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![Page 1: 01/20151 EPI 5344: Survival Analysis in Epidemiology Introduction to concepts and basic methods February 24, 2015 Dr. N. Birkett, School of Epidemiology,](https://reader036.fdocuments.net/reader036/viewer/2022062309/56649f285503460f94c4073a/html5/thumbnails/1.jpg)
01/2015 1
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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01/2015 2
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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01/2015 3
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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01/2015 4
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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01/2015 5
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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01/2015 6
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
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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)
01/2015 7
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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%
01/2015 8
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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
01/2015 9
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01/2015 10
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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01/2015 11
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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01/2015 12
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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01/2015 13
D
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01/2015 14
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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01/2015 15
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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01/2015 16
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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01/2015 17
Time ‘0’ (3)
• ‘point source’ exposure
– Use date of event• Hiroshima atomic bomb
• Dioxin spill (Seveso, Italy)
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01/2015 18
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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01/2015 19
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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01/2015 20
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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01/2015 21
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01/2015 22
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01/2015 23
Study course for patients in cohort
2001 2003 2013
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01/2015 24
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01/2015 25
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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01/2015 26
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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01/2015 27
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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01/2015 28
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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01/2015 29
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.
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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
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Dates in SAS (3)
01/2015 31
Obs # ptid surgdate
1 13725 13061
2 25422 13580
3 34721 12667
4 11111 -1670
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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
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Dates in SAS (5)
01/2015 33
Obs # ptid surgdate
1 13725 05OCT1995
2 25422 07MAR1997
3 34721 06SEP1994
4 11111 06JUN1955
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01/2015 34
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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01/2015 35
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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;
01/2015 36
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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;
01/2015 37
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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;
01/2015 38
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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;
01/2015 39
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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;
01/2015 40
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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;
01/2015 41
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01/2015 42
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01/2015 43
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.
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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
01/2015 44
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01/2015 45
Not the right axis for a survival curve
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01/2015 46
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.
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Sample Data: Mortality, no losses
01/2015 47
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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01/2015 48
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01/2015 49
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
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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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