Introduction to
Survival Analysis
Rich Holubkov, Ph.D.
September 23, 2010
Today’s Class
• Introduction to, and motivation for,
basic survival analysis techniques (and
why we need advanced techniques)
• Presented at a “newbie Master’s
statistician” level
• I will not assume you’ve seen survival
data before, but necessarily will go
through this material extremely quickly
• I do assume you know biostats basics
like heuristics of chi-square statistics
Motivation for survival analysis
• Let’s say we have an event of interest,
such as death or failure of a device.
• We may be interested in event rates at
a particular timepoint (1 year), or
assessing event rates over time as
patients are being followed up.
• Could just calculate one-year rates…
– Losses to follow-up before one year?
– Patients not followed for one year as yet?
– What if most events happen within 30
days?
Motivation for survival analysis
• In studies where people are followed
over a period of time, we usually have:
– people in the study with different lengths
of follow-up (whether had event or not)
– people dropping out of the study for
various reasons at different times
• We want to figure out rates of an event
(like death, or liver transplant failure) at
a particular time point (or all
timepoints), taking this differential
follow-up into account.
Motivation for survival analysis
• We will often want to compare survival
data between treatment groups
(logrank test), quantify effects of
factors on outcomes (Cox model and
alternatives), sometimes in settings
where patients can have competing
outcomes (competing risks).
• Today, I will talk mainly about KaplanMeier curves and their comparison via
logrank tests. No parametrics!
Motivating Example (briefly)
• My Registry has enrolled 1000 kids
undergoing liver transplants from 1985
till 2010 (now). I want to estimate the
chance of a child being alive at 10
years post-transplant.
• The problem is, Registry has a child
who has been followed for 20 years, a
child followed for just one year,…
• How can I use every child’s
information?
Makes sense to look year by year!
(“Actuarial life tables”)
• I’ll see how many kids I had at the
beginning of each year of follow-up
(1000 enrolled in my Registry, 950 were
around at beginning of two years, etc.)
• If a child dropped out of the analysis
without dying (let’s say, during first
year of follow-up, 25 withdrew or just
weren’t around for long enough yet),
makes sense to count that child as
being around and “at risk” for half a
year.
Actuarial Life Tables
• Of my 1000 enrolled kids, at 1 year, 950
were alive, 25 dead, 25 dropouts/not
followed long enough. 1-yr survival is:
(950+12.5)/(950+25+12.5)=97.5%
• Examine the 950 left at beginning of
Year 2 in same way. At end of 2 years,
900 were alive, 20 more dead, and 30
dropouts/not long enough. Additional
1-yr survival for these 950 is then
(900+15)/(900+20+15)=97.9%.
• 2-yr survival=97.5% X 97.9%=95.4%
Extend this idea, event by event!
• I can do this for 10 years, year by year.
• But, I can do this much more finely,
effectively calculating an event rate at
every timepoint where a child dies.
Just need to know number left in the
study (“at risk”) at each of those
timepoints. This way of generating
survival curves is called Kaplan-Meier
analysis or the product-limit method.
A Kaplan-Meier Example
http://www.cancerguide.org/scurve_km.html
Let’s say we have just seven patients:
Patient A:
Patient B:
Patient C:
Patient D:
Patient E:
Patient F:
Patient G:
died at 1 year
dropped out at 2 years
dropped out at 3 years
died at 4 years
still in study, alive at 5 years
died at 10 years
still in study, alive at 12 years
So, three deaths at 1, 4, and 10 years.
Kaplan-Meier calculation
http://www.cancerguide.org/scurve_km.html
Interval
(StartEnd)
# Who
# At Risk # Censored # At Risk
Died at
at Start of During
at End of
End of
Interval Interval
Interval
Interval
Proportion
Surviving at
End of
Interval
Cumulative
Survival at
End of
Interval
0-1
7
0
7
1
6/7 = 0.86
0.86
1-4
6
2
4
1
3/4 = 0.75
0.86 * 0.75 =
0.64
4-10
3
1
2
1
1/2 = 0.5
0.86 * 0.75 *
0.5 = 0.31
1/1 = 1.0
0.86 * 0.75 *
0.5 * 1.0 =
0.31
10-12
1
0
1
0
This is our Kaplan-Meier curve!
(It’s smoother if more observations)
For basic survival analysis,
we need for each patient:
• A well-defined “time zero”, the date that
follow-up begins
• A well-defined “date of event or last
contact”, the last time we know if the
patient had an event or not
• Above two elements are equivalent to
“time on study”
• An indicator of whether or not the
patient had an event at last contact.
Simple Analysis Dataset
For our seven patient example:
Event
A: died at 1 year
1
B: dropped out at 2 years
0
C: dropped out at 3 years
0
D: died at 4 years
1
E: in study, alive at 5 years
0
F: died at 10 years
1
G: in study, alive at 12 years
0
Time
1
2
3
4
5
10
12
Dropouts: a critical assumption
• Statistically: “any dropouts must be
noninformative”. Dropout time should be
independent of failure time.
• Practically: A child who drops out of the
analysis cannot differ from a child who
stays in analysis, in terms of chances of
having an event as follow-up continues.
• Is this assumption ever reasonable?
• Is this assumption formally testable?
Simple Example
• A registry follows kids receiving
transplants in rural states
• Kids whose transplants are not doing
as well tend to move to urban areas to
be close to major medical centers.
Let’s say they are lost to follow-up.
• My registry now has relatively healthier
kids left as follow-up gets longer
• Thus, my long-term follow-up data will
give rosier picture of follow-up for
children living in rural states.
A Variant…
• Recent improvements in surgical
techniques/post-surgical treatment have
improved prognosis. So, kids
transplanted decades ago have worse
long-term prognosis. But, these are the
only kids in my registry with long-term
follow-up.
• So, my long-term follow-up data will give
unnecessarily grim picture for recently
treated kids. It’s just like healthy kids
dropping out early!
• Changes in survival probability over time
of enrollment is like nonrandom dropout.
A Variant I’ve Seen
• A surgeon keeps his own registry. His
research assistant regularly followed
patients until about 5 years ago.
• He still regularly updates the registry
database for important outcomes.
Specifically, whenever he is notified that
a patient has died, that patient’s survival
data are updated.
• What will be wrong with the database for
survival analysis?
• How can it be fixed?
Informative Censoring
• If we stay nonparametric, noninformative
censoring cannot be tested using
observed data of failure/censoring times!
• Parametric methods do exist, for
example jointly modeling dropout/events.
• Can compare entry characteristics, risk
profiles of censored versus uncensored
• Should look for time trends in long-term
follow-up studies or RCTs
• If censoring rate is appreciable, usually
concern regarding potential bias.
How can we compare
two (or more) survival curves?
• The standard approach to comparing
survival curves between two or more
groups, is termed the “logrank test”,
sometimes known as the Mantel-Cox
test.
• The test can be motivated in several
ways. I present a “chi-squared table”
approach, sort of like Mantel (1966).
How a logrank test works!
• At every timepoint when one or more
kids have an event, we calculate the
expected number of events in both
groups assuming identical survival.
• If one child has event when there were
25 kids in Group A and 75 kids in
Group B, we “expect” 0.25 events in
Group A and 0.75 in Group B.
• Do this for all events, see if sum of the
“Observed-Expected” for a group is
large, like in the chi-squared test.
Heuristic Derivation
•
•
•
•
j indexes times of events, 1 to J
N1j, N2j number at risk at time j, Nj= N1j+N2j
O1j, O2j # of events at time j, Oj= O1j+O2j
Consider as 2 x 2 table for each time j:
Arm 1
Arm 2
Total
Dead
O1j
O2j
Oj
Alive
N1j-O1j
N2j-O2j
Nj-Oj
N1j
N2j
Nj
At Risk
How a logrank test works!
• If two arms truly have same survival
distribution, then O1j is hypergeometric,
so has expectation
E1j =Oj(N1j/Nj)
and variance
V1j =(N1jN2jOj(Nj-Oj))/(Nj2(Nj-1))
• Now get a statistic summing over the J
event times, treating as sum of
independent variables
(this is very heuristic!!!)
How a logrank test works!
• Under null, with E1j =Oj(N1j/Nj) and
V1j =(N1jN2jOj(Nj-Oj))/(Nj2(Nj-1)),
(∑j(O1j-E1j ))2/∑jV1j has a chi-squared
distribution with 1 d.f. This is the standard
logrank test for testing equality between
two survival curves.
• Readily generalizes to more than 2
groups, and to subgroups (strata) within
each group to be compared.
Why this derivation?
• Using our E1j =Oj(N1j/Nj) and
V1j =(N1jN2jOj(Nj-Oj))/(Nj2(Nj-1)), we can
apply weights wj≡w(tj) at each event time
tj, j=1,…,J. Then, the weighted statistic
(∑j(wj(O1j -E1j ))2/∑jwjV1j still has a chisquared distribution with 1 d.f.
• But, using wj ≡1 yields a test that is
uniformly most powerful against all
alternatives with proportional hazards
(where “risk of event in Arm 1 vs. Arm 2”
is constant throughout follow-up)
Why is this of interest?
• This is why we generally use a
standard logrank (Mantel-Cox) test!
• But, programs like SAS will give or
offer you other variants!
– wj = Nj (Gehan-Wilcoxon or Breslow test,
gives more weight to earlier observations)
– wj = estimated proportion surviving at tj
(Peto-Peto-Prentice Wilcoxon test, fully
efficient for alternatives where odds ratio of
event for Arm 2 vs. Arm 1 constant over time)
– There are classes of these test types…
Which test to use?
• For the practicing statistician, main thing
is to be aware of these various test
statistics and how they differ.
• We usually use the standard Mantel-Cox
logrank test when planning a trial
comparing survival curves, but whichever
test is selected has to be prespecified.
• Can’t go hunting for significant p-values
after an RCT. What about before?
Variance of Kaplan-Meier
Estimates
• Using the delta method, if estimated
survival at time t is S(t), an estimate of the
variance of S(t) is S(t)2Σ(t <=t)Oi/((Ni-Oi)Ni).
i
• This estimate (Greenwood’s formula),
used by SAS, is only modified at
timepoints when events occur. Thus,
estimated variance may not be
proportional to subjects remaining at risk.
Product-Limit Survival Estimates
days_180
0.000
7.000*
15.000*
17.000
18.000
39.000*
42.000*
42.000*
44.000*
59.000*
59.000*
70.000*
92.000
120.000*
130.000*
162.000*
174.000*
Survival
1.0000
.
.
0.9756
0.9512
.
.
.
.
.
.
.
0.9215
.
.
.
.
Failure
0
.
.
0.0244
0.0488
.
.
.
.
.
.
.
0.0785
.
.
.
.
Standard
Error
0
.
.
0.0241
0.0336
.
.
.
.
.
.
.
0.0438
.
.
.
.
Survival
Number
Failed
0
0
0
1
2
2
2
2
2
2
2
2
3
3
3
3
3
Number
Left
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
Variance of Kaplan-Meier
Estimates
• You can implement Greenwood’s
formula to get confidence intervals of a
Kaplan-Meier rate at a particular
timepoint, or to test hypotheses about
that rate.
• Be aware that using the Kaplan-Meier
nonparametric estimation approach will
give substantially larger standard error
estimates than parametrically based
survival models.
Variance of Kaplan-Meier
Estimates
• Peto (1977) derived alternative formula: if
estimated survival at time t is S(t) and
number at risk at time t is N(t), Peto
estimate of the variance of S(t) is
S(t)2[1-S(t)]/ N(t).
• Easy to compute, heuristically makes
sense, perhaps(?) better when N smaller
• Worth knowing about and examining if
testing an event rate is your primary goal!
• Neither great with N small/high censoring.
Interval Censoring
• A patient dropping out of analysis
before having an event is “right
censored”.
• A patient who is known to have an
event during some interval is called
“interval censored”. For example, a
cardiac echo on 12/1/2009 shows a
valve leak. Previous echo was on
11/1/2008, and the leak could have
started anytime between the two echos.
Interval Censoring
Left Censoring
• A patient who is known to have an
event, but not more specifically than
before a particular date, is called “left
censored”.
• For example (Klein/Moeschberger), a
study looking at time till first marijuana
use may have information from some
kids who report use, but cannot recall
the date at all.
• Special case of interval censoring.
Left Censoring (#1, #4)
Interval Censoring
• Turnbull (1976) developed a
nonparametric survival curve estimator
applied to interval-censored data.
Basically, an EM algorithm is used to
compute the nonparametric MLE
product-limit curve iteratively.
Recently, more efficient algorithms
have been developed to compute
Kaplan-Meier type estimates.
Interval Censoring
• Just something to be aware of, if you
encounter this type of data.
• SAS macros %EMICM and %ICSTEST
allow construction of nonparametric
survival curves with interval-censored
data, and their comparison using a
generalized version of the logrank test.
• R also has routines (“interval” package)
Curves with Interval Censoring
Not a Pop Quiz, but…
• A statistician familiar with survival
basics may sometimes encounter data
that didn’t turn out quite as expected.
• Here is an example I’ve certainly seen,
though usually less extreme. I show
hypothetical data on a six-month study
with about 40 patients in each of two
treatment groups…
What’s Your Conclusion? Why?
What’s Your Conclusion? Why?
Logrank test p=0.64
Gehan-Wilcoxon test p=0.04
Crossing Curves Example
• If this were a pilot observational study,
logrank test is clearly inappropriate for
a future trial if data are like this.
• Can consider a different test , shorter
follow-up, or otherwise varying design
of future trial, depending on setting
(e.g., survival after cancer therapy)
• If these are results of an actual RCT
with a prespecified logrank test, you
are at least partly stuck.
Basic Survival Analysis in SAS
• To do Kaplan-Meier and logrank in SAS:
proc lifetest data=curves plots=s;
time days*event(0);
strata group;
run;
• plots=s option asks for a Kaplan-Meier plot
• time statement: days is time on study,
event is outcome indicator, with 0 as the
value for patients who are censored
(versus those who had an event).
• strata statement: compare levels of group
Basic Survival Analysis in R
• To do Kaplan-Meier curves in R:
> library(survival);
> mfit=survfit(Surv(days,event==1)~group,
data=curves); plot(mfit);
• In the survival library, Surv() creates a
survival object. First argument is follow-up
time, second is a status indicator (if true, event
occurred, otherwise censoring). survfit() then
computes Kaplan-Meier estimator.
• Logrank test: survdiff(Surv(days,event==1)
~group, data=curves)
Summary
• I have presented basic, nonparametric
approaches to analysis of survival
data. Kaplan-Meier curves and the
logrank test should be a part of every
biostatistician’s toolbox. Same is true
for proportional hazards models, to be
discussed next week by Nan Hu.
Summary
• The survival analysis assumption of
“noninformative censoring” usually
cannot be formally tested and should
be assumed not to hold. So, if a study
has substantial dropout, there may be
bias. Compare the characteristics of
dropouts to others to get an idea of
how bad the situation could be.
Summary
• The usual logrank test can be viewed as
just one member of a class of tests with
different weightings of each event time.
This test weights all event times
equally, and is usually preferred as it’s
uniformly most powerful when one
treatment’s benefit (in terms of relative
risk) is constant throughout follow-up.
Variants may be preferred for
nonstandard scenarios.
Bibliography
• Fleming R, Harrington DP. Counting Processes and
Survival Analysis. 1991. Wiley.
• Kaplan EL, Meier P (1958). Nonparametric estimation
from incomplete observations. JASA 53:457–481, 1958.
• Klein JP, Moeschberger ML. Survival Analysis:
Techniques for Censored and Truncated Data. 2nd ed.
2003. Springer.
• Mantel, N (1966). Evaluation of survival data and two new
rank order statistics arising in its consideration. Cancer
Chemotherapy Reports 50(3): 163–70.
• Peto R, Pike MC, Armitage P, et al. (1977) Design and
analysis of randomized clinical trials requiring
prolonged observation of each patient, Part II. British
Journal of Cancer 35: 1-39.
• Turnbull, B (1976). The empirical distribution function
with arbitrarily grouped, censored and truncated data.
JRSS-B, 38: 290-295.
Next Seminar
• Nan Hu, Ph.D. will be discussing
the Cox proportional hazards
model, one week from today on
September 30th.