MAXIMUM UTILIZATION
ANALYZING SURVIVAL
SIDNEY J CUTLER, M.A.,
OF THE LIFE TABLE METHOD
IN
AND
(Receivedfor publicationAug. 18, 1958)
FRED EDERER, B.S.
BETHESD~L,MD.
of patient survival is necessary for the evaluation of
44EASUREMENT
treatment of usually fatal chronic diseases. This is particularly true for
I
cancer. The American College of Surgeons, recognizing this, requires the maintenance of a cancer case registration and follow-up program for approval of a
Acceptance of survival as a criterion for measuring
hospital cancer program.’
the effectiveness of cancer therapy is also attested to by the very large number
of papers published every year reporting on the survival experience of cancer
patients.
Although the proportion of patients alive 5 years after diagnosis (S-year
survival rate) is the most frequently used index for measuring the efficacy of
therapy in cancer, an increasing number of investigators are reporting on the
manner in which patient populations are depleted during a period of time, e.g.,
survival curves. A popular and relatively simple technique for describing survival experience over time is known as the actuarial or life table method. Whereas
the method and its uses have been admirably described by a number of authors,2-6
A principal adone important aspect has received relatively little attention.
vantage of the life table method is that it makes possible the use of all survival
information accumulated up to the closing date of the study. Thus, in computing
a S-year survival rate one need not restrict the material to only those patients
who entered observation 5 or more years prior to the closing date.
We will
show that patients who entered observation 4, 3, 2, and even one year prior to
the closing date contribute much useful information to the evaluation of 5-year
survival.
Let us consider a group of patients entering observation continuously beginning with Jan. 1, 1946. Sometime early in 1952, we decide to analyze the survival experience of these patients to obtain a 5-year survival rate. We choose
Dec. 31, 1.951, as the closing date, i.e., the follow-up status and survival time of
each patient is recorded as of that date.
699
CliTLEK
Of the
patients
entering
1951, only those diagnosed
5 years.*
The exposure
is shown in Table I.
the study
during
YEAR
the
in 1946 were exposed
time for patients
6 >.ears ended
in each of the calendar
OF DIAGNOSIS
YlT4RS
OF I<SPOSURE
5
4
3
2
1
Less
be supposed,
intuitively,
that
the patients
from 1947 to 1951 are of no value in computing
31, 1951, since each of these patients
This,
however,
is not true.
Merrell
for whom less than the required
should
not
be discarded
through
maximum
reliable
S-year
expcsure
survival
time
from
utilization
is just
to show how partial
the
analysis.
for a large
of 5 years.t
survival
>.ears
even
has
is available
demonstrated
it is possible
when
primary
the
objective
can be included
Data
out that patients
information
method,
series
observation
rate as of Dec.
for less than 5 years.
have pointed
survival
OF DYING
to 6
to 5
to ‘I
to 3
to 2
than 1
survival
Wilder’
The
information
show how much is gained by doing so.
ter are used for illustrative
of years’
of the life table
rates
short
and Shulman5
TO RISK
who entered
a S-year
was under observation
number
31
I
1946
1947
1948
1949
1950
1951
It might
on Dec.
to the risk of dl-ing for at least
entering
‘I‘ADLE
CALENDAR
J. Chron. Dis.
December, 1958
.\ND EDEKEIZ
that,
to compute
longest
possible
of this paper
in the life table
from the Connecticut
is
and to
Cancer
Regis-
purposes.$
THlS ANA\TOMY Oli THE I,IFE T.W,E$
Table
II provides
patients
with
through
1951.
localized
the basic
cancer
facts,
The cases are divided
from the date of diagnosis
example,
a patient
died during
of patients
column
the third
)‘ear after
that left observation
(3, 4, or S), according
Jan.
diagnosis,
column
20, 1946,
i.e., during
to the reason
126 male
the period
1946
one for each year of diagnosis.
of one year,
during each interval
concerning
during
here.
(x to x+2).-This
in intervals
who was diagnosed
31, 1951,
diagnosed
into 6 cohorts,
The columns of Table II are described
Column 1. EVeearsAfter Diagnosis
elapsed
as of Dec.
of the kidney,
the
time
F-or
and died on Oct. 5, 1948,
interval 2-3. The number
is entered
for removal
gives
i.e., O-1, 1-2, etc.
in the appropriate
from observation.
Columns 2. Alive at Beginning
of In,terval (1,).-The
entry on the first line
of this column indicates the number of cases alive at diagnosis,
i.e., the initial
number
of patients
Column
3.
in the cohort.
Died
During
Interval
(d,).-
Tn practice, other
*In this example, date of entry into the study is dnfirxxl as data of diagnosis.
refnrrnce dates, such as date of initiation of a particular course) of therapy, may be used.
tin the snrirs reported by Wilder, the range of exposure time was from one day to, but not including.
5 years.
$We wish to thank Dr. Matthew H. Griswold, Director,
Division of Canrrr and Other Chronic
Diseasw. Connecticut
State Department
of Health, for his courtesy
in making these data available.
BWe borrowed the phrase “anatomy of the life table” from Pearl’s2 excellent textbook Biometry and
Medicnl
Stntistics.
Volume
Number
8
6
LIFE TABLE
‘~‘A1~1.11 II.
(126 hlalc
YEARS
(1)
X TO >: +
ALI\‘Ii
NING
OSIS
SI~R\TVAT.
1 jA7.A
FOR
SINGLIi
E’KAR
AT
COHORTS
HI<(;INDCrRINC;
19-I-h-1951 aml
Ikz~Iosetl
,1
OLLOW-UP
OF INTIiR\‘AI.
701
SURVIVAL
Resitlents \I’ith Localizetl Kidnev Ca~~rcr:
Follo\rwl ‘l‘hrough I kc. 31,‘1951)
Conwcticrlt
AI’TI’R
DIAGb
METHOD IN ANALYZING
INTERVAL
WITHDRAWN
(4)
1
ALIVE
DITRINC; INTERVAL*
(5)
w x
ux
Patients diagnosed in 1946 (1946 cohort)
9
o-1
1-2
2-3
3-4
4-5
S-6
t
-
:
-
4
4
1
-
1
-
._____Patients diagnosed in 1947 (1947 cohort)
0-l
l-2
2-1
3-2
4-5
18
1:
10
f,
6
Patients diagnosed in 1948 (1948 cohort)
21
10
7
O-I
l-2
2-.3
3-4
I
11
1
7
-
I
7
-
Patients diagnosed in 1949 (1949 cohort)
2”;
O-l
l-2
2-3
12
3
1
16
,
”
15
Patients diagnosed in 1950 (1950 cohort)
_______
19
13
o-:!
l-2!
5
1
:
11
Patients diagnosed in 1951 (19.51 cohort)
O-l
1
25
*Alive at closing date of study.
(
8
1
2
~
I
1.5
alive
:Five-year
survival
date
table
computing
rate.
of the study.
for
5 of this
needed
2 through
z
21
10
4
126
DUR-
obtained
DUR-
rate;
by
cell,
2 -
here
the
NUMBEI
TO THE
merely
survival
‘E
116.5
51.5
30.5
COL.
M
‘
of the
of the initial
.
(9)
pX
Bow
. .x
0.60
0.54
0.50
0.44
0.441
P?X
126 patients.
II.
(P,X
THROUGH
were
Px)
END
DIAG-
PROPORTIOS
FROM
OF INTERVAL
NOSIS
SURVIVING
C C’MULATIVE
Dec. 31, 1951)
of Table
0.60
0.90
0.93
0.88
1.00
cohorts
the account
6 yearly
0.40
0.10
0.07
0.12
0.00
(8)
PK
COL. 7)
(1 -
3f
COL. 6)
(7)
qx
(COL.
SURVIVING
PROPORTION
DYING
‘ROPORTION
to complete
data
5)
COL.
OF DYING
- w
-( COL.
RISK
EXPOSED
E.FFECTIVE
it is included
cell
1.5
w*
W,
VG INTERVAI
ALIVE
WITHDRAWN
summing,
survival
by
2
4
6
NG INTERVAL
LOW-UP
LOST TO FOL-
the 5-year
were
INTERVAL
OF INTERVAL
DIED
DURING
AT
BEGINNING
line is not
at the closing
iThis
*Columns
O-l
i-2
2-3
3-4
4-5
5-q
(1)
XTOXf
1
AFTER
DIAGNOSIS
YEARS
ALIVE
III.
COMBINED
LIFE TABLE AND COMPUTATIONOF S-YEAR SURVIVAL RATE
Residents With Localized Kidney Cancer Diagnosed 1946-1951 and Followed Through
TABLE
(126 Male Connecticut
0
2
Volume 8
Number 6
LIFE TABLE
Column 4.
enter
the
Dec.
31,
alive.
1951,
of
patients
whose
was unknown.
The
is the time elapsed
Thus,
line, i.e.,
a patient
interval
observed
under
from the analysis
method
Dec.
follow-up.
The
interval
on their
the third
II,
but
In contrast,
to assuming
of patients
year
known
during
Although
only
by pooling
by summing
after
diagnosis,
In practice,
the entries
the data
contributed
diagnosis
in 1949,
and
3 years
after
A statistical
after
measure
be discussed
later.
in Columns
1 through
COMI’l_‘TATION
Similarly,
diagnosis.
Thus,
however,
5 of Table
OF SCRVIVAL
II,
column
for all
Table
summing
III
for
was
For
cell by cell.
3 for each yearly
who died within
tabulations
one year
for 6 individual
information
resulting
from
year
some
after
three
in 1947,
in 1948,
of 5 years
of
Tables
in 1948,
diagnosed
diagnosed
contributes
a period
we will explain
each
by comparing
to have died in the second
18 patients
cohort
during
cohort
information
3), one was diagnosed
in precision
III
for each
of 126 cases would be tabulated
known
each
diagnosed
that,
survival
For example,
of the
from obser-
in this
to show how much
data.
we
date,
from observation
line of Column
cohort
column
withdrew
all 6 cohorts.
of the 21 patients
survival
of the gain
First,
on
of 47 cases
2, Column
diagnosis;
of patient
provided
sur-
follow-up
all patients
Note
2-3.
of
cases
on the closing
as withdrawals
by summing
procedure
Line
to that
of lost
this
alive
patients
in Table
of the 5 patients
III,
one in 1950.
to our knowledge
than
subsequent to
similar
omission
are entered
(1946)
the total
to the pooled
(Table
were alive 4 years
alive
lost
to be
on the fourth
that
was
For example,
on the first
rather
\JTe used the latter
II and II I, we find that
after
these
information
for the pooled
III,
assumed
cases
been
interval
all the information
as in Table
have
by dashes)
cohort in Table 11, we obtained
of diagnosis, shown in Table III.
the cohorts
is entered
Interval (w,).-In
to
one of the cohorts
example,
cohorts.
patient
from date of diagnosis the
that
which
we used the available
obtained
directly,
for each
we
date,
to date last known
complete
31, 1951, are recorded
zeros (symbolized
the last.
a full 5 years,
column
closing
to that for cases with complete
date of diagnosis.
in 1949 alad alive on Dec.
in Table
intervals
of lost
Withdrawn Alive During
5.
number
31, 1951.
during
it is usually
of lost cases was similar
depends
this
as of the
of observation
experience
is equivalent
vival experience
information.
vation
length
status
for 3 years and 4 months
the life table
cases remaining
the
survival
3-4.
In applying
Column
703
SURVIVAL
from date of diagnosis
date of last contact, the survival
enter
IN ANALYZING
Lost to Follow-up During Interval (u,)“. -In
number
to follow-up
METHOD
6
7 were
information
diagnosis.
this procedure
will
how the basic data summarized
are used to compute
survival
rates.
RATES
The first step in preparing a life table is to distribute
the deaths, losses, and
This
withdrawals
with respect to the interval in which they left 0bservation.t
__~__.
*WF!al’e using the letter “u” to represent “untraced” cases, rather than the letter “1” which comes
to mind as it symbol for “lost” cases, because “1” is a standard life table notation for “alive at beginning
of interval.”
tFor a detailed account of the mechanics of recording and tabulating survival data, see Berkson and
Gage,” pp. 4-5.
704
CUTLER
iuformatiou
entries
which
is summarized
iu (‘olumus
is entered
this column
iu (~oiumns
3, 4,
wd
5
For
year
the
by
according
4 +
number
subtracting
alive
from
Column 6.
that patients
at the
the
11, +
it1
W,).
beginning
number
losses,
of the second
alive
at the
and withdrawals
equally
year
beginning
during
(60)
was
of the
first
the
first year
to assume
number
during
that the date of diagnosis
exposed
7.
as the probability
-----
the calendar
for one-half
to risk
of the year,
the year.
year
during
For
31, 1951
for these
example,
(withdrawn
15 patients
1951 and that,
on
the
year.
is obtained
one-half
by subtracting
from
the sum of the number
the
lost
(UX + w,) /2.
Proportion Dying During Interval (q,).-This
of dying
is assumed
were exposed
Thus,
1,’ = 1, Column
the interval.*
15 were alive on Dec.
during
was observed
alive at the beginning
and withdrawn
for each follo\v-
during an interval
for one-half
iu 1951,
distributed
each patient
effective
from observation
on the average,
diagnosed
It is reasonable
was roughly
The
(d, +
E$ective Number Exposed to Risk of Dying (I,‘).-It
lost or withdrawn
of the 25 patients
number
in the study,,
Sue-c-cssive clltries
is completed
by a series of four computations
6 through 9).
to the risk of dying,
average,
The sum of the
11 I.
of cases
15).
The life table
up year (Columns
alive).
uumber
to the formula:
(126)) the sum of the deaths,
(47 +
the total
the first liue of (~01um11 2 (126 (~;\sw).
on
are obtained
example,
3, 4, and 5 of Table
equals
Ix+1 = 1, obtained
J. Chron. Dis.
December, 1958
AND EDERER
the interval.
It is obtained
is also referred to
by dividing the
*The computing
procedure given here is based on the assumption that, for cases wit,hdrawn alive
and cases lost to follow-up,
survival subsequent to date of last contact is similar to that for cases with
complete follow-up
information.
For cases withdrawn
alive, this assumption
introduces no bias, brcause there is no reason to believe that patients alive on the closing date are different from patients
observed for a longer period.
However, for cases lost to follow-up,
this assumption
may introduce
a bias.
Patients lost to follow-up
were alive when last observed, and whether their survival experience
is better than, worse than, or equal to the survival of patients remaining under follow-up
is highly
Farspeculative.
For example, cancer patients may be lost to follow-up
for a variety of reasons.
advanced cases may leave their usual place of residence to enter the household of a relative; successfully
treated patients may stop reporting to t.he tumor clinic, because they feel that no further medical care
is required.
It is therefore important to keep the proportion of cases lost to foliow-up
at a minimum.
Survival rates based on a series in which a substant,ial proportion of paGent have been lost to follow-up
are of highly questionable value, because it is impossible to determine the extent to which they are biased.
Some investigators,
such as Paterson and Tod8 recommend
that lost cases be counted as dead
“to avoid undesirable uncertainty.
although (it) may result in a slight bias against t,he ellicacy of
treatment.”
Other investigators,
such as Ryan and his colleagues9 omit lost cases from the analysis
of survival.
The latter procedure involves the assumption that from date of diagnosis the survival
experience of lost cases is similar to that of cases with complete follow-up.
We prefer the first of the several possible assumptions regarding lost cases, namely that subsequent
The complete
to date of last contact their survival is similar to that for cases with complete follow-up.
The
omission of lost cases from the computation
of survival rates discards available
information.
Registry
assumption that lost cases died immediately
after the date of last contact is cont.rary to fact.
experience with intensive field investigation
of lost cases, which resulted in recovery of some, indicates
that such patients often live for several years beyond the initial date of last contact.1D
Although cases withdrawn alive and cases lost to follow-up
are treated alike in the computations
(1) it is import,ant
described here, we distinguish between the two in the life table for reasons mentioned:
to be aware of the number of cases lost to follow-up because of their potential bias, and (2) other computational
methods may treat the two groups differently.
Volume
Number
8
6
number
LIFE
of deaths
TABLE
METHOD
by the effective
IN ANALYZING
number
exposed
d,._
70.5
SURVIVAL
to risk:
qx= 1,”
To express as a percentage,
Column
ternately
multiply
Proportion
8.
as the probability
is obtained
by 100.
Surviving
the In,terval &,).-This
of surviving
by subtracting
the proportion
px =
l‘o
express
as a percentage,
Column
9.
is obtained
multiply
is generally-
by cumulatively
Note that successive
survival
entries
cumulative
rates
Although
vals of one year after
of days,
intervals
the
first
intervals.
rates
the proportion
ps x
It
from unity:
Diagnosis
surviving
give the l-year,
II I).
a survival
of
rate.
It
each interval
:
The
2-year,
3-year,
successive
4-year,
cumulative
curve.
in Table
the date of diagnosis,
to End
survival
. . . px.*
(Table
illustrated
III werecarried
the life table
out in inter-
may be set up in terms
weeks, months, years, etc.
In fact, the life table may be organized in
of varying length.
For example, one might record experience
during
::ear
in monthly
This
type
intervals,
of presentation
during
occur
described
here may be used whatever
Gz\IN IN I.‘TILI%ING
Istandard
the first year.
EXPERIENCE
error
and
experience
The
method
thereafter
when
a large
of computing
in annual
proportion
survival
rates
the size of the intervals.
OF (‘OHORTS
provides
the
ma)- be desirable
of deaths
The
From
to as the cumulative
pz x
in drawing
the interval
to al-
rate.
qx.
in this column
the computations
during
Surviving
referred
p1 x
survival
are plotted
1 -
is referred
or the survival
by 100.
multiplying
px =
and S-year
dying
Proportion
Cumulative
(I;‘,).-This
Interval
the interval,
WITH
a measure
PARTIAL
of the
FOLLOW-UP
confidence
with
which
one
may interpret a statistical
result.
Thus, the standard error of the survival rate
indicates the extent to which the computed
rate may have been influenced by
sampling
error
variati0n.t
to and from
per cent
confidence
the same conditions
errors
on either
For example,
the computed
by adding
survival
and subtracting
rate,
one obtains
twice the standard
an approximate
9.5
interval.
This means that in repeated observations
under
the true survival rate will lie within a range of two standard
side of the computed
rate,
an average
of 95 times
in 100.
Thus, the computed
5-year survival rate for male patients
with localized
cancer of the kidney is 44 per cent.
The standard
error, computed
according
to the method explained iu the Appendix, is 6 per cent.
It is therefore likely that
*This f~xmula is based on the assumption that the various interval survival probabilities
are statistically independent.
tThe 126 cases of localized cancer of the kidney are in effect a sample from a population of male
patients with localized kidney cancer.
An illus,tration of sampling variation may be drawn from baseball.
A 0.250 hitter may, in four times
“at bat,” get one hit. Frequently,
though, he will ge6 no hits or two hits.
And not too infrequently
he will get three hits.
If we watch a game and see a player get two hits in four times “at bat,” it is
difticult for us to judge how good a hit,ter this player really is. We have to watch this player for many
games before we can get a reliable estimate of his batting average.
Survival rates are similar to batting averages in the sense that they are relative frequencies,
i.e..
the numerator is part of the denominator.
For each hit there must be at least one time “at bat.” and
for each death there must be at least, one case exposed to the risk of dying,
CUTLER
the true 5-year
survival
AND
EDEKEK
rate is not smaller
than
32 per cent
not larger
and
than
56 per cent.
Admittedly,
the computed
rate
does not yield
a veq.
firm estimate
true survival
rate, but we must bear in mind that it was based
126 patients
and only 9 of these
patients
were diagnosed
on
a
of the
series of onl?
a full 5 >.ears prior to
the date of study.
Furthermore,
whereas the survival rate based on all informaon these 126 patients provides at least a rough idea of the true rate
tion available
(one-third
to one-half),
discarding
follow-up
information
explained
in the discussion
The
would result
computing
in Table
III,
method
can be applied
fore used it to compute
larger
that
patient
the information
on the cohorts
in an extremely
unreliable
with
partial
estimate.
This
is
follows.
applied
to the total
to any selected
a series
series
portion
of S-year
of 126 cases,
of the group.
survival
rates
illustrated
We have thereon successively
based
cohorts.
A S-year survival rate was computed
for the 9 patients
diagnosed in 1946, all of whom had a S-J-ear exposure time.
\Ve then added the
1X patients diagnosed in 1947, who had a 4-year exposure time, and computed a
S-year
survival
This
procedure
was
utilized
and their
Table
rate
based
the
on
was continued
in
estimating
corresponding
available
until
the
the
S-year
standard
information
known
for these
experience
survival
27 patients.
of all 126 patients
rate.
The successive
rates
in the uppermost
section of
errors are shown
IV.
The
1946 cohort
a standard
of 9 cases yielded
The
error of 17 per cent.
very unreliable
estimate;
a 5-year
survival
large standard
the true rate is probabl>-
rate of 53 per cent,
error
tells
between
us
that
with
this is a
19 and X7 per cent,*
a very wide range.
The combined experience of the 1946 and 1947 cohorts yielded
rate of 46 per cent, with a standard error of 10 per cent.
Thus, the
a survival
addition
error
of information
on cases with 4 full years of exposure
reduced
17 per cent
to 10 per cent,
of 43 per cent.
of the available
information
from
addition
a relative
on cohort
decrease
1948
(3 full years
the standard
The
of exposure)
reduces the standard error to 7.5 per cent, etc.
The utilization
of all available
information
on
all the cohorts results in a standard error of 6.0 per cent.
Thus,
the standard
error
of the survival
per cent less than the standard
We
series
then
computed
of successively
groups
of patients:
breast
cancer,
rate
survival
enlarged
kidney
in women;
based
error based
rates
cohorts
cancer
breast
on
and
corresponding
of patients
with regional
cancer
with
of varying
size and with
in Fig. 1.
rate
than
based
on
varying
mortality
In ever)’ instance,
the combined
the corresponding
experience
standard
are the
95 per cent
confidence
53
*
Z(17).
errors
for
additional
in women;
IV).
\I’e did this ill order to
experience
for patient groups
The
results
are shown
error of the 5-year
1946 through
error for the 1946 cohort
limits:
is 6.5
in men; localized
involvement,
experience.
of cohorts
of four
involvement,
the standard
--__*These
information
standard
for each
regional
and cancer of the lip, both sexes combined
(Table
illustrate
the advantage
of utilizing all available
graphically
all available
on cases with a full 5 J-ears of exposure.
1951
by at least
survival
is smaller
one-third
\‘olume
8
Number 6
LIFE
TABLE
METHOD
IN ANALl-ZING
707
SURVIVAL
'I‘ABLE IV. FIVE-YEAR SUR\-IVAL KATES AXD -THEIR STANDARD
ERRORS FOR FIVE GROWS OF
CANCER PATIENTS, SHOWING THE ~?DuCTIOK IN STANDARD ERROR \%‘ITHINCREASE IN COHORT SIZE
COHORT
NUMBEROF
CASES
DIAGNOSED
5-YEAR SURVIVAL
RATE
STANDARD ERROR
OF 5-YEAR
SURVIVAL RATE
PER CENT REDUCTION INSTANDARD
ERROROF~-YEAR
SURVIVAL RATE
Kidney, localized
1946
1946m-1947
1946.-1948
1946~-1949
1946m.1950
1946-.1951
0.53
0.46
0.43
0.43
0.45
0.44
2:
48
82
101
126
h
1946
1946-.1947
1946~-1948
1946m-1949
1946~-1950
1946m.1951
idney,
regional
0.18
0.33
0.28
0.25
0.23
0.24
11
23
30
39
0.171
0.098
0.075
0.064
0.063
0.060
0.116
0.101
0.091
0.071
0.069
0.070
13
22
36
41
40
Breast, localized
1946
1946m.1947
1946- 1948
1946~-1949
1946&l 950
1946~-1951
0.64
0.64
0.64
0.64
0.65
0.65
225
454
695
963
1,227
1,490
Bread,
1946
1946m.1947
1946- 1948
1946- 1949
1946--1950
1946m.1951
208
443
708
0.033
0.025
0.023
0.022
0.022
0.021
24
30
33
33
36
regional
0.42
0.38
0.39
0.035
0.025
0.021
0.020
0 020
0.020
29
40
43
43
43
Lip
1946
1946.-1947
1946~-1948
1946--l 949
1946--1950
1946.-1951
61
109
i69
224
283
332
0.71
0.65
0.68
0.68
0.68
0.67
0.060
0.048
0.042
0.040
0.040
0.039
20
30
33
33
35
J. Chron. Dis.
December, 1958
708
The
advantage
of utilizing
>rears of exposure
informatiol1
was greater
This is because:
(1) particularly
groups.
were diagnosed in the first >.ear (1946),
in each of the subsequent
of follow-up
(0.40)
years;
was much
on patient
for localized
kidney
cohorts
cancer
for
few cases (9) of localized
compared
with
and (2) the mortalit>,
larger
lvith less than
than
than
of 23 cases
rate during
in succeeding
l’ears
5
other
kitlne~~ ca1lcer
average
an
the
the
(annual
first >.ear
average
of
Thus, because of this mortality
pattern,
the information
on
surviv:il
0.07).
during the first year after diagnosis contributed
ver). substantially
to the information on survival over a S-year period.
Five of the 6 annual cohorts
(1946-1950)
contributed
complete information
In general, the relative
agnosis.
cohorts
with
relative
increase
of
added
the
mortality
partial
survival
and
lung or stomach
that
little
results.
tends
cancer,
is gained
and
diseases,
to taper
(3)
the
may
data
be inadequate,
pattern
may
at a later
For
relatively
some
before
be unnecessary
to evaluate the effects
important
information.
for only 5 years
(1) the
completeness
magnitude
of
the
such
within
evaluating
as
one year
therapeutic
to wait until a .5-year sur-
of therapy.
A I-year,
In other
because
high shortI>
diseases,
be so depleted
vival rate can be computed
or 3-year rate may provide
mortalit)-
relative
is often
more than one year
it may frequently
with:
intervals.
mortality
group
directi),
(2) the relative
off thereafter.
the patient
by waiting
Therefore,
will vary
the first few follow-up
and in other
diagnosis
information
size of the cohort*;
information;
during
In cancer,
after
follow-up
in the initial
rates
regarding survival during the first year after digain in utiliziug survival information
on patient
instances,
of significant
2-year,
survival
changes
in the
time.”
DISCUSSION
A category
tionally
chosen
all available
of patients
information
of localized
kidney
cause
it is frequently
groups
in combination
of patients
desirable
with radiation,
Similarly,
with
of survival
if survival
breast
find that
therapy
126 cases
This
was done be-
experience
cancer
of relative11
in evaluating
the
treated
by surgery
in any one year
the number
is small.
As an illustration,
in 1946 were treated
is to be evaluated
was intenof utilizing
rates-only
in 6 years.
the survival
localized
in Connecticut
per year
the advantage
if we were interested
we would
the combined
of the 225 cases diagnosed
therapy.
to describe
For example,
of patients
receiving
few new cases
to illustrate
in men were diagnosed
of patients.
experience
relatively
example
for the computation
cancer
small
survival
with
as the principal
onlv 25
b,. the combined
for a specific
subgroup
with
respect to age, the number of cases per year would usually be small.
Therefore,
in order to increase the reliability
of survival rates computed for various patient
groups of clinical
interest,
It is of paramount
computing
trial.
survival
For example,
in a clinical
___--
trial.
*See the Appendix
rates
it is important
importance
if the rates
a 3-year
survival
It may be possible
for a discussion
to utilize all available
to use all available
information.
survival
information
are going to be used as criteria
rate may
have been selected
to determine
of effective
sample size.
in
in a clinical
as a criterion
which of the several
treatments
VoIume
x
Kumber
709
1.1~~ TABLE METHOD IN ANALYZING SURVIVAL
6
KIDNEY
KIDNEY,LOCALIZED
,REGIONAL
1
1.00 [
‘.O(’ 1
.90 -
90
F
.eo -
2
,712 -
2
70
t
&I3 ,,
2
.60
-
-I
2
2
.5’l
w
r-
1
-
2
40-
:
w
>
A
.SD
-
20
-
80
-
&
.50
2
40-T
5
w
>
30-
In
20-(,
IO
J
1946
l946-
1946-
,946-
,946-
,946-
1947
1946
1949
1950
1951
BREAST,
-
”
1946
LOCALIZED
l946-
l946-
1946-
l946-
lS46-
1947
1946
1949
1950
1951
BREAST
,REGIONAL
1.00 I
.90
w
-00
5
lx
.70 i
4’
>
.60
2
50-
z
oz
40-
g
3o
1;1 .20
0-
-
-
I
0’
1946
l946-
l946-
1946-
l946-
l946-
1947
1948
1949
1950
1951
l946-
1946-
l946-
l946-
l946-
1946-
1947
1946
1949
1950
1951
LIP
.J
I I I:
i
0'
1946
1946-
lS46-
1946-
,S46-
,946-
1947
1948
1949
1950
195,
Pig. I.-Decrease in the 9-5 per cent confidence interval for the B-year survival rate as cases with
less than 5 years’ exposure to the risk of dying we added. (The 95 per cent confidence interval is ohtained by adding -2 standard errors to the survival rate.)
Source: Table Iv.
710
CGTLER
being tested yields the best survival
death
or for a full 3 years.
the earliest
possible
survival
including
data
survival
information
to survive
of reduction
patients
the life table
for cancer
had the opportunity
terms
before all patients
inferior
treatments
have
been
followed
would be discontinued
to
at
point.
We have illustrated
S-year
Thus,
J. Chron. Uis.
December, 1958
AND EDERER
in standard
in this paper,
method
patients,
for computing
emphasizing
on cases which
survival
the advantage
entered
rates
gained
with
b)
the series too late to have
a full 5 years.
error
the reduction
The advantage
is measured
in
of the survival rate.
For the five series of
in standard
error
ranged
from one-third
to
two-thirds.
REFERENCES
1.
2.
Manual for Cancer Programs, Bull. &4m. Coil. Surgeons 38:149, 1953.
Pearl, I~.: Introduction to Medical Biometry and Statistics, Philadelphia and London,
1
j:
New York, 1956, Oxford liniversity
Press.
Berkson, J., and Gage, R. P.: Calculation of Survival Rates for Cancer, Proc. Staff Meet.,
Mayo Clin. 25:270, 1950.
Merrell, M., and Shulman, L. E.: Determination
of Prognosis in Chronic Disease, Illustrated
by Systemic Lupus Erythematosus,
J. CHRON. Drs. 1:12, 1955.
Griswold, M. H., Wilder, C. S., Cutler, S. J., and Pollack, E. S.:
Cancer in Connecticut,
1935-1951, Hartford,
1955, Connecticut
State Department
of Health, pp. 112-113.
Wilder, C. S.:
Estimated
Cancer Survival Rates Confirmed, Connecticut
Health Bulletin
70:217. 1956.
Paterson, 1C.l and Tod, hl. C.:
Presentation
of the Results of Cancer Treatment,
Brit.
J. Radlol. 23:146, 1950.
Ryan, A. J., et al.: Breast Cancer in Connecticut,
1935-1953, J..4.M.ii.
167:298, 1958.
Griswold, M. H.: Personal communication.
Cutler, S. J., Griswold, M. H., and Eisenberg, H.:
An Interpretation
of Sure-it-al Rates:
Cancer of the Breast, J. Nat. Cancer Inst. 19:1107, 1957.
Greenwood, M.:
Reports on Public Health and Medical Subjects,
No. 33, .L\ppcndis 1,
The “Errors of Sampling” of the Survivorship Tables, London, 1926, H. M. Stationer)
Office.
5.
6.
7.
8.
9.
:::
12.
1923, W. B. Saunders Company.
Hill, A. B.: Principles of Medical Statistics.
APPENDIX
Contputing
the S’tandnud
Error
of the 5-Ymv
Snrvhd
Kate.-The
method
for con,puting
the standard
error of the S-year
survival
rate was developed
by Greellwood
(see ref. 12) and is also described by Merrell and Shulman (see ref. 5). The formula is
Sa =
5
9
rq
J
qx
-__ x = 1 l’,
where sg is the standard
k-year survival rate is
d, = ”
i
-__
(II
\ If1 -
error of the S-year survival
(12
__
dl +
rate.
;-
+
d?
In general,
.
. + A__,
1’; the standard
d;
error of the
Columns 10 and 11 of Appendix Table I show how the calculation of the standard error of
the S-year survival rate is carried out as a continuation
of the computation of the sur\-ival rate.*
The first 9 columns are a replica of Table III. (1) Subtract d, from I’, for each line (Column 10);
(2) divide g, by I’, - d, for each line (Column 11); (3) total the entries in the first 5 lines of Col__--_
*The standard error computed in this illustratkxl is, itself, only ai11estimate of the true standard wror.
For PXAnd, since it is based on relatively small numbers of cases, it is not a very reliable estimate.
ample, had there been, due to sampling variation, one death in the last interval, rather than rmne, the
computed standard error would be 0.0216 rather than 0.0187.
-
1
OF
x
126
60
38
21
10
4
p’
INTERVAL
NING
ALIVE AT
BEGIN-
;
-
47
5
TER\‘AL
IN-
-
it
2
-
(4)
Ur
TERVAL
ING IN-
UP DUR-
TO
DURING
LOST
FOLLOW-
DIED
WITH-
J
15
11
15
7
6
4
(5)
WX
DURIN(
OF DYING
COL. 5)
(6)
I’,
116.5
51.5
30.5
16.5
7.0
w
NUMBEI R
TO THE
-
2 - yi COL. 12
RISK
EXPOSED
F:FFECTI\‘E
; ( COL.
I NTERVP LL
-
ALIVE
DRAWh
-
0.40
0.10
0.07
0.12
0.00
3t
COL. 6)
(7)
qx
(COL.
DYING
1PROPORTION
0.60
0.90
0.93
0.88
1.00
PX
(8)
(1- car..
7)
SURVIVING
PROPORTION
-
THROUGH
. .x
0.60
0.54
0.50
0.44
0.44*
(PI x p2 x .
‘r”)
x
Px)
END
DIAG-
PROPORTIOr J
FROM
OF INTERVAL
NOSIS
SURt’IVING
C UMULATIVE
COMPUTATION OF THE ~-YEAR SURVIVAL RATE AND ITS STANDARD ERROR.
(Data from Table III)
3)
69.5
46.5
28.5
14.5
7.0
_____-
(10)
I’,-d,
COL.
(COL. 6-
0.0187t
0.0058
0.0022
0.0024
0.0083
0.0000
(11)
10)
7f
q x/I .‘-dx
COL.
(COL.
*Five-year survival rate.
tThis is the sum of the five entries in Column 11. The square root of this number, when multiplied by the 5-year survival rate, yields the standard error
of the 5-year survival rate: s = (0.44) 0.0187 = (0.44) (0.37) = 0.060.
$See footnote *, Table III.
1-2
2-3
3-4
4-S
5-61
o-1
(1)
XTOX+
YEARS
AFTER
DIAGNOSIS
-
;~PPEKDIX TABLE I.
712
CUTLER
AND
J. Chron. Dis.
December,
1958
EDERER
U”lll
11: 0.0187; (4) take the square root of this number:
40.0187
= 0.137; (5) multiply the
result by Ps: 0.137 X 0.44 = 0.060. This is the standard error of the S-year survival rate.
The standard error of survival rates for end-points other than 5 years is computed similarly.
For example, to compute the standard error of the 3.year survival rate, the first three entries
in Column 11 must be totaled, the square root taken, and multiplied by P,.
Effective Sample S&.-The
concept, effectiue sample size, provides another way of assessing
the benefit of including in the life table cases with partial survival information.
The concept
relates to the fact that the reliability
of a statistical
result depends on the size of the sample,
i.e., the number of cases observed.
For example, the standard error of a survival rate, P, when
all cases have been followed until death or for the required time interval (i.e., no losses from
observation or withdrawals alive prior to the cut-off date) is given by the binomial formula
In formula
where 1, is the sample size, i.e., the initial number of cases.
is inversely proportional to the square root of the sample size.
‘r.lI3LE
;IPPENDIS
126
1,4%
Kidney, localized
Kidney, regional
Breast, localized
Breast, regional
Lip
*Since
5 years
the
cut-off
date
1,531
332
was
Dec.
31,
rligihla
for
1951,
cases
diagnosed
the standard
error
I I
SAMPLE
1946-1951
COHORT*
(l),
SIZE
1946
EFFECTIVE
SAMPLE
SIZE
COHORTj
9
11
225
208
61
68
37
516
595
14.5
in 1947 or later
were
eligible
for
less
Lhan
of observation.
tActua1
number
of cases
5 years
of observation.
Let us consider the 1946-1951 localized kidney cancer cohort (.\ppendix Table I), for which
the survival rate is 0.44, and its standard error, 0.060.
Of the initial 126 cases in this cohort,
IVe now ask how
a substantial
number were withdrawn alive less than 5 years after diagnosis.
large a cohort, with a S-year survival rate of 0.44 and with all cases followed to death or for a
full 5 years, would have a standard error equal to 0.060.
To answer this question, we solve
equation (1) for I,, placing a circumflex over the I, to indicate that this is a hypothetical
value:
Substituting
P = 0.44 and s = 0.060, we obtain
Had we started with
The result, 68, is the eyective sample size, which we interpret as follows.
about 68 cases (instead of 126) and followed them all until death or survival for 5 years and found
that 44 per cent survived 5 years, then the standard error would have been equal to that we
actually obtained in our cohort of 126 cases.
Thus, the survival rate we obtained is as reliable
as one based on 68 cases. This is in sharp contrast to 9 cases which were eligible for 5 years of
observation.
These three values are compared for the five cancer groups discussed in the text.
In each instance, the effective sample size based on the 1946-1951 cohort is substantially
larger
than the number of cases eligible for 5 years of observation (1946 cohort).