5

00

.75

1.0

0

0 5 10 15time in years

Kaplan-Meier survival estimate

0.0

00

.25

0.5

00

.75

1.0

0

0 5 10 15analysis time

sex = male sex = female

Comparing survival by group using Kaplan-Meier graphs

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

. sts test sex, logrank failure _d: mastat == 3 4 5 6 analysis time _t: wave exit on or before: mastat==3 4 5 6 . id: pid Log-rank test for equality of survivor functions | Events Events sex | observed expected -------+------------------------- male | 98 113.59 female | 136 120.41 -------+------------------------- Total | 234 234.00 chi2(1) = 4.25 Pr>chi2 = 0.0392

Log-rank test example

Significant difference between men and women

The Cox regression model

Event History with Cox ModelEvent History with Cox regression 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

Some hazard shapes

Increasing Onset of Alzheimer's

Decreasing Survival after surgery

U-shaped Age specific mortality

Constant Time till next email arrives

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

Cox regression equation

).......exp()()( 22110 inniii xxxthth

)(0 th

)(thi

is the baseline hazard function and can take any formIt is estimated from the data (non parametric)

is the hazard function for individual i

inii xxx ,....,, 21

n ,....,, 21

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

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

Cox regression in Stata

. stcox female failure _d: mastat == 3 4 5 6 analysis time _t: wave exit on or before: mastat==3 4 5 6 . id: pid Cox regression -- Breslow method for ties No. of subjects = 1357 Number of obs = 12337 No. of failures = 234 Time at risk = 12337 LR chi2(1) = 4.18 Log likelihood = -1574.5782 Prob > chi2 = 0.0409 ------------------------------------------------------------------------------ _t | Haz. Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- female | 1.30913 .1734699 2.03 0.042 1.009699 1.697358 ------------------------------------------------------------------------------

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

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%

0.0

5.1

.15

.2.2

5

0 5 10 15_t

sex = women sex = men

Estimated cumulative hazard: men vs. women

.01

2.0

14

.01

6.0

18

.02

Sm

oot

hed

haza

rd fu

nct

ion

4 6 8 10 12analysis time

hazard function varying over time

Cox proportional hazards regression:

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

Cox regression with time dependent covariates

. xi: stcox female i.agecat i.agecat _Iagecat_0-7 (naturally coded; _Iagecat_0 omitted) failure _d: mastat == 3 4 5 6 analysis time _t: wave exit on or before: mastat==3 4 5 6 . id: pid Cox regression -- Breslow method for ties No. of subjects = 1357 Number of obs = 12337 No. of failures = 234 Time at risk = 12337 LR chi2(8) = 78.44 Log likelihood = -1537.4472 Prob > chi2 = 0.0000 ------------------------------------------------------------------------------ _t | Haz. Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- female | 1.3705 .1842481 2.34 0.019 1.05304 1.783666 _Iagecat_1 | .5838602 .1883578 -1.67 0.095 .3102449 1.098786 _Iagecat_2 | .311325 .1039311 -3.50 0.000 .1618279 .5989281 _Iagecat_3 | .2136714 .0737986 -4.47 0.000 .1085813 .4204725 _Iagecat_4 | .2225187 .0811395 -4.12 0.000 .1088888 .4547261 _Iagecat_5 | .4770023 .1691695 -2.09 0.037 .238035 .9558732 _Iagecat_6 | 1.203702 .4306775 0.52 0.604 .5969856 2.427023 _Iagecat_7 | 1.644141 .9677715 0.84 0.398 .518688 5.21161 ------------------------------------------------------------------------------

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

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

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

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.

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

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)

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

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

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

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

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