1. School of Epidemiology, Public Health &
EPI 5344: Survival Analysis in Epidemiology Introduction to concepts and basic methods February 24, 2015
Dr. N. Birkett,
School of Epidemiology, Public Health &
Preventive Medicine,
University of Ottawa
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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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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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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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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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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7. 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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8. 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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9. 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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10. Survival concepts (9) Cumulative Incidence Incidence density (rate)
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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11. Measuring Time (1) Units to use to measure time ‘scale’ to be used
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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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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13. D C C D
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14. Time Scale (2) For now, focus on ‘study time’ as the time scale
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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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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16. Time ‘0’ (2) No best time ‘0’ for all situations RCT of Rx
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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17. Time ‘0’ (3) ‘point source’ exposure Use date of event
Hiroshima atomic bomb
Dioxin spill (Seveso, Italy)
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18. Time ‘0’ (4) Chronic exposure Issues to consider 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.
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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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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21. D C C D
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22. D C C D
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23. Study course for patients in cohort
2001
2003
2013
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24. 01/2015

25. Time ‘0’ (7) Can be interested in more than one ‘event’ An Example:
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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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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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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28. 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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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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30. Dates in SAS (2)
data dates; input ptid surgdate mmddyy8.; datalines; /5/ /7/ /6/ /6/55 ; run; proc print data=dates;
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31. Dates in SAS (3) Obs # ptid surgdate 1 13725 13061 2 25422 13580 3
34721
12667 4
11111
-1670
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32. Dates in SAS (4)
data dates; input ptid surgdate mmddyy8.; datalines; /5/ /7/ /6/ /6/55 ; run; proc print data=dates; format surgdate date9.;
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33. Dates in SAS (5) Obs # ptid surgdate 1 13725 05OCT1995 2 25422
07MAR1997 3
34721
06SEP1994 4
11111
06JUN1955
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34. Read the date data using a ‘date format’
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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36. 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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37. 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

38. 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

39. 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

40. 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

41. 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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42. 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.
01/2015

44. 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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45. Not the right axis for a survival curve
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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.
01/2015

47. Sample Data: Mortality, no losses
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
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
2002
2003
2004
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48. 01/2015

49. Survival curve (2) Previous graph has a problem Instead
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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51. 01/2015

52. 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 65
And so on
01/2015

53. 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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54. Histogram of death time Skewed to right pdf or f(t) CDF or F(t)
Area under ‘pdf’ from ‘0’ to ‘t’
F(t) t
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55. 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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56. Survival S(t)
1-S(t)
S(t)
Deaths CI(t) t
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57. Consider these 2 survival curves
‘Rate’ of dying
Consider these 2 survival curves
Which has the better survival profile?
Both have S(3) = 0
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58. 01/2015

59. 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
01/2015

60. 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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61. Survival Curves (5)
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
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62. S(t0) t0 t0
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63. Hazard (instantaneous) Force of Mortality Incidence rate
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 Mortality
Incidence rate
Incidence density
Range: 0 ∞
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64. 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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65. 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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67. 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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69. Hazard curves (2)
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70. Hazard curves (3)
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71. 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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72. 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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74. 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
01/2015

75. 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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76. 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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77. 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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78. D D D
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79. 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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80. 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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81. 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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82. Study course for patients in cohort
2001
2003
2013
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83. 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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84. Censoring (8) Informative vs. uninformative censoring Strong bias
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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