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• CONFIDENTIAL
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Estlmution o~ loss o~ liTe in r~lutlon to a dlse~se or to
factor causln~ it; wit], particL~lar reference to s~doking
Author : P.N. Lye
Date : 1.2.7B
le
Introduction
Following the publication of the latest Royal College of Physicians
Report on Smoking and I{ealth (R.C.P., 1977), considerable attention was
given in %he press, both in the United Kibgdom and abroad, to the claim
contained in it that "on average the time by ~hich a habituzl cigarette
smoker's li~e is shortened is about 5~ minutes for each cigarette smoked -
which is not much less tba/1 the time he spends smoking it." This claim
was first made by Diehl (1989), who based his ealculatlo|~s on tables
provided by Hammond (1969) giving the loss of llfe expectancy o£ U.S.
men of various ages smoking different numbers of cigarettes.
Such a claim is only one ~mon~ s number of ways in which ~e Loss cf
life due to smokin~ czn be q~zntified. The aim of this paper is to look
at a number of methods of general application in estimating loss o2 li~
in relation to a disease or to a factor causing it, and to apply the ones
thought most useful to obtain estimates relevant to the population of
England and Wales of the loss of life due to smoking and to diseases
associated with it.
This paper starts, in Section 2, by looking at the theory bel, ind
estimation of loss of llfe in the experimental situation. The concept
of the life table is introduced sad the advantages and disadvEnt~Ees of
a number of alternative statlstlcs describing differences in survival
between exposed and non-exposed groups are discussed. In practice, human
data on the relationship between smoking mld mortality is coll~cted
ob~ervstionally rather than e~orimentslly. Th~ problems involved ~n
collecting relevant data are discussed in Sectien 3 along with the
assumptions required in extrapolating results obzalned to the current
smoker in England ~ud ~a!es. Following discussion of what d~ta actually
are available (Section 4), calculations of the loss of llfe due to
smoking and some smckinK-associaT.ed diseases are made in Section 5.
The conclusions of the paper are theu discussed in S~ctlon 6 z~d
summarized in Section 7.
C-~
O

2.
-2-
2
Estimation of loss of life in the experimental situation
2.1 The two ~roup experiment
In an ideal world, to determine the loss of life related to a
particular factor, one would take a population and randomly allocate
it into two groups. One group would then be exposed to the factor
of interest while the other group would not. The mortality of each
group would then be monitored for the rest of its life by noting the
time at which each member died. Statistics describing the difference
between the mortality experience of the two groups would then be
computed and could be ~aken to he relevant to the effect the factor
would have on the loss of life of other people typical of the origiual
population.
In the real'world, such an experimental epproach is only usually
possible with animals and inferences have to be made about the effects
of a factor from other types of data. The problems involved and
ass~ptions required to make such inferences are discussed later (Section
S); for the moment our interest is centred on what are, and are not,
useful statistics to describe the effect of a factor on mortality and
for this it is convenient to stay with our ideal situation. To further
simplify discussion of method we assume, firstly, that exposure to the
factor of interest is at a regular rate throughout lifetime, and,
secondly, that the original population are all of the same age at the
beginning of the experiment.
2.2 Functions describin~ survival
Survival data are data of times to death. The distribution of
survival times can be characterized by three equivalent functions
(Gross and Clark, 1975):
a) Death Density Function f(t)
f(t)dt is the probability that a person will die in the time
Interval (t, t + dr). If we assume that the experiment starts
at time zero it follows that
f(t) is n0n-negative.
0
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b)
c)
Survivorship Function S(t)
S(t) is the probability that a person will survive tO at least
time t (t>O). It follows that
S(t) = f(T)dT
t
and that f(t) = -S'(t)
S(t) is a non-negative decreasing function starting from I at
time zero.
Hazard Function k(t)
~(t)dt is the probability that a person will die in the time
interval (t, t + dr) ~iven he has survive~ to time t. This
ftmotion, which is also known as the failure rate or the Fo~ ~ ~,c..l,~
tneldence rate, satisfies the condition
~(t) = f(t)/s(t)
A(t) is non-negatlve but may be increasing (such as in the
Weibull distribution k(t) = btk where b and k' are constants),
•
constant (such as in the exponential distribution ~(t) = a)
or have other more erratic shapes.
....°
2.3 Cohort life-tab le
For absolute precision one observes, as mentioned above, the actual
time at which deaths occur. In practice, especially for human popu-
lations, it is usually convenient to group the data into certain
defined time intervals rather than actual points of time. We shall
assume, for our purposes that the data we have for each group consists
of information relevant to n time intervals (i ~ I, .... n) as follows:
ti Age of population at beginning of interval i
Ai "Number alive at beginning of interval i
Di Number dying from all causes in interval i
Li Number dying from a particular cause o~ interest in interval i
Yi "~idpolnt" of interval'i.
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Such information is known as a lifo-table. According to the
nomenclature of Gross and Clark (1975), the particular type o£ llfe-
table we are deallnE with here is a cohort llfe-table, a "cohort"
being a group of individuals born at about the same time. Later on"
(Section 3.5), we consider other forms o~ life table.
I
We note that, because we have assumed all people are followed until
death, Ai - Di = Ai + i for all i and that tI = 0 and An = Dn. Yi' the
"midpoint" of interval i, can usually, if the interval is suffleiently
small, be taken as the actual midpoint of the age-lnterval considered.
If more accurate answers are required the actual average age at which
deaths in the interval occur should be substituted. Pi.= Ai/A1 estimates
S(ti) the survlvorshlp function at age ti.
Description of the mortality of a population by life-tables has a
lone history dating back to the pioneer work of Halley (1693 - sic)
Hortality indicators in general have been reviewed in the literature
on a number of occaslons (e.E. Woolsey (1943), Haenszel (lg50), Logan
and Benjamin (1953), Eitagawa (1966), Benjamin and Haycocks (1970),
Romeder and McWhlnnle (1977)). In this, and the sectlons that follow,
we discuss the merlts of a number of statistics that have been suggested
to summarize the main features both of the information contained in a
llfe-table and of the differences between the two llfe-tables being
compared. We start by looking at some of the more simple statistics
that have been employed in the past.
2.4 Measures of the number dyin~
One obvious type of statistic to look at is the proportion d~In~.
Clearly if one is looking at total mortality then the proportion
dying over the whole experiment will be 100% and will offer no dis-
crimination between the groups. However it can be useful to compare
the proportion dying between two particular time points tj and tk,
especially if the time period represent~ in some sense, "premature"
deaths. This statistic, QI' i~ defined by
ql
If one is interested in mortality from a particular cause then
the total proportion dying from this cause
s(tk) Aj - Ak ~ ~ ~.~, ~ ,~
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can give some indication of the magnitude of the problem caused by the
disease. It is; however, limited in its usefulness by tile fact that
it gives no information as to .when the deaths occur.
There are two other types of indicator which have the same objection L.~,
that they concentrate on numbers of deaths and ignore when deaths occur.
The first of these arJstandardlsed death rates. They can be calculated
by two methods, the direct and the indirect method. In the direct method,
the rate, QS' is a weighted sum of the individual crude death rates (Ri),
with the weights representing the populations in each age-group in some
standard population. Thus, if wi are the weights, Q3 is defined by
n
~-~ ","%e~.- Q3 = Z wiRi
i=l
In the indirect method, the number of deaths from the cause of
interest observed (Oi) in an interval is compared with that expected (Ei)
if some standard death rates (Ri ) from the cause had existed. The sum
of deaths observed from all intervals is divided by the tots/ expected
to give a 'Standardized Mortality Ratio', Q4" Q4 is defined by
Q4 = --
n n "~
z oi Li
i=l =~-'i 1
r z AiRi
i=i i=l ~- . ~+ ~ ~-~+C~~
,~~
A problem with both these indicators, as was pointed out by
Yerushalmy (1951), is that they are markedly affected by relatively
small differences in mortality in older ages when deaths ~re frequent
and little affected by large proportional differences in early years
which cause great loss of life.
The final type of statistic, quite popular in quantifying the effect
of smoking (e.g.R.C.P. (1971)) is the number of deaths associated with
the factor. To calculate this statistic, QS' the number of deaths that
actually occur in an interval in the exposed group are compared with the
number that would have occurred had the exposed group had the same number ~_~
at risk in the interval but the death rates of the non-exposed group. ID C~
o.~her words ---~

: J2
0 . d
q-
vii
Q5 = E
I=1 2i A2i" ~/
where the first subscript refers to the group (I = non-exposed, 2 =
exposed) and the second to the time interval. Apart from the £act
that Q5 gives no indication at all of life-shortening, the main defect-
with this statistic is that it carries with it the implicatlon that,
had the factor not existed, this number of deaths would, In some sense,
have been avoided. Clearly everyone dies once, so what is the real
implication? As normally used, the number of deaths associated with a
• factor is a~tached to a time scale, e.g. "50,000 deaths a year are asso-
ciated with smoking" but what does this mean? As calculated, if the time
intervals were years, and if in £act this calculation was carried out on
population data rather than our idealized cohort data, the statistic would be
a reasonable estimate of the numbers of deaths that would not have occurred
in the year following a universal giving up of smoking, assumin~ (and
there is evidence to show that for some diseases, e.g. lung cancer (Doll
(1971)), this is not the case) that on giving up smokers age-speclfic
death rates reverted at once to those of never smokers. However, it would
only be accurate for the first year and would be @ecreasingly inaccurate
for subsequent years. The reason being, of course, that in later years,
due to the lower mortality immediately following mass givlng-up (on the
assumption quoted), there would be more survivors at higher ages snd
consequently more deaths than the current age-distribution would suggest.
As shown in Appendix A, it can be estimated (under certain further
assumptions) that, on m-~ss giving up of smoking at the end of 1975, 80,000
less male deaths in England and Wales would have occurred the first yzar
afterwards than had no giving-up occurred. However this number would be
half as much by 1988 and down to 12,000 by the year 2,000.
2.5 Life expectation and average ate at death~ "
Another simple statistic that has been used to assess mortality is
average a~e at death. Average age at death of the whole population from
all causes is identical to expectation of life at birth; expectation of
life at age t, Q6' being defined by the expression
and measuring the average number of additional years people alive at ~>
age t live on average, 6~ [~9~00 [

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h_
il
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-7-
Average age at death of those people dying only of the cause of
interest, Q7' is an alternative statistic which has been used by some
workers. It is defined by
n n
Q7 =i I~ (LiYi)/iZiLi=
Though it can be of value in some circumstances to compare such
an average for one cause of death with a slmilarly computed average for
another cause, it only measures when the disease occurs and no~ how
many people die of it. Furthermore it is not a ve~j useful statistic
to measure llfe shortening. It might be thought that, a cause of
death resulting in an average age of death x years less than the average
age of death from all causes is in some sense an Indication that the
cause takes x years off life. That this reasoning is incorrect can be
seen if one considers a cause of death with an average age greater than
the expectation of llfe at birth. On the implied llne of reasoning this
cause adds years onto llfe, which is, of course, nonsense.
Average age at death can also be a very misleadlng statistic to use
when comparing groups exposed to different levels of a factor of interest.
If, for example, the cause of death of interest is the only one affected
by a factor and is relatively rare, and if the effect of the factor is
simply to multiply the age-specific incidence rate from the cause by an
age-independent constant, it can be easily seen that, though the pro-
portion of cases of the cause of death in the group more exposed to the
factor will be greater than in the group less exposed, the distribution
of times of death from the cause, and hence the average age at death from
the cause, will be virtually identical in the two groups. If, further-
more, the average ages at death from the cause are compared in cross-
sectional data. where the age distribution o~ the more and less exposed
groups are different, It is not surprising that fairly meaningless results
can be obtained. For example, Passey (1962) studied successive hospital
lung cancer patients and observed that the average age of death of the
heavy smokers did not differ £rom that of the light smokers. He-concluded
that there was an anomaly to be explained, but as Pike mld Doll (1965)
pointed out, fol]owing out general llne of argument above,if was only
the poor choice of statistic that had led to the apparent anomaly.
O
CXD
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--8--
2.6 Measures of los.q of llfe expectation
In the preceding sections it should have become clear that any
statistic not taking into account both the frequency and the time of
occurrence of death Is not an adequate description of the effect a
factor has on loss of llfe. A better approach, and one that has been
tried by various workers over the last 30 years, is to quantify the
effect in terms of numbers of years lost. Some of these attempts have
tried to take into account to at least some extent the fact that the
loss of years of life at young ages may be of more importance ~o an
Individus/~ or to a societyj than the loss of a similar number of
years in old age.
Thus, a number of workers, e.g. Murray ~d Axtell (1974), Romeder
and McWhinnle (!977), have estimated the number of years of "active
life" lost. Though the critical age differs (usu~11y between 60 and
70)4 the same essential method of calculatlou has been used; it has
e
been assumed that any person dying before the critical age has lost a
number of active years equal to the difference between the critical
age and the yean of death.
Other workers have used deaths occurring at all ages nnd have
counted zears lost to llfe expectancy, e.g. Dempsey (1947) who used
llfe expectancy at birth and Dickinson and Walker (1948) who used life
expectancy at age of death.
Both these types of measure have objections. The years of "active
llfe" lost measures, as described above, are over-estlmates as it is
clear that some of those dying early would still not have reached the
critical age had they not died when they did. Hakulinen ~nd Teppo
(1976) got round this objection by using adjusted life-table procedures
(see Section 2.10) to estimate the suhsequentsurvival pattern of the s
"reincarnated" population, i.e. the survival of those who would not have
died had the cause of death of interest been removed. An alternative
method would be to compare the years of "active life" lost in the
exposed and non-exposed groups. However these measures all have the
disadvantage that an essentially arbitrary choice of erltical age has
to be made and that information on people of age greater than this Is
ignored.
The loss of years to life expectancy measares mentioned above have
the disadvantage that they do not take account of the fact that, had
• the cause of death been removed, the life expectancy itself would have
CD
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