01/20151 epi 5344: survival analysis in epidemiology introduction to concepts and basic methods...

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EPI 5344:Survival Analysis in

EpidemiologyIntroduction to concepts and basic methods

February 24, 2015

Dr. N. Birkett,School of Epidemiology, Public Health &

Preventive 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.

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Survival concepts (2)

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

• Death– Any cause

– 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 (3)

• 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

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Survival concepts (4)

• Cohort studies– 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 (5)

• 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

Survival concepts (6)

• Common epidemiological approach to the analysis of cohort studies– Most common outcome measure is:

• Incidence proportion• Cumulative incidence

– Select a point in time as the end of follow-up.– Compare groups using t-test– Logistic regression is commonly used– Produces a CIR (RR)

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Survival concepts (7)

• Issues with this approach 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 (8)

• Alternate method uses 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 (9)

• 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)

• Units to use to measure time– Normally, years/months/days

– Time of events is usually measured using dates on a calendar

– 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 using ‘calendar dates’

• Can be represented in graphic display by ‘time lines’– The 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

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C

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Time Scale (2)

• In RCT’s, focus is commonly on ‘study time’– How long after a patient starts follow-up do their events

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

starts accumulating for each patient.– Frequently used as the ‘default’ in observational research

• Most epidemiologists recommend using ‘age’ as the time scale for etiological studies– More in Session 6

• For now, focus on ‘study time’ as the time scale

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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

– Use 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– For age as time scale, time ‘0’ is date of birth

• Issues to consider– 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.

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Time ‘0’ (5)

• Here is our sample time line data• Convert for analysis by defining a time ‘0’

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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Time ‘0’ (6)

• Calendar time can be very important– Uses the actual date of the event

– Studies of incidence/mortality trends

– Normally uses Poisson or similar models

• 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

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D

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C

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C

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Study course for patients in cohort

2001 2003 2013

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Time ‘0’ (7)

• Can be interested in more than one ‘event’– More than one ‘time to event’

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

• Time to relapse;• Time to survival

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Time ‘0’ (8)

• SAS code to compute time-to-event.

• Surgical treatment for breast cancer

• Four time points:– Date of surgery

– Relapse

– Death

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

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Time ‘0’ (9)

• Time ‘0’: Date of surgery

• Event #1: Relapse– Earliest of relapse/death/end

• Event #2: Death– 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 (or weeks, months, years) from

time ‘0’ for each person• Let’s talk some SAS

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Dates in SAS (1)

• Multiple ways to get date data into SAS– I commonly use three variables for each date:

• Day• Month• Year

– Facilitates data entry and editing– Requires more complicated manipulation later

• Stored as SAS date variables– Multiple formats available for data entry– Always stored as # days since Jan 1, 1960.

Dates in SAS (2)

data dates; input ptid 1-5 @7 surgdate mmddyy8.; datalines;13725 10/5/9525422 3/7/9734721 9/6/9411111 6/6/55;run;

proc print data=dates;run; 01/2015 30

Dates in SAS (3)

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Obs # ptid surgdate

1 13725 13061

2 25422 13580

3 34721 12667

4 11111 -1670

Dates in SAS (4)

data dates; input ptid 1-5 @7 surgdate mmddyy8.; datalines;13725 10/5/9525422 3/7/9734721 9/6/9411111 6/6/55;run;

proc print data=dates; format surgdate date9.;run; 01/2015 32

Dates in SAS (5)

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Obs # ptid surgdate

1 13725 05OCT1995

2 25422 07MAR1997

3 34721 06SEP1994

4 11111 06JUN1955

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Time ‘0’

• Read the date data using a ‘date format’• If the event didn’t happen, then the date is ‘missing’

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SAS code to create event variables

Data melanoma; set melanoma;/* surv -> Alive at the end of follow-up */ if (date_of_death = .) then survevent = 0; else survevent = 1;

Run;

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SAS code to create event variables

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

Run;

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SAS code to create event variables

Data melanoma; set melanoma;/* 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;

Run;

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SAS code to create event variables

Data melanoma; set melanoma;/* 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;

/* dfs -> Died or relapsed */ if ((date_of_relapse = 0) and (date_of_death = .)) then dfsevent = 0 else dfsevent = 1;

Run;

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SAS code to create event variables

Data melanoma; set melanoma;/* 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;

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

Run;

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SAS code to create event variables

Data melanoma; set melanoma;/* 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;

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

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 ‘an event happened’• You need to know:

– type of event – time to 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)

• Censoring (censored outcome)– 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

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Survival curves (5)

• DEFINITION of hazard– 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.

• That might not be true:– Cures occur– Immunity

• Mixture models

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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 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

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Censoring (3)

• Three main kinds of censoring– 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 (4)

• 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

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Censoring (5)

• Right censoring is most commonly considered– 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 (6)• Right censoring is most commonly assumed• At the end of their follow-up, subject has not had event.

– 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 (7)

• Standard analysis ignores method used to

generate censoring.

• Type 1/2 methods work 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 (8)

• 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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