Transcript Survival analysis - University of British Columbia

Some survival
basics
Developments from the
Kaplan-Meier method
October 29 2007
Kaplan-Meier Method
• Nonparametric Estimation from
Incomplete Observations
E. L. Kaplan, Paul Meier
Journal of the American Statistical
Association, Vol. 53, No. 282 (Jun., 1958),
pp. 457-481
Kaplan-Meier plot
1.00
0.95
Survival Probability
• Method estimates
survival probabilities
while accounting for
withdrawals from the
sample before the
final outcome is
observed
0.90
0.85
0.80
• Graphical display to
show the survival
probabilities of ≥ 1
groups
0
1
2
3
4
5
6
7
Time (months)
8
9
10 11 12
Example: Time-varying covariate
Time
Kaplan-Meier plot
Example
• After a heart attack, compare patient
survival for those that underwent a
procedure (Treatment group) vs. those
who did not (No Treatment group)
• The follow-up begins at the time of the
heart attack but sometimes the procedure
(Treatment) is not done immediately
Survival curves
• Cox PH model with time-varying covariate
to test but how to graphically display
survival curves?
• Some of the different approaches used:
– Final covariate value
– Reset start time
– Extended Kaplan-Meier
Data example: Approach 1
*
*
Time
Approach 1: Final covariate value
• Categorize patients by treatment
completed by the end of follow-up
• Some issues to consider:
Treatment bias  patients not yet receiving the
treatment are assigned to the treatment group
Survival bias  survive long enough to receive
treatment
Approach 1: Final covariate value
*
*
Time
Approach 1: Final covariate value
1.00
Survival Probability
0.95
0.90
0.85
No Treatment
Treatment
0.80
0
1
2
3
4
5
6
7
Time (Months)
8
9
10 11 12
Data example: Approach 2
Time
Where:
=
Time*0 for Treatment group
Approach 2: Reset start time
• Include patients in the No Treatment group
until they receive treatment
– Censor them from No Treatment group
– Add to Treatment group on day of treatment
• The start time (t0) for No Treatment group
is the beginning of follow-up.
• The start time (t0 ) for Treatment group is
date of treatment.
– Underlying assumption: hazard rate in
treatment group is constant over time
Approach 2: Reset start time
1.00
Survival Probability
0.95
0.90
0.85
No Treatment
Treatment
0.80
0
1
2
3
4
5
6
7
8
Time (Months)
9
10 11 12
Data example: Approach 3
Time
Approach 3: Extended Kaplan-Meier
• Start patients in the no treatment group and switch the
patient over after treatment
• The start time (t0) is first day of follow-up for both groups.
– No Treatment Group:
• Include patients in the risk set until they receive
Treatment
• Censor patients at time of Treatment.
– Treatment Group:
• Add patients on the day they receive Treatment.
• Patient at risk at time ti in Treatment group is
Ni = Ni-1 – deaths – censored + new Treatment
Approach 3: Extended KM
1.00
Survival Probability
0.95
0.90
0.85
No Treatment
Treatment
0.80
0
1
2
3
4
5
6
7
8
Time (Months)
9
10 11 12
Comparison of approaches
1.00
Survival Probability
0.95
0.90
0.85
No Treatment (Approach 1 - Final covariate value)
Treatment (Approach 1 - Final covariate value)
No Treatment (Approach 2/3 - Reset start time/Extended KM)
Treatment (Approach 2 - Reset start time)
Treatment (Approach 3 - Extended KM)
0.80
0
1
2
3
4
5
6
7
8
Time (months)
9
10 11 12
Approach 3: Extended KM
• Advantage:
– Consistency in Approach 3 and
Cox proportional hazards model with timevarying covariate
• Disadvantage:
– Disadvantage of Approach 3: if too few
patients with treatment at start of follow-up 
unstable estimate
Survival Tip #1: Proper Napping Position
(from Worst Case Scenarios online http://www.worstcasescenarios.com/)
Survival Tip #2: How to Fend Off a Shark
• Hit back. If a shark is
coming toward you or
attacks you, use anything
you have in your
possession—a camera,
probe, harpoon gun, your
fist—to hit the shark's eyes
or gills, which are the areas
most sensitive to pain.
• Make quick, sharp,
repeated jabs in these
areas.
(from Worst Case Scenarios online http://www.worstcasescenarios.com/)
Example: Competing Risk
Event
(Competing Risk)
Starting condition
Event of Interest
Introduction to Competing Risks
• Interested in recurrence of an event in a time-toevent analysis
• Prior to recurrence, death could occur  Death
is a competing risk to recurrence event
• How to specify the outcome if recurrence is the
event of interest?
– Composite outcome - First event of either
recurrence/death, censor on last follow-up
– Censor on death and last follow-up
Assumptions for KM method
•
•
•
Survival probabilities are the same for
patients entering into the study early or
late
Actual event time is known
Patients who are censored have the
same survival probabilities as those who
continue to be followed
Non-Informative Censoring
• The rate of event is similar for those who
experience the event as those who did not
due to censoring
• What happens if there is informative
censoring?
Example: Informative Censoring
• Assume all event times are known for 10
patients
• Patients either experience recurrence
(event of interest) or death (competing
risk)
• No patients are lost to follow-up
From: Grunkemeier G, Anderson R, et al. 1997. Time-related analysis of nonfatal heart valve complications: Cumulative
incidence (Actual) versus Kaplan-Meier (Actuarial). Circulation. 96(9S):70II-74II.
Example: Informative Censoring
D
D
R
R
D
R
D
D
R
R
D
D
D
D
D
R
R
D
D
R
D
R
R
D
D
R
D
0.6
0.4
0.2
0.0
Freedom from Recurrence
0.8
1.0
D
D
R
0
2
4
6
Time
8
10
Note about informative censoring
• Censored patients  ‘withdrawn’ from risk
set at time of censoring in KM method
– However, censored patient is assumed to still
have the same probability of experiencing the
event of interest (non-informative censoring)
– If patient is censored at time of death, KM
estimation assumes this patient has the
possibility of the recurrence
Competing Risks
• When a patient can experience >1 type of
event and the occurrence of one type of
event modifies the probability of other
types of events
• Graphing based on using cumulative
incidence function estimator
Example: Informative Censoring
D
D
R
R
D
R
D
D
R
R
D
D
D
D
D
R
R
D
D
R
D
R
R
D
D
R
D
0.8
0.6
0.4
0.2
0.0
Cumulative Incidence Function
1.0
D
D
R
0
2
4
6
Time
8
10
Cumulative incidence calculation in
the presence of competing risks
• Step 1:
Calculate overall survival probability of
being ‘event-free’, S(tj)
– KM survival probabilities for assuming events
include both event of interest as well as
competing risk event(s)
Cumulative incidence calculation in
the presence of competing risks
• Step 2:
Calculate cumulative probability of
experiencing event of interest
Step 2a:
Calculate probability of failure for event of
interest:
h(tj)=1-(nj-dj)/nj=dj/nj
where:
nj=# patients at risk before time tj
dj=# events of interest occurring at time tj
Cumulative incidence calculation in
the presence of competing risks
Step 2b:
Calculate incidence of the event of interest:
h(tj)*S(tj-1)
• Cumulative incidence estimator at the end
of the time,t = sum of the incidence in this
interval and all previous intervals:
^
^
^
F(t) = Σ h(tj) * S(tj-1)
all j, tj ≤ t
Cumulative incidence function
approach to competing risks
• The probability of any event happening
can be partitioned into the probabilities for
each type of event
• For example,
^
^
^
Frecurrence(t) + Fdeath(t) = 1- S(t)
0.20
Recurrence + Death
Recurrence
Death
0.15
^
^
^
F1(t) + F2(t) = 1-S(t)
0.05
0.10
^
F2(t)
^
F1(t)
0.00
Cumulative Incidence Function
0.25
Cumulative Incidence Function Plot
0
100
200
Time (days)
300
Competing Risks
• Graphical display
– Comparing cumulative incidence functions for
competing risk and event of interest
– Comparing two or more groups for event of
interest
• Testing for differences
– Tests to compare cumulative incidence
between groups (similar to log-rank test)
– Modeling to adjust for covariates
(modification to Cox PH model)
Software
• Cmprsk Package in R
– Provides functions to plot, estimate, test,
model
• SAS Macro
– Provides similar functions as in R
• Downloadable from
www.uhnresearch.ca/hypoxia/People_Pintilie.htm
References
Survival curves with a time-dependent covariate
•
Simon R and Makuch RW. 1984. A non-parametric graphical representation of the relationship between survival
and the occurrence of an event: application to responder versus non-responder bias. Statistics in Medicine. 3:3544.
•
Feuer EJ, Hankey BF, et al. 1992. Graphical representation of survival curves associated with binary nonreversible time dependent covariate. Statistics in Medicine. 11:455-474.
•
Snapinn SM, Jiang Q, Iglewicz B. 2005. Illustrating the impact of a time-varying covariate with an extended
Kaplan-Meier estimator. The American Statistician. 59(4):301-307.
•
Austin PC, Mamdani MM, et al. 2006. Quantifying the impact of survivor treatment bias in observational studies.
Journal of Evaluation in Clinical Practice. 12(6):601-612.
Competing risks
•
Pintilie, M. 2006. Competing Risks: A Practical Perspective. John Wiley & Sons Ltd. West Sussex, England.
•
Satagopan JM, Ben-Porat L et al. 2004. A note on competing risks in survival data analysis. British Journal of
Cancer. 91:1229 -1235.
•
Grunkemeier G, Anderson R, et al. 1997. Time-related analysis of nonfatal heart valve complications: Cumulative
incidence (Actual) versus Kaplan-Meier (Actuarial). Circulation. 96(9S):70II-74II.
•
Southern DA, Faris PD, et al. 2006. Kaplan-Meier methods yielding misleading results in competing risk scenarios.
Journal of clinical epidemiology. 59:1110 -1114.