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EPI 5344:Survival Analysis in

EpidemiologyIntroduction to concepts and basic methods

February 25, 2014

Dr. N. Birkett,Department of Epidemiology & Community

Medicine,University of Ottawa

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Survival concepts (1)

• Cohort studies– Follow-up a pre-defined group of people for a period of time which can

be: • Same time for everyone

• Different time for different people.

– Determine which people achieve specified outcome.

– Outcomes could be many different things, such as:• Death

– Any cause or cause-specific

• Onset of new disease

• Resumption of smoking in someone who had quit

• Recidivism for drug use or criminal activity

• Change in numerical measure such as blood pressure– Longitudinal data analysis

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Survival concepts (2)

• Cohort studies– Traditional approach to cohorts assumes everyone is followed for the same time

• incidence proportion

• logistic regression modeling

– If follow-up time varies, what do you do with subjects who don’t make it to the

end of the study?• Censoring

– Cohort studies can provide more information than presence/absence of

outcome.• Time when outcome occurred

• Type of outcome (competing outcomes)

– Can look at rate or speed of development of outcome• incidence rate

• person-time

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Survival concepts (3)

• Time to event analysis– Survival Analysis (general term)– Life tables– Kaplan-Meier curves– Actuarial methods– Log-rank test– Cox modeling (proportional hazards)

• Strong link to engineering– Failure time studies

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Survival concepts (4)• Analysis of Cohort studies (from epidemiology)

– Incidence proportion (cumulative incidence)• Select a point in time as the end of follow-up.• Compare groups using t-test, CIR (RR)• Issues include:

– What point in time to use?– What if not all subjects remain under follow-up that long?– Ignores information from subjects who don’t get outcome or reach

the time point– What is incidence proportion for the outcome ‘death’ if we set the

follow-up time to 200 years?» Will always be 100%

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Survival concepts (5)

• Analysis of Cohort studies (from epidemiology)– Incidence rate (density)

• Based on person time of follow-up• Can include information on drop-outs, etc.• Closely linked to survival analysis methods

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Survival concepts (6)

• Cumulative Incidence– The probability of becoming ill over a pre-defined

period of time.– No units– Range 0-1

• Incidence density (rate)– The rate at which people get ill during person-time of

follow-up• Units: 1/time or cases/Person-time• Range 0 to +∞

– Very closely related to hazard rate.

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Measuring Time (1)

• Need to consider:– Units to use to measure time

• Normally, years/months/days• Time of events is usually measured as ‘calendar time’• Other measures are possible (e.g. hours)

– ‘scale’ to be used• time on study• age• calendar date

– Time ‘0’ (‘origin of time’)• The point when time starts

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Time Scale (1)

• Time of events is usually measured as ‘calendar time’

• Can be represented by ‘time lines’ in a graph• Conceptual idea used in analyses

Patient #1 enters on Feb 15, 2000 & dies on Nov 8, 2000

Patient #2 enters on July 2, 2000 & is lost (censored) on April 23, 2001

Patient #3 Enters on June 5, 2001 & is still alive (censored) at the end of the follow-up period

Patient #4 Enters on July 13, 2001 and dies on December 12, 2002

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D

D

C

C

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Time Scale (2)

• In survival analysis, focus is commonly on ‘study

time’– How long after a patient starts follow-up do their events occur?

– Particularly common choice for RCT’s

– Need to define a ‘time 0’ or the point when study time starts

accumulating for each patient.

• Most epidemiologists recommend using ‘age’ as the

time scale for etiological studies– We’ll focus on time since a defining event but, remember

this for the future.

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Origin of Time (1)

• Choice of time ‘0’ affects analysis– can produce very different regression

coefficients and model fit;

• Preferred origin is often unavailable• More than one origin may make sense

– no clear criterion to choose which to use

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Time ‘0’ (2)

• No best time ‘0’ for all situations– Depends on study objectives and design

• RCT of Rx– ‘0’ = date of randomization

• Prognostic study– ‘0’ = date of disease onset– Inception cohort– Often use: date of disease diagnosis

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Time ‘0’ (3)

• ‘point source’ exposure• Date of event

– Hiroshima atomic bomb

– Dioxin spill, Seveso, Italy

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Time ‘0’ (4)

• Chronic exposure• date of study entry

• Date of first exposure

• Age (preferred origin/time scale)

– Issues• There often is no first exposure (or no clear date of 1st

exposure)

• Recruitment long after 1st exposure– Immortal person time

– Lack of info on early events.

– ‘Attained age’ as time scale

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Time ‘0’ (5)

• Calendar time can be very important– studies of incidence/mortality trends

• In survival analysis, focus is on ‘study time’– When after a patient starts follow-up do their events occur

• Need to change time lines to reflect new time scale

Patient #1 enters on Feb 15, 2000 & dies on Nov 8, 2000

Patient #2 enters on July 2, 2000 & is lost (censored) on April 23, 2001

Patient #3 Enters on June 5, 2001 & is still alive (censored) at the end of the follow-up period

Patient #4 Enters on July 13, 2001 and dies on December 12, 2002

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D

D

C

C

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D

D

C

C

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Study course for patients in cohort

2001 2003 2013

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Time ‘0’ (5)

• Can be interested in more than one ‘event’ and thus more than one ‘time to event’

• An Example– Patients treated for malignant melanoma– Treated with ‘A’ or ‘B’– Expected to influence both time to relapse

and survival

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Time ‘0’ (6)

• Some studies have more than one outcome event

• Let’s use this to illustrate SAS code to compute time-to-

event.

• Four time points:– Date of surgery: Time ‘0’

– Relapse

– Death

– Last follow-up (if still alive without relapse.)

• Event #1: earliest of relapse/death/end

• Event #2: Earliest of death/end

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Time ‘0’

• How do we compute the ‘time on study’ for each of these events?• Convert to days (weeks, months, years) from time ‘0’ for

each person• SAS reads date data using ‘date format’

• stored as # days since Jan 1, 1960.

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SAS code to create event variables

Data melanoma; set melanoma;/* dfs -> Died or relapsed */ dfsevent = 1 – (date_of_relapse = .)*(date_of_death = .);

/* surv -> Alive at the end of follow-up */ survevent = (date_of_death ne .);

if (survevent = 0) then survtime = (date_of_last – date_of_surg)/30.4; else survtime = (date_of_death – date_of_surg)/30.4;

if (dfsevent = 0) then dfstime = (date_of_last - date_of_surg)/30.4; else if (date_of_relapse NE .) then dfstime = (date_of_relapse - date_of_surg)/30.4; else if (date_of_relapse = . and date_of_death NE .) then dfstime = (date_of_death - date_of_surg)/30.4; else dfstime = .E;

Run;

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Survival curve (1)

• What can we do with data which includes time-to-event?

• Might be nice to see a picture of the number of people surviving from the start to the end of follow-up.

Sample Data: Mortality, no losses

Year # still alive # dying in the year

2000 10,000 2,000

2001 8,000 1,600

2002 6,400 1,280

2003 5,120 1,024

2004 4,096 820

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Not the right axis for a survival curve

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Survival curve (2)

• Previous graph has a problem– What if some people were lost to follow-up?– Plotting the number of people still alive would

effectively say that the lost people had all died.

Sample Data: Mortality, no losses

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Year # still alive # dying in the year Lost to follow-up

2000 10,000 2,000 1,000

2001

2002

2003

2004

Year # still alive # dying in the year Lost to follow-up

2000 10,000 2,000 1,000

2001 7,000

2002

2003

2004

Year # still alive # dying in the year Lost to follow-up

2000 10,000 2,000 1,000

2001 7,000 1,400 800

2002 4,800 960 500

2003 3,340 670 400

2004 2,270 460 260

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Survival curve (2)

• Previous graph has a problem– What if some people were lost to follow-up?– Plotting the number of people still alive would

effectively say that the lost people had all died.

• Instead– True survival curve plots the probability of

surviving.

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Survival Curves (1)

• Primary outcome is ‘time to event’• Also need to know ‘type of event’

Person Type Time

1 Death 100

2 Alive 200

3 Lost 150

4 Death 65

And so on

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Survival Curves (2)

• Censored– People who do not have the targeted outcome (e.g.

death)• For now, assume no censoring• How do we represent the ‘time’ data in a

statistical method?– Histogram of death times - f(t)– Survival curve - S(t)– Hazard curve - h(t)

• To know one is to know them all

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t

dxxftF0

)()(

Histogram of death time- Skewed to right- pdf or f(t)- CDF or F(t)

- Area under ‘pdf’ from ‘0’ to ‘t’

t

F(t)

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Survival curves (3)

• Plot % of group still alive (or % dead)

S(t) = survival curve

= % still surviving at time ‘t’

= P(survive to time ‘t’)

Mortality rate = 1 – S(t)

= F(t)

= Cumulative incidence

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Deaths CI(t)

Survival S(t)

t

S(t)

1-S(t)

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‘Rate’ of dying• Consider these 2 survival curves• Which has the better survival profile?

– Both have S(3) = 0

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Survival curves (4)

• Most people would prefer to be in group‘A’ than group ‘B’.– Death rate is lower in first two years.

– Will live longer than in pop ‘B’

• Concept is called:– Hazard: Survival analysis/stats

– Force of mortality: Demography

– Incidence rate/density: Epidemiology

• DEFINITION– h(t) = rate of dying at time ‘t’ GIVEN that you have survived to

time ‘t’

– Similar to asking the speed of your car given that you are two hours into a five

hour trip from Ottawa to Toronto

• Slight detour and then back to main theme

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Conditional Probability

h(t0) = rate of failing at ‘t0’ conditional on surviving to t0

Requires the ‘conditional survival curve’:

Essentially, you are re-scaling S(t) so that S*(t0) = 1.0

Survival Curves (5)

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S(t0)

t0 t0

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S*(t) = survival curve conditional on surviving to ‘t0‘

CI*(t) = failure/death/cumulative incidence at ‘t’ conditional on surviving to ‘t0‘

Hazard at t0 is defined as: ‘the slope of CI*(t) at t0’

Hazard (instantaneous)Force of MortalityIncidence rateIncidence density

Range: 0 ∞

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Some relationships

If the rate of disease is small: CI(t) ≈ H(t)If we assume h(t) is constant (= ID): CI(t)≈ID*t

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Some survival functions (1)

• Exponential– h(t) = λ– S(t) = exp (- λt)

• Underlies most of the ‘standard’ epidemiological formulae.

• Assumes that the hazard is constant over time– Big assumption which is not usually true

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Some survival functions (2)

• Weibull– h(t) = λ γ tγ-1

– S(t) = exp (- λ tγ)• Allows fitting a broader range of hazard

functions• Assumes hazard is monotonic

– Always increasing (or decreasing)

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Hazard curves (2)

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Hazard curves (3)

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Some survival functions (3)

• All these functions assume that everyone eventually gets the outcome event. Suppose this isn’t true:– Cures occur– Immunity

• Mixture models– S(t) = exp(-λt) (1-π) + 1 π– S(t) π as t∞

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Some survival functions (4)

• Piece-wise exponential– Divide follow-up into intervals– The hazard is constant within interval but can differ

across intervals (e.g. ‘0’ for cure)

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Some survival functions (5)

• Piece-wise exponential– Divide follow-up into intervals– The hazard is constant within interval but can differ

across intervals (e.g. ‘0’ for cure)• Gompertz Model

– Uses a functional form for S(t) which goes to a fixed, non-zero value after a finite time

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Censoring (1)

• So much for theory• In real world, we run into practical issues:

– May only know that subject was disease-free up to time ‘t’ but then you lost track of them

– May only know subject got disease before time ‘t’– May only know subject got disease between two exam dates.– May know subject must have been outcome-free for the first

‘x’ years of follow-up (immortal person-time)– Can’t measure time to infinite precision

• Often only know year of event

– Exact time of event might not even exist in theory

Censoring (2)

• Three main kinds of censoring– Right censoring

• The time of the event is known to be later than some time

• Subject moves to Australia after three years of follow-up– We only know that they died some time after 3 years.

– Left censoring• The time of the event is known to be before some time

– Looking at age of menarche, starting with a group of 12 year old girls.

– Some girls are already menstruating

– Interval censoring• Time of the event occurred between two known times

– Annual HIV test

– Negative on Jan 1, 2012

– Positive on Jan 1, 2013

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D

D

D

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Censoring (3)

• Right censoring is most commonly considered– Type 1 censoring

• The censoring time is ‘fixed’ (under control of investigator)

– Singly censored• Everyone has the same censoring time

• Commonly due to the study ending on a specific date

– Type 2 censoring• Terminate study after a fixed number of events has happened

– most common in lab studies

– Random censoring• Observation terminated for reason not under investigator’s control

• Varying reasons for drop-out

• Varying entry times

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Censoring (4)

• Right censoring is most commonly considered– Event of interest is death but at the end of their follow-up,

subject is still alive.• Administrative Censoring• Loss-to-follow-up

– A patient moves away or is lost without having experienced event of interest

• Drop-out– Patient dropped from study due to protocol violation, etc.

• Competing risks– Death occurs due to a competing event

• We know something about these patients.– Discarding them would ‘waste’ information

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Study course for patients in cohort

2001 2003 2013

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Censoring (5)

• Standard analysis ignores method used to generate censoring.

• Type 1/2 methods are fine• ‘Random’ censoring can be a problem.

– Informative vs. uninformative censoring• Standard analyses require ‘uninformative’

censoring– The development of the outcome in subjects who are

censored must be the same as in the subjects who remained in follow-up

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Censoring (6)

• Informative vs. uninformative censoring– RCT of new therapy with serious side effects.

• Patients on this Rx can tolerate side effects until near death. Then, they drop out.

• Mortality rate in this group will be 0 (/100,000)

– Control therapy has no side-effects• Patients do not drop out near death.

• Strong bias

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Type of Censoring May Violate Assumption of Independence of Censoring/Survival

If assumption is violated, likely direction of bias on CIR estimate

Deaths from other causes when there are common risk factors*

Yes Underestimation

Failure to follow-up contacts

Yes Underestimation

Migration Yes Variable

Administrative censoring Unlikely§ Variable

* In cause-specific incidence or mortality studies§ More likely in studies with a prolonged accrual period in the presence of secular trends.

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