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-\ / ° . • CONFIDENTIAL %. 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 0 Cr~ t~ ~Jn

"7"' --3-- • % 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. CD CZ) C~

--4-- • • 0 L i 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 ~ ~ ~.~, ~ ,~ C~

L 1 L-- n 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 [

J i" I • L-- I " t f °! h_ il j: I! " I i :. ~.. -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 Cr~ t~4

--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 CY~

-9- :I b i t d" . ~" .,,' ~%-J~ 2.7 Recommended methods for comparison of two life-tables • ~,%~ Probably the most informative method of quantifying the loss of life / due to a factor is to compute the difference in life ezpectation of the i ! , i "! [. [ L 1 I t__ been altered. Furthermore, the resulting statistic eau be rather difficult to interpret especially if it is calculated on a per decedents rather than a per head of population nt risk of basis. ~hus, as we shall show later, the average years lost to llfe expectancy of lung oancer decedents ~or some populations is in fact less than the average years lost for all decedents. Is lung cancer a Eood th~ng • therefore? It can also be shown that (considering cross-sectional data rather than life-table type data), even had smoking no effect on mortality at all, the average loss of llfe expectancy of smokers who die in a given time period would be greater than ~hat of non-smokers who die in the same period, simply because smokers are younger than non-smokers. I~ is clear that such statistics are liable to misuse by the uninitiated. exposed and non-exposed groups from the start of the experiment. If I~ is desired to place a different value, V(t), on life at different ages • then one :could calculate the difference between the two groups in their "expected value of life", Qe' where Q8 is defined by Q8 = ~T f(T)Z(T)dT where Z(T) the total value of llfe up to time T is given by • -° An alternative good method is to compare the proportions dying over some special age range of interest (~i). This method has, for example, been used by the R.C.P. (19~7) to quantlfy the effect of smokinE. They pointed out that, in the study of British Doctors (see • Section 4.2) a male smoker of 25 cigarettes or more da.%ly aged 35 had a 40~ chance of dying by age 65 whereas ~ over the same period a non- smoker only had a 15~ chance. ' I In particular circumstances, other comparisons of life-tables cau be extremely informative. For example, if the'effect of the factor is simply to transform the death density function so tna~ either f2(~) = fl(t + a) or f2(t) = fl(bt) CD CD t~4 Cr~

..... -I0- holds for all t (where a and b are constants and the subscripts 2 m,d I refer to exposed mad non-exposed respectively) tlten one could m::.ke. statements such as "the probability of an exposed person dying is the same as that of a non-exposed person a years older'! or "exposed people have a probability of dying the s~nme as non-exposed-people b times as old". Alternatively, if the factor multiplies the hazard function by a constnnt c, so that ~2(t) = C~l(t) one might usefully make statements such as "if you are exposed you have c times the chance of dying at any instant as a non-exposed person of the same age." There is quite a lot of evidence that, for particular diseases the effect of some factors can be a multiplication of the hazard function (e.g. Peto and Lee (1873)) but such a simple relation- ship seems unlikely to hold for total death rates ~or more than at most a very few factors. 2.8 Quant!f/in~ loss of life expectation in terms of dose app!i~d None of the statistics which we have discussed so far to quantify the difference between llfe-tables representing exposed and not exposed groups have expressed the differences in terms of the dose applied to the exposed group. We now look at some that do. If all we wish to say is something along the lines "exposure to X units a day for life results in a loss of llfe expectation of Y years" then, of course, the methods of the last section are directly applicable. If we wish to generalize thls statement so that we can make inferences sbodt what would happen if lifetime exposure was at some other daily level we would need to have information on additional groups exposed at different dose levels in order to build up a dose-response relationship, but no new statistical treatment of this llfe-table would be needed. It is not so straightforward, however, if from the results of an exposed group given X units a day for life we wish to infer tile effect on life shortening of a single exposure to X. Such an inference has to be made to arrive at Diehl's (1969) claim "On average the time by which a habitual cigarette smoker's li~e if shortened is about 5~ minutes fop each cigarette smoked." <ZD t>4

• ° -11- The simple method to do ~hls calculatlon, and the one usod by Diehl (1969) is to compute the averngc torn1 exposure during the llfe of the exposed group and to divide this into the estimated loss of life expectation. But is this correct? Normally, in working out such an average, one computes the llfe shortenin~ per exposure for each Indlvidual and then averages the answers over the individuals; in other words if LSj is the life shortening for individual J and Ej his exposure up to ,time of death one would calculate N 1 Q9 = ~ Z (LSj/Ej) J--1 where N is the number of individuals, and not as Diehl did, = LS / Z Ej Q10 =1 J=l As is well known, unless LSj/Ej is constant for every individual, these expressions will differ. The reason why Q9 has not been calculated, of course, is that it is not possible, on an individual basis to measure llfe shortening~ Another worrying thinE about the whole concept is the fact that one might be averaging effects which are very different at different times of life. In fact, as we show below, the assumption that each exposure has an equal effect, implies a particular relationship between the death density functions of the exposed and non-exposed'groups. The death density function, f(t), in the non-exposed group can be seen as the proportion of people who have "tickets to die" at time t. (Or more precisely f(t)dt represents the propo~ion with tickets to die in (t, t + dr)). Suppose that every unit of time that elapses the exposed group take R elf their criglnal life expectation; in other word§ they take R off the value of their ticket. It follows that, at time T, the f(T + RT) people who originally had tickets to die at time T + RT then hold tickets to die at time T. It follows that the survivors at time T are only those who originally had tickets to die at least at time T ÷ RT. The survivorship function in the exposed group at time T, $2(~), is therefore given by .° C7~

.. -12- S2(T) =I~ ~(u)du = SI(T(1 + R)) (I+R) In other words, at any time the surviving proportion in the exposed group is the same as the surviving proportion in the non-exposed group at a time a factor of R + 1 as great. It follqws that, unless such a con- dition holds at least approximately for some R, expression of the effect of a single exposure as an average tends to be rather uninformative, 2.9 The one group .experiment On some occasions information is available on the survival of a group exposed to a factor or subject to a cause of death, but no such data is available on a comparable non-exposed control group. Can inferences then be "made about the effect the factor or cause of death has had on loss on life from this "one group experiment"? Provided the cause of death of each member of the group is recorded, then under certain assumptions which we shall discuss in the next section it is possible to estimate the loss of llfe caused by a certain cause of death. Inferences cannot in general be made about the effect of a factor unless, from other knowledge, it is possible to label certain deaths as having been caused by the ~actor. 2.10 Adjusted life-tables When information on a single group only is available the re- commended procedure is to calculate the life-table that would have existed had the particular deaths (from ~he cause of interest or classed as due to the factor) not occurred. This li~e-table is known as the "adjusted life-table" and inferences about loss of llfe can then be made by comparison of the actual and" adjusted file-tables in exactly the same way as Lave been described for the comparison of life-tables for the exposed and non-exposed groups in the two group experiment. flow is the adjusted life-table calculated? The'most common approach to this problem is to assume that those people who die of the cause of death of interest would have had the same chance of dying from the other causes of death had the cause of interest not existed. Under this assumpT- tion of independence the adjusted life-table is estimated as follows. Consider the ith time interval. Let the lenzth of this i,tterval be Ti and let us assume it to be small enough for the fo__rce of mqr__ta_li__%..y from causes other than the one of ~n~rest to be taken as co,~stant (-- =i), and that f1"om the cause el Interes~ also to be taken as cons~.~tlt (= Bi). _~_~ l~~.

-13- Now the total survival from all causes exp (- (=i + 8i) Ti) and the relative mortality =ilSi is estimated by (Ai - Di)/Ai by (Di - Li)/Li (Li are the deaths from the cause(s) to be adjusted for) It follows that DI) Di - LI exp (-=i T i) = - ~i Di and Ii D~ilLi exp (-8i Ti) = - exp (-=i Ti) is the proportion that would survive the interval if the cause oZ death of interest had not existed. The survivorship function of the adjusted life-table is thus built up by starting with 100% sur- vivors (i.e. S(O) = i) at the beginning of the first interval and successively multiplying S by the estimate of exp (-=i Ti) in each consecutive interval. The formula for exp (-=i Ti) has been derived previously by Chiang (1961). An alternative formula ~,~ -~ --~'~- Di - Li ~tt~i ~.t C.-C~ , • exp !-=i Ti) = 1 Ai - Li/9- .~j~ ~ ~.~k~ . ~_~l ~ ~.~. has been attributed to Berkson by Schwartz and Lazar (1961). As Lee ~v~- ~\ (1970) shows these two formulae give virtually identical answers provided the proportions dying in the interval are reasonably small. Although most work done in this area makes The.ass__wnpti_on we made at the beginning of the previous sectioo, it is clear that it is only likely to hold under special circumstances. It may well be, for example, that the sort of person who dies early from one Cause may be generally 'beak" and, were this cause to be removed, he would in fact have a greater chance than average of dying from other causes subsequently. In a recent paper, Wong (1977), considered this problem. .His approach was, instead of assuming the relative susceptibility of those dying in an interval to be the same as those surviving, to assume that it was different by a general proportional factor F. In Appendix B, we discuss this alternative not,,assumingindep_e_nde___nc___eein more detail. ~lere we <ZD CD C~ Cr~

.... ° . --ALA-- @ show that there are some objections to the way in which Wong carried out his calculations and derive a somewhat different scheme along Wong's basic idea. The method described in Appendix B could be generalized to the situation where there are more than 2 groups with differing sus- ceptibility. Indeed the population could be given some defined continuous susceptibility distribution inltlally. Such generalizatlons are not investigated in this paper, the simpler situation dealt with in Appendix B being sufficient to allow illustratlon of the sort of effect that variations in susceptibility of the population have on estimates of loss of life expectation due to elimination of a particular cause of death. 2.11 Estimation of loss of life in a multif~ctorial situation All the estimates of loss of llfe described in this section deal wlth quantifying the effect a single factor has on loss of llfe. In I practice many diseases are multlfactorial in origin. It follows that any conclusions made about the factor studied only apply to a popu- lation with the same levels of other relevant factors as in the groups studied. ~or example the true loss of life related to smoking may be much higher in a group of asbestos-exposed workers than that estimated from a study of the normal population. By studying more than 2 groups it would be possible in a single experiment to obtain inferences on the loss of life related to more than 1 factor. For example, by carrying out a 4 group experiment with groups exposed to both, one only or neither of 2 factors, inferences can be made about the loss of life related to each factor. In these circum- stances it can be convenient, if it is possible to do so, to choose a statistic to'measure loss of llfe which allows independent expression of the effect of each factor. For example, if~f~actor A causes 3 years loss of llfe expectation, factor B 4 years and the~two together 7 years it would be convenient to use loss of life expectation as a measure as the conclusions about the two factors Individually hold whether or not the other factor is present. Of course, if in this example, there was no~loss of life in the group exposed to both factors it would be misleading ~o say simply.that factor A causes ~ years loss of life expectation and factor B 4 years as this only holds if the other factor is not present. C~

• " -15- Although we do not intend to pursue the multifactorlal situation in detail in this paper, we note, finally that clear thinking is needed in the interpretation of the findings. Rose (1977)', carreing out an analysis on data relating coronary heart disease mortality to a number of factors previously described by Reid et al (1976), found that the '~number 9~ deaths.associated" with each"factor, when added -together, exceeded the total number of deaths occurring. Todd (1977), considering these results, felt that there was something wrong with the "number of .deaths associated" approach. So there may be, as we showed in section 2.4, but it is not the choice of statistic that caused the apparent problem. The problem lay in addinE together a number of results, each of which represented the effect of removing a.partlcular'factor in %he presence of.all,he .others..''If-%he Individualde~ths"'assoclated'wlth each factor • are to add to the total, on6"should add-together the number ofdeaths associated with factor A in the presence of B, C .... N + the number associated with B in the presence of C N (and not A) + the number associated with C in the presence of D ... N (and not A or B) + etc. etc. Of course, if it came to attribution of deaths,, or rather claims related to them in a legal case, then one should be aware of the fact that. the order in which the associations with the factors are calculated (i.e. the order in which the factors are successively eliminated) may affect the answers. For example, if a disease only occurs if two conditions are met, then whichever factor is considered first in the analysls will appear $o be wholly responsible. But this is a problem for lawyers and not statistlcisms, whose ~ob is to present the facts in an unblassed and meaningful way ..... CD t~ f5~ CO

~e -16- Estimation of loss of life in the human smokin.g situation - problems involved and assumptions required . 3.I Randomization ~id causalitT. In the real life situation smokers and non-smokers are not ra~Ldomly allocated from a single group, but .themselves choose whether or not they smoke. For this reason excess death rates from various diseases observed in smokers may, in theory, to at least some extent, represent differences due to the smoke______r rather than to smoking. In applying the methods described in the previous section to smoking we choose to ignore this problem, however, and make the simplifying assumption that any differences in death rates observed between smokers and non-smokers of the same age and sex are due to smoking itsel~. While medical authorities (R.C.P., 1977) generally believe that this • assumption is approximately tr%~e for some causes of death, there are particular causes of" death where at least some opinion holds that this assumption is false. Thus, the excess dea.th rate" of cirrhosis of the liver amongst smokers is generally held to be attributable %0 the fac~ that cirrhosis of the liver is caused by smoking and that virtually all heavy • one['-gn-~--- drinkers smoke (Doll and Hill 1964)), while, on the other hand, study (Truett et al, 1957) has shown that, after taking differences be- tween smokers and non-smokers in blood pressure, serum cholesterol, glucose tolerance and left ventricular hypertrophy on the electrocardio- gram 'into account, the excess death rate from ischaemic heart disease among smokers is a slight underestimate of the true dilference in risk related to smoking./~Furthermore, though this view is generally disavowed / by the medical authorities (R.C.P., 1977), some workers (Fisher (1959), Burch (1976)) believe the excess lung cancer death rate of smokers is not caused by their smoking at all, but by a common genetic tendency to smoke and to get lung cancer. "To argue-wh~t the true proportion of the excess deaths among smokers that are .actually caused by-~moking is beyond the scope of this paper. Indeed, the R.C.P. (1971) when considering this problem, said it was not possible to Ei.ve a precise estimate, sayinE only "there - can be little doubt that at least half c~ the estimated 31,000 excess deaths among male smokers aged 35-64 in the United Kingdom (in 1988) were due to smoking". Rather, we simply note that there is no practlcal difficulty in adjusting the estimates we make, ~f different assumptions are used. 3.2 Problems of cohort data ........ • .. Even asstuming that cohort data on self--selected L~roups of smokers anfl non-smo~ers can be used in the s~.me wn.y ~s d~ta ou ~'au~o,r,~[~' selected ~,r~)t~ps, 0 0 C'-- ~.,-d

• ° .... -17- there still remain a number of reasons why the interpretation of results from such data is not straightforward. Firstly, for information over the whole of its lifetime to be available, the cohort must have been born before about 1875 and much ~-~ of the information on cause of death will of necessity refer to ~o~u~s when diagnostic practices may have differed substantially from those used today. It is generally believed that, for lung cancer, standards of diagnosis have changed dramatically since the beginning of the century, accounting for a considerable proportion of the observed rise in incidence (R.C.P., 1977). Even if death rates from the cause of interest could be adjusted in some suitable way for changes in diagnosis (and attempts have been made to do this for lung cancer (ibid)), there would still remain a second practical objection~ This is that, if there have been marked trends in time with respect ~o the level and age-distribution of the cause of death, statistics summarizing the effect a cause has had on a past cohort may have little relevance to people in cohorts still alive today. Although, for particular purposes it may be useful to quantify what effect a particular cause of death used to have, our main interest is really in trying to produce statistics that quantify the effect it has on presen%-day man. A third problem in studying the effect of smoking is that cigarettes themselves have changed over the years. In particular there has been a general switch from smoking plain to smoking filter cigarettes. Since 1955 when 98% of cigarettes smoked in the U.K. were plain, the percentage has dropped to 13% in 1975 (Lee, 1976). It seems likely that the health risks associated with filter cigarettes are significantly less than those associated with plain cigarettes (Hammond et al, 1978; Dean et al, 1977; Bross and Gibson, 1968; Wynder e t.t al, 1970) and consequently inferences about the effect of smoking on loss of life based on old data may markedly overestimate the true effect on current populations of smokers. A fourth difficulty with cohort data lies in the fact that in practice • an appreciable number of smokers modify the amount they smoke during their lives or even give up. At first sight, provided regular Informatiou on the smoking habits of the cohort is recorded, this does not seem much of a problem as one can exclude from analysis any people whose habits deviate from those that one originally intended to study. However, if "modifiers" or "givers-up" are not represen,tntive of "continuers" bias cnn occur in estimating loss of llfe. In par~iqular, if some diseases related tO C~ C~

smoklnE cause people to cut down or glvo up, and ~f such diseases are rela~ed to the subsequent prob~billty o~ death,I a situation where "smokin[, causes disease" interacts with "disease causes smoking"~; great care should be taken in the interpretation of the findings. This is discussed again later in this paper. Finally, as we have mentioned previously, one should take care before assuminE that estimates of loss of life obtained from cohort studies of special populations are necessarily relevant to other pop- ulations. The relevance to current smokers in England and Wales of some of the data that is available on smoking and mortality is discussed in Section 4. 3.3 Population and prospective llfe-tables Theoretically, it mlgl~t be possible to construct a more up-to-date life-table by using available data from a more recent cohort and, by extrapolation from the trend of age-specific mortality rates in previous cohorts, filling in estimates of death rates in ages not yet reached by the cohort. In view of the unreliability of extrapolation for more then a short time ahead, such an approach is unlikely to be very t:seful. It seems better to look for another method more referable to current experience and not involving extrapolation or adjustment for changes in diagnosis. The alternatlve approach, which, though not without objections, is perhaps the best available to illustrate the current effect of a cause of death, is to construct a life-table usinE up-to-date cross- sectional data. Data is said to be collected cross-sectionally if, for each age-group, it is collected over the same small period of time. For our purposes we would need to collect, for each age-group of interest, estimates of the population alive and of the number of deaths occurring, both in total and due to the ~ause of interest. Such data does not, of coursej refer to one cohort (it can be seen as a snapshot of the current status of a number of consecutive cohorts) and- does not form a life-table directly (the population allve in one see group may, for example, be higher than that in a younger age group due to variation in the birth rate over time). However, a life-table can be constructed from it by starting with some arbitrary number alive (Ay, usually taken as I00,000) and reducing it, for each successive ti age group, by multiplying it by (I - di) where di Is the cross- sectional death rate per year ~rom all c~tuses in age group i and ti is the length of the interval in years. C~ CD C~x

In the terminology used by Gross und Clark (1975), the life-table so constructed is a population life-t~ble. It is the l~fe-table of a hypothetical cohort having at each age of its life the mortality rates of the whole population observed cross-sectionally at that time. ~iis should be borne in mind in considering statistics summarizing this experience. Thus, the fact that the expectation of life ~or such a cohort may be 72 years does not imply that a child born at that time will, on average, llve 72 years. To obtain an estimate of how long children born then will live on average requires more than just know- ledge of the age-specific death rates then. It also requires knowledge of how the rates are going to change in the future. And this one can- not reliably predict for more than a short distance ahead. ~_~.-~ ~ In some human data, as we shall see in the next section, a population @~'~ ~ (consisting of a number of different birth cohorts) is followed for a "- ~ umber of years in a' "prospective" study. It can be useful, on occasion, to combine all the survival experience from the study into a single life- table. To do this, the average death rate for a particular age interval is calculated by dividing the total number of people in the study who died in the age interval by the total number of "man-years at risk". Man-years at risk is the total length of time spent by the study popu- lation in the age interval during the study. Given these death rate estimates a life-table can be constructed in the same way as for the population life-table. For clarity, we christen this a prospective life- table. An objection to a prospective life-table is that, in £ormlng age- specific death rate estimates, one is combining together survival experience from a number of years. Such an averaging will tend to conceal any trends in survival over the period o£ the prospective s tudy. ~j L' J C.~cf ~,~ C~ F~

........... -~20- 4. Data available on smoking and mortality 4.1 Introduction Having, in the last 2 Sections, discussed the theoretical merits and demerits of some statistical techniques to measure life shortening and of applying such techniques to human populations, we now turn our attention to using the methods discussed to quantify the loss of llfe due to smoking and to diseases associated with it. As noted before our main interest is centred on obtaining estimates relevant (under" certain assumptions) to the current population of England and Wales. Before embarking on the calculations (Section 5) it is convenient to consider what data is actually available to us on smoking and mortality. . ~ ~J-'~ prospective study and it is not surprising therefore that no actual ,~w~s ~Jstudy carried out has followed the mortality of cohorts of smokers and non-smokers for more than about 20 years. Thus no cohort life- 4.2" Data from prospective studies It was not until the early 1950's that evidence started to accumulate from retrospectiye studies (e.g. Doll and Hill (1952)) to suggest a strong association between smoking and lung cancer mortality. i Before that time there would have been no real reason to carry out a tables can be-calculated directly. It would not be appropriate to consider here all the prospective studies on smoking and health that have been started in the last 25 years. Some are only on small populations, others have been running for "only a relatively short period of time and others have not reported results in a form suitable for our analyses. We will restrict ourselves to mentioning three, one British and two AmericPm which have followed large populations for a long period of time and have reported results in detail. One of the most important of the prospectiv@ •studies is the British Doctor's Study started in 1951 by Doll and Hill, for which 20 year follow- up results for males have been given by Doll and Pete (Ig76). In this sSudy all doctors were asked to complete a questionnaire on smoking h~its in 1951, 69% rbsponding. Subsequcntly participating doctors were requestlonned in 1957, 1966 "and 1972 and their mortality continuously monitored. Only a handful, 103 out of 34,440 were lost to ~ollow-up of the male doctors who completed tl~e original questionnaire. <In CD Cr~ ~J

-21- There are a number of reasons why doctors are not typical of the general population. Doctors are financially hotter off than the averng8 and members of social class 1, a group known to have levels of mortality for many diseases less than the natlonal average. And as Doll and Pete (1976) pointed out, many more doctors.than would have been expected from national smoking figures gave up smoking in the last 20 years and, as the authors clalm, their total mortallty in consequence decreased markedly relative to national figures. However, despite these differences, most medical opinion holds that the relative mortality of smokers to non- smokers found in the British Doctor's Study from the various causes of death tabulated is a reasonable indicator of the relative mortality of smokers to non-smokers in the British population at large. It would seem reasonable, however, that even i~ this opinion is correct, relative mortalities from the British Doctor's Study are only likely to be appli- cable to the British population in about 1961 (the centre-point of the 20 year follow-up period) and not to the British population in 1978, because of the large switch to filter cigarettes that has taken place since 1961 (see Section 3.2). The second prospective study worth considering here is Hammond's million person study for which the latest results have been reported by Hammond et al in 1976. In this study, 1,078,894 people in 25 American States were interviewed by American Cancer Society volunteers between October 1959 and February 1960 and subsequently requestlonned at 2 yearly intervals until 1965 to obtain details of smoking habits. Mortality has been followed-up for 12 years. The study populatlon is not typical of the U.S. population, being of markedly higher than average soclal class. However, as for the British Doctor's Study, it has been used as a vehicle for providlng estimates for the population at large of the relatlve risk of smoking for many diseases or groups of diseases. The third large prospective study is the Dorn Study of U.S. Veterans. In this ease the population studied, 198,834 in total, was drawn from" policy holders of U.S. Government Life Insurance, an insurance available to those who served in the U.S. Armed Forces between 1917 and 1940. Like Hammond's million person study there were very few non-whltes studied and the higher social classes were over represented~ the reason for this being that poorer members of society tend less to be able to afford insurance policies. In addition, Dorn's Study, results for which were reported in Kabn (196G), consisted almost completely of males. 0

-22- 4.3 National morta!ity data Although information is not nationally available on mortality for "smokers and non-smokers separately,~it is available~ many countries publish regularly information on mortality for the population as a whole. For England and %Yales, for example, the Registrar General publishes annually tables giving estimates o~ the number living end the number dying from each main cause of death by 5 year age group. 4.4 National smoking data Imperial Tobacco Limited have .carried out annual surveys of smoking habits in the United Kingdom since 1948, The results of these surveys have been published at regular intervals by the Tobacco Research Cow lcil in their Research Paper 1, "Statistics of Smoking in the United Kingdom", the latest edition Lee (1976) giving figures up to 1975. Currently the surveys obtain information on smoking habits from about I0,000 people a year and the puhllshed results give details, inter alia, on the per- centage of people who are non-smokers, ex-smokers and smokers o~ vsryiu~ numbers of cigarettes a day by sex and age group. In recent yearz, estimates of smoking habits in the population have also been published in %he tabulations of the General Household Survey (O.P.C.S. (1975)). Where comparable figures are presented, the degree of agreement between the surveys is, on the whole, very close, f ~" ~2".J~%.=~-~ 4.5 Data used in the estimations In the examples considered in Section 5 estimations are made of loss of llfe to males in three situations: a) due to smoking in Hammond's study b) due to lung cancer in England and Wales in 1971-75 c) due to smoking in England and Wales in 1971-75. The data for the first situation was drawnfrom Hammond (1969) who gave a life-table showing the survivorship of men aged 25 in relation to current number of cigarettes smoked per day which is reproduced here as Table I. The life-table (a population llfe-tsble - see Section 3.3) was constructed from the -deaths occurring in the 5 year period July I, 1960 to June 30, 1965 oZ 447,196 male subjects born between 18G8 and 1927 classified into 5 year age-groups 33-37, 38-42," ... 88-92. As describe0 in the paper the llfe-table has been adjusted to the 1959-61 U.S. life- table rates ~or all males and extrapolations have been made to cover the age range 25-34. O CD [_n

• J -23- Table 2 gives estimates of the populatlon, total deaths per year and deaths from lung cancer per year for males in England and Wales ever the period 1971 to 1975. These estimates were calculated by averaging Zigures given for individual years in Tables 1 (pepulatlon) and 17 or 2 (deaths) of the Registrar General's Statistlcal Revlew of England and Wales. For the years 1971 to 1973, where the living population data by age group ended wlth an 85+ group, the distribution into ~5-89 and 90+ was estimated by assuming the same ratio of population in the two age groups for the years 1974 and 1975, where the data was given. The data required for estimation of the effect of smoking in England and Wales is more difficult to obtain. As cohort data on the mortality ~f smokers and non-smokers in this population is not available, it has to be estimated indirectly using data on the relative risk of mortality associated with smoking obtained from the prospective studies and on the distribution of smoking habits of the EngLand and Wales population from national smoking data. To be useful such mortality and smoking data should be given by similarly defined age and smoking hablt group. Considering the mortality data first, we have to decide firstly which prospective study data to use. Table 3 glves-,estlmates of the relative mortality of cigarette smokers only and never smokers by five I0 year age groups taken from Doll and Pete (1976) o Hammond (1966) and Kahn (1966). It shows that there is very little difference between the relative fish estimates from the three studies and suggests that using data from any one of the studies should give fairly similar answers. In practice, information on cigarette only smokers and never smokers is not enough as we have to consider the total population in some of our analyses. For this reason we must look for a subdivision of the total population by smoking habit for which information by age.group is available not only on relative risk from one of the prospective studies but also on distribution in the England and Wales populatio~ in the national smoking data. After looking at the data available, it was decided that it was convenient to divide the population into four smoking habit groups: never smoked b) current smokers of cigarettes (with or without plpes or cigars) c) current smokers of pipes and/or cigars (but not cigarettes) d) ex-smokers. Data on relative mortality by five I0 year age groups for these four smoking hablt groups, tshen from the Dorn Study, the only one presentlng dat8 in the form required, is given in Table 4. C~ C~

. -24- AS can be seen Table 4 gives information only for men in th~ age-' range 35 to 84. For our purposes somc assumption will have to be made as to relative risk of mortality for men outside this raugs. In our calculations we have made two somewhat arbitrary assumptions. ~e first is. that above the age of 84 the same relative risks apply as in the age group 75-84. Inaccuracies in this assumption will not have a great effect as deaths of people aged 85 or more form only about 9% of total deaths (see Table 2). The second assumption is that below the age of 35 the four smoking habit groups have equal relative risk of mortality. Of total deaths in the 25-34 age range almost half are due to accidents, ~while only about 7% (unlike the 30~ or so in the 35-44 age range) are due to ischaemic heart disease, the cause of death contributing most strongly to premature deaths associated with smoking. In view of this, and the fact that deaths in the age range 10-34 (it seems inconceivable that any earlier deaths are due to smoking) form only about 2% of total deaths, any error associated with this approximation is bound to be small compared with errors associated with other assumptions made. The figures given in Table 4 are based on deaths occurring over a period 1954 to 1962, and represent the effect of smoking for a period when many of the population were smoking high-tar plain cigarettes. To make inferences from such data and apply it to people living today is to assume that the risks associated with smoking are the same as they were 20 years ago. We call this assumption Assumption A. As noted before (Section 3.2)some evidence has recently accumulated all indicating that the. hea~th risks associated with filter cigarettes are significantly less than those associated with plain cigarettes so it is very likely that inferences under Assumption A would overestimate the loss of life smoking modern cigarettes causes a modern smoker. Unfortun- ately much of the evidence on the retative effects of plain and filter cigarette smoking considers lung cancer mortali~y only (Bross aud Gibson (1968), Wynder et a1 (1970) or only selected diseases (Dean e_~t a~l (1977)) and the one paper that considers total mortality (Hammond (1976)) base~ information on smoking habits at latest in 1966, defining "low" T/N as less than 1.2 mg of nicotine and normally less than 17.6 mg of tar, 8 borderline which includes many cigarettes classed today as "low to middle tar" or "middle tsr" in the Government Chemist's League Tables. Hammond's estimated relative risk for total mortality for male deaths in 1966-72 of "low" T/N smokers as against "high" T/N smokers is 0.81, a ratio much higher than the relative risks Dean et al (1977) found when comparing O O Cr~

-25- retrospectively the lung cancer, coronary heart disease, chronic bronchitis and stroke death rates of people who had smoked filters in 1954, 1964 and 1969 compared with continuing plain smokors (0.39, 0.49, 0.58 and 0.53 respectively). As Dean's study is open to doubt for a number of reasons discussed in it, and as the latest death considered is only in 1972, a year when the sale of "low tar" cigarettes had hardly started in England, it is clear that there is no reliable basis for cal- culating an alternative assumption. However to illustrate the effect alternative assumptions to Assumption A have on the statistics derived, and to give some feel of what might be f the true number of deaths associated with smoking nowadays, we have used a somewhat arbitrary second assumption, Assumption B, in some calculations. In this assumption we have assumed that only 60% of the excess relative risk of current cigarette smokers to never s~okers indicated by Kahn's figures applies today. We assume (in the absence of knowledge either way) that the relative risk of current pipe and/or cigar smokers is the same as uhder Assumption A. Also that ex-smokers, most of whom will have smoked wholly or a majority of plain cigarettes when they smoked, have the same relative risks as under Assumption A. The relative risks involved in Assumption B are given in Table 5. Having obtained the mortality data required for part 3 of our example we now consider the data needed to give the distribution of the male popu- lation of England and Wales by smoking habit group and age. Lee (1976) does not give figures in precisely the form required. However, by methods described in detail in Appendix C, i.t was ~ossible $o derive a sufflciently good approximation to the data required for our purposes indirectly. The resultant data are given in Table 6. Figures for men aged less than 35 are omitted as they are not required since it has been assumed that smoking does not affect mortallty rates below 35. C~ C~

-26- 5, Calculations of loss of life due to smokin~ and some smoking-associated diseases 5.1 Loss of life due to smokin~ in Hammond's study The analyses described in this section are based on Table I, which, as noted in section 4.5, shows the survivorship o£ men aged 25 in relation to current number of cigarettes smoked per day based on the results of Hammond's study. It is effectively the same as a life- table differing only in the'fact that the entries are multlplied by i00, so that it measures percentage rather than proportion surviving. It also only refers to ages 25 or over taking 100% as the percentage surviving at age 25. To calculate life expectation, QII' for a particular group of men the formula "" n. qll = Z Yi(zl- Zi+l)/lOO i=l is used (which is equivalent to that for Q6 for continuous data) where Yi is the midpoint of interval i and Zi is the entry in the survivorship table. If, as Hammond did, one assumes that, within each 5 year period, those men who die do so at the centre of ~he period, the successive mid- points are 5 years apart and the formula reduces to i n Qll = 27.5 +30 Z Z i=2 i The expectations of life for men aged 25 for all men and the various smoking groups were calculated by Hammond (1969) and are presented in Table I. As can be seen it is estimated that a smoker of 40 or more cigarettes a day who continues to smoke can expect to live for 8.3 years less than someone who has never smoked regularly. The loss of life expectation reduces as the number of cigarettes smoked is smaller. Another useful way of looking at the data is'to consider the proportions survivln~ to some age of interest. For example, taking age 65 as of interest as being the customary age of retirement it can be seen that slightly over three quarters (77.7%) el those who have never smoked regularly-can expect to live to age 65 whereas only Just over a half (54.0~) of 40+ a day cigarette smokers can. Putting it another way the probability of death by 65 for 40+ a day smokers (4G.0~) is almost double that for never smokers (22.3%). C~ CZ~

-27- To calculate the proportion of those alive at one given age who survive ~o another one simply divides the survivors in the given 'column at the t~wo relevant ages. R.C.P. (1971) presented figures, based on Doll's study, showing the proportion of men aged 35 who will survive to age 65. These are reproduced in Table 7, together with comparable figures based on Hammond's data in Table I. As can be seen the figures in the two studies are in the main reasonably comparable, though Hammond's percentages in general are slightly higher. We next consider loss of life expectation per cigarette smoked. To : :~calculate this from the losses of life expectation given in Table 1 two additional pieces of data are required, the average age of starting to smoke of cigarette smokers and the average number of cigarettes smoked per day within the broad categories 1-9, 10-19, 20-39 and 40+. FromTable 33M of Lee (1976) it can be seen that, in every survey year considered (1965 to 1975) the median age of starting to smoke of those who have ever smoked and know when they started lles in the age group 16-17. Though this figure refers to all smokers in the U.K. we will assume that it applies to Hammond's cigarette smokers and take the centre of the age group 17.0 years as the average. Similarly, using data in Table 22M from the same source, one can estimate that the average number of cigarettes smoked in the 1-9 a day group is about 4 and in the 10-19 group about 13. Information on the other groups is less reliable, as Lee (1976) only goes up to 30+ a day In his tables, and we have somewhat arbitrarily assumed the midpoints are 25 of the 20-39 group (it is dominated by 20 a day smokers) and 50 of the 40+ group. Based on these assumptions (and the further one that deaths between the ages of 16 and 25 axe not affected by smoking) Table 8 shows the calculations involved and the results obtained~in estimating the loss of life per cigarette smoked. The figures show that, for a 40+ a day smoker, the loss of llfe expectation is 4.8 minutes, somewhat different from the figure of "almost 6 minutes per cigarette" given by Diehl (1969) who, presumably used slightly different zssumptions of age at starting or average number smoked within the 40+ a day group. It is noteworthy that the estimates of loss of life per cigarette smoked increase 'continuously with decreasing amount smoked. Particularly noteworthy is the very high figure of 31.2 minutes ~or 1-9 a day smokers. This may partly be due to an artefact related to the fact that Hammond's C~ C~ C~

• . -28- life-table was not based on a study of men who actually smoked th~s number all their lives but on the mortality experience over a limited .period of men who happened to be smoking 1-9 a day at the time. It could well be t~at a proportion of the 1-9 a day smokers contained men who had previously smoked more but had cut down due to health problems. Such people would presumably be at a higher risk than actual continuing 1-9 a day smokers. It should be polnted out, however, looking increase risk of back at Table 7j that Doll's data does not show~an • in death at the lowest level of smoking as Hammond's ~'~ Doll's 1-14 a day doctors had a 39~ increase in probability of dying before 85 (given survival to 35) as compared with never smokers whereas Hammond's 1-9 a day men had a 48~ increase despite presumably smoking about half as much as the doctors. We have shown that loss of life expectation per cigarette smoked depends on the number of cigarettes smoked. We can also see from Table 9 that the implicit assumption required for this measure to be useful, as discussed in section 2.8, does not hold. As we showed there, the assumption that each cigarette has an equal effect implies that the ratio of the ages at which smokers and never smokers" reach any given proportion surviving is constant. As can be seen, this is not so, ~he ratio increasing steadily over the period for any of the smoking groups considered. In fact what is more nearly constant is the difference in ages to reach a given proportion surviving'. 5.2 Loss of llfe due to lung cancer in England and Wales in 1971-75 Table 2, given previously, shows the population, total deaths and deaths from lung cancer by age for males in England and ~ales based on 1971-75 data. To estimate loss of llfe due to lung cancer we first construct a population llfe-table (see section 3.3) so that we can see what the effect of lung cancer would be on~'a ~opulatlon having at each age of its life the average mortality rates present in England and Wales in 1971-75. This life-table is given in Table I0. As can be seen from Table I0, lung cancer would be responsible for 8.4% of total deaths in such a population, slightly less than the proportion, 8.8%, in the actual population. From the population life- table it is possible to calculate, using adjusted life tables (see sectlon 2.10) the expe%tation of life of the population if lung cancer did not exist and to compare it with that in the adjusted life-table. C~ Q t24 C~, tz~

-29- The results, given in Table 11, are based on calculations made under more than one assumption. Firstly we assumed that the population was homogeneous, i.e. those dying of lung cancer would have the same chance of dying from other causes had lung cancer not existed. Secondly, we assumed (see Appendix Bi that the population consisted of two groups (proportions P, l-P) with the first group having F times the general level of susceptibility as the first. As Table II shows, under the first assumption, lung cancer is responsible for a loss of llfe expectation equal to Just about a year per head of the population or equal to almost 12 years in those actually dying of it. Under the second assumption, the loss is smaller, as would have been expected. However unless one postulates both a fairly large relative susceptibility factor (F) and also a susceptible proportion (P) which is neither very small nor very large, the difference is fairly .marginal. Even when th~ factor is 10-fold and the proportions equally divided lung cancer is still responsible for just over 2/3 of a year of loss of life. Although a different sort of assumption, in which lung Cancer decedents in particular are assumed to have been virtually certain to have died of some other cause very soon after their age o2 death had they survived, would have reduced the estimate of loss of life due to lung cancer this does not seem very plausible. It thus seems reasonable to assume that if deaths from lung cancer ceased to exist, the population at large would live the best part of a y@ar longer. Based on Table 12, which gives numbers of deaths from ischaemic heart disease (ICD 410-414) and from all neoplasms (ICD 140-239) for the same population as for Table I0, we calculated, for comparison with lung cancer, the effect .these two major cause of death categories have on loss of life. The results are summarised in Table 13. It can be seen that, per head of population, loss of ~i£e expectation due to lung cancer is about a third of that due to all neoplasms and just over a quarter of that due to ischaemic heart disease. In the previous paragraphs ~e have been computing average loss of expectation of life due to cause of death by comparing the expectations of life in unadjusted and adjusted life tables. This loss can be related to the people dying from the cause by dividing the total years lost by the total number dying from the cause. As noted in section 2.6, other workers have computed average years lost to life expect~Icy o~ CD C~

-30- people dying from a cause by averaging the remaining llfe expectancy of those people at the time they died. Table 14 gives, based on the population life-table of Table I0, estimates of remaining llfe expectancy by a~e and illustrates the method of calculation used. Applying the figures in column 3 to the numbers of lung cancer deaths ~- > in Table I0 one can then show that the average loss of llfe expectancy of lung cancer decedents is 11.91 years. Applylng them to the numbers of total deaths in Table i0, the average loss, as noted previously, is lower - 11.42 years. 5.3 Loss of life due to smoking in England add Wales in 1971-75 For the purposes of the analyses in this sectlon we wish to compare the estimated llfe table of men who have never smoked with that for the total 1971-75 England and Wales male population. The life table of never smokers is not available dlrectly, but can be estimated from the information available by age group on population and total deaths (Table 2), distribution of smoking habits (Table 6) and relative total death rates by smoking habit group (Table 4 - Assumption A; Table 5 - Assumption B). The first stage in the calculatlon is to estimate, for each age group, death rates within each smoking habit group. The method used to do this is illustrated in Table 15 taking the 60-64 age group as an example and using assumption A. The results obtained are glven in Table 16 (Assumption A) and Table 17 (Assumption B). From the estimates of death rates of never smokers, their survivor- ship functions can easily be calculated. The functions under both assumptions, and that for the total population (taken from Table I0) are given in Table 18. From Table 19, which gives certain characteristics of these survlvorship functions, it can be seen that, assuming cigarettes have as much relation with mortalltyin 1971-75 as they did some 15 years earlier in Dorn's study (Assumption A), if the whole male populatlou of England and Wales were never smokers then their llfe expectation (72.5 Fears) would be 3} years greater than that of the current population (69.0 years). On the alternative assumption, that the excess risk related to smoking cigarettes in 1971-75 is only 60% Of what it was in Dorn's study, the difference is about 2½ years. It is also of interest to compare the estimated li~e table of men who have never smoked wlth that of continuing clgarettc smokers and C~ C~ .<w:

, i ' -31- continuing pipe and/or cigar only smokers. These three life tables are given in Table20 (Assumption A) and Table 21 (Assumption B) based on the death rate estimates given previously in Tables 16 and 17, and their mortality is compared in Table 22. It can be seen that under Assumption A the expectation of life of continuing cigarette smokers is 5.50 years less than that of never smokers while that of pipe and/or cigar smokers is 0.70 years.less. Under Assumption B these differences are 3.76 years and 0.80 years respectively. Table 23 gives estimates of the average annual consumption of manufactured cigarettes per male smoker by 5 year age groups. These figures are averages for data for the years 1971-75 given in Lee (1976), with the figures for the highest 2 age groups estimated by extrapolation. Taking these estimates as being applicable to the average annual consumption of all cigarette smokers (data for hand-rolled smokers is not available) one can then apply them to the estimated life tables for cigarette smokers given in Table 20 (Assumption A) and Table 21 (Assumption B) to produce estimates of lifetime cigarette consumption. Thus, under Assumption A, the loss of expectation of life of cigarette smokers of 5.50 years can be related to an estimated lifetime average consumption of 397,000 cigarettes to yield an estimate of loss of life per cigarette of 7.3 minutes. Similarly, under Assumption B, the estimated llfetime average'consumption of 401,000 cigarettes gives an estimate of loss of life per cigarette of 4.9 minutes. C~ C~

% 6. Discussion -32-~ From the preceding sections it can be clearly seen that.there are right ways and wrong ways to quantify the effect of a disease or a factor causing it on loss of life. That this is not generally recognised can be seen from Miller's recent (1977) observation that people who smoke filter cigarettes die two to four years sooner than smokers of plain cigarettes, and his subsequent attempt to explain this finding in terms of higher blood carbon monoxide levels in the blood of people who smoke filter cigarettes. His study was based on a cross-sectional . observation of deaths occurring between 1972 and 1974 in Pe~ylvauia's .~ Erie County, and the difference in average age at death observed of filter and plain smokers can be explained by the fact that the average age of living filter smokers (due to the swi{ch to filters oecurrin~ more in younger smokers) is markedly lower than that of plain smokers. Had.Miller compared relative death rates or expectations of life based on life-tables (both of which would have required observation of filter and plain smoking habits in the living population also) he would have doubtless found, in line with the studies we quoted in section 3.2, %hat filter cigarettes were in fact safer, aud not the reverse. However, though it is easy to see in some cases that particular statistical approaches are wrong it is not so easy, in the case of • smoking, to arrive at estimates of loss of life which are right. Death is a fairly rare event, and, therefore, to get information on adequate numbers of deaths of people of different smoking habits, it has been necessary in all the major prospective studies to study deaths over a fairly long period of time. When one additionally takes into account the time required to report the results of these studies, the fact that the tar yield of the average cigarette smoked is reducing rapidly and the evidence that the lowering of average tar yields has been beneficial to health (see section 3.2) it is clear that estimates of the relative death rates of smokers and non-smokers derived from the major prospective studies are virtually certain to be markedly too high, though by how much one cannot know precisely. It is also possible that the fact that all three of the major prospective studies considered here (see section 4.2) studied nationally unrepresentative populations might have caused a bias in rel~tive risk estimates. However, though the actual levels of mortality in these studies are known to be different from n~tional levels, the consistency CZD

-3S- of the estimates of relative levels of mortality of never smokers and .cigarette only smokers in the three studies (Table 3), makes it seem unlikely that the relative levels of mortality in the U.S. and U.K." populations around the time of the three studies of these two groups differed much from those found in the studies. Unrepresentatlveness does not seem a major problem therefore. A more difficult problem in quantifying the relationship of smoking to mortality is the fact that many smokers modify the amount they smoke during their lives. If, as is often the case, this modification is due to symptoms associated with smoking, it can be seem that construction of life-tables using prospective study data on risk of smoking by age may be somewhat misleading inasmuch as the data used on men in younger age groups is likely to include some Information on men who fail to reach older age groups, not because they die, but because they give up. For this reason alone if one took a random sample of the population and forced them to smoke throughout their lives (the analogue of animal experiments) the actual loss of llfe expectation observed in this group as compared with a control group of continuing non-smokers may well be different to the estimates we calculate. It would of course also be likely to be different because smokers are clearly not a random sample of the whole population in so many ways. Bearing all these reservations in mind, one can calculate that, assumlnE the relative risks of smokers and never smokers are as found by Kahn (196S), (Assumption A), and assuming total death rates are those observed i~ England and Wales in 1971-75, the average male cigarette smoker can expect to live 5} years less than the average male never smoker. Put another way, one can show that 23% of male cigarette smokers are likely to die before age 60,.if they continue to smoke, as compared with 13½% of male never smokers. .If the relative risks of cigarette smokers to never smokers are assumed to be 60% of those given by Kahn (Assumption B), as may be the case nowadays due to the switch to lower tar cigarettes, then the loss of life expectation related to smoking falls to 3.76 years. This loss of life expectation can be expressed on a per cigarette smoked basis by dividing by the estimated average total lifetime cigarette consumption of cigarette smokers (about 400,000 cigarettes) to give figures of 7.3 minutes (Assumption A) and 4.9 minutes (Assumption B). However, apart from the fact that it assume~, which i~ unlikely to CD O Cr~

-34- be the case (see section 3.1), that excess death rates of smokers are wholly due to their smoking, this estimate is open to'criticism on two grounds. Firstly, as noted in section 2.8, the theoretically proper way to compute average loss per cigarette smoked is to calculate, for each person, his loss per cigarette and then to average it (which is not practical as one cannot estimate life expectation on an individual basis) and not as we have done to divide average loss by average number of cigarettes smoked. Secondly, and more importantly, the argument of • sections 2.8 and 5.1, show clearly that in fact each cigarette does not take an equal amount off life expectation. The use of this statistic to quantify loss of llfe related to smoking cannot be recommends d. CX> 0 Cr~ -..j

t__. i ' u_.. "i 7. -35- SummaIT The merits of a number of different estimators of loss of life in relation to a disease or to a factor causing it are discussed. It is concluded that those based on life-tables, "such as "expectation of llfe" or "percentage dying between given ages", are likely to be the most useful. Among statistics to be avoided are "average age at death" based on cross-sectional observations, which cml lead to gross bias, and "number of deaths attributable to a factor" which is diffioul~ to interpret and gives l~ttle useful information. The problems involved in inferring loss of life expectation due to a single exposure to a factor from observations on a population exposed at regular intervals throughout their 1lees are noted, and it is shown that the assumption that each exposure leads to an Identlcal los~ of life expectation implies the existence of a particular mathematical relationship between thelife-tables of exposed and non-exposed populations. I The choice of appropriate data to use in constructing llfe-tables, where full lifetime survival history of a population is not available, is discussed and it is concluded that life-tables constructed from up- to-date cross-sectional data are most useful as a basis for quantifying the current situation. The problems of constructing such llfe-tables for assessing loss of life due to smoking are noted, partlcularly important being the fact that the major sources of reliable information by age on the relative risk of death of smokers and non-smokers all refer to a time period when the average cigarette smoked had a much higher tar yleld than it does today. A number of applications of the methodology are made to estimating loss of llfe related to smoking. For example, it is sho~n that, assuming the whole of the excess of the death rates o£ smokers over non-smokers is due to their smoking, and assuming that their relative risks of death by age are as found in the U.S. Veteran's study, the li£e expectation of the male populatlon of England and Wales would be 3} years greater if none of them had ever smoked. The difference in llfe expectation of never Smokers and continuing smokers under these same assumptions is 5} years which, related to the estimated lifetlme cigarette consumption of smokers of about 400,000 clg~re~tes, means an estimated average loss of llfe per cigarette of 7.3 minutes. However comparison of the ll/e-tables of smokers and non-smohers makes it quite clear that each cigzrette smoked does not have the same effuct on loss of life. t ~j~J

APPENDIX A - "Deaths associated with smoking" - an estimate of the reduction in deaths that would occur over 25 years if the populatlo_~n had never smokers death rates As we point out in Section 2.4, the "number of deaths associated with smoking" as calculated by, e.g.R.C.P. (1971), is, in fact, ~ estimate of the number of deaths that would not have occurred in the year following a universal giving up of smoking assuming that on giving up, smokers age-specific death rates reverted at once to those of never smokers. This statistic ignores completely the fact that everyone must die at some time and only measures a short-term effect. The purpose of this Appendix is to estimate, under certain assumptions, the number of male deaths that would occur in England and Wales each year for 25 years given the population at large had non-smokers death rates and to compare these numbers with those that would occur if the population kept to present death rates. .... The present (1971 - 1975) average distribution of the Englsmd and Wales population and current numbers of deaths by 5 year age groups has been given previously in Table 2. Estimated death rates of non-smokers under two assumptions A and B (see Section 4.5) have also been given in Tables 16 and 17. To compute, for a particular set of death rates, expected numbers of deaths'occurring in each of the following 25 years, calculations were made for single year cohorts. For each cohort the number alive at the beginning of the 25 year.:period was calculated ~from the population figures for the corresponding age group in Table 2 assuming that the age-distribution within the population was uniform. (Thus the cohort aged 32 years old initially was assumed to number 301,404 initially, one fifth of the 1,507,020 people aged 30-34 in Table 2). The numbers in the cohort were then successively reduced by applying the death rate for that 5 year age group, t~:ing into account that in each successive year the cbhort would be- one degree older. Cohorts born after the begianing of the period were started off with 356,300 alive (i.e. assuming the birth rate of Table 2 remained constant). Cohorts reaching ages of 90 or over had the death rate of the 90+ group applied to them thereafter. Table A1 shows the numbers of deaths that would be expected to occur assuming I) death rates stayed constant at present rates if) death rates stayed constant at never smokers rates estimated under Assumption A --.a CZD q>4 CY~

iii) death rates stayed constant at never smokers rates estimated under Assumption B. It also shows the differences between (ii) and (i), and between (ill) and (i), i.e. in the "deaths associated with smoking" under the two" assumptions. The estimates show that, assuming historical figures of the relative risk of smoking derived from past prospective studies apply nowadays (Assumption A), 80,000 less deaths of males would occur in England and Wales in 1976 if the population at large had never smokers death rates. However if the population continued to have non-smokers death rates, these estimates of deaths associated with smoking decrease steadily with time, reaching 40,000 by 1988 and 12,000 by the year 2,000. Under Assumption B, where one is assuming that the relative risk of smoking applying nowadays is rather less than that under Assumption A due to the switch to lower tar filter cigarettes, the estimated deaths associated with smoking are consistently about three quarters of those under Assumption B. It should be noted that, in fact, as time passes over the period 1976 - 2000 an increasing number o2 smokers (and ex-smokers) will have smoked predominantly lower tar filter cigarettes during their lifetime and that it is somewhat unrealistic therefore to assume both constancy of death rates assuming that no smokers give up and as high relative risks as those used for Assumption B." We have not made calculations taking this fact into account as the data is not available to do this at all accurately. However it is clear that the effect of such a calculation would be to decrease the deaths associated by a proportion that would increase over the time period considered. In other words the drop-off in "deaths associated" would be even steeped th~nwe suggest in Table A1. C~ C~ (_/~l C~ C~

APPENDIX B - Calculatlon of adjusted llfe-tnbles not assuming independence Giv~ a llfe-table for a single ~roup in which, for each time interval i (i = 1 ... n) we observe the number alive at the beginnlng of the interval Ai, the total number dying Di and the number dying • roma particular cause of interest Li. We wish to estimate %he life- table that would have existed had deaths from the particular cause of interest not occurred, i.e. the adjusted life-table. We assume that the population consists of two groups of individuals, one groups ("susceptibles") having F times the susceptibility to death from all causes of the other group ("normals"). We also assume that both F and the proportion of susceptibles at the beginning of the first interval Pi are known. '" Wong (1977) derived the following formula for the probability, Pi' of dying in the ith interval given the cause of death of interest is rem6ved. (Di-Li) Li (Di-Li) Pi = AI + F. Ai AI Now (DI-Li)/Ai is simply the crude probability of dying from causes other than the one of interest, and Li/Ai is the crude probability of dying from the cause of interest. It follows his formula "reincarnates" the whole of the Li people dying from the cause to be removed and applies the former crude probability multiplied by F to then~ ignoring the fact that these decedents are already known to have survived the other causes of death for a part (on average, about half) of the interval. Further- more, Wong does not take into account in his formulae the fact that in subsequent intervals the reincarnated survivors will have a greater risk of dying assuming their extra susceptibility continues. In the following paragraphs~ therefore, we consider thee estimation afresh. Consider the ith time interval ~nd let the proportion of the popu= lation who are susceptibles be Pi" Let the length of this interval be Ti and let us assume it. to be small enough for the force___of mortality in the normal population from cnuses other than the one of interest to be t~en as constant (= =i)~ ~Id that from the cause of interest also to be taken as constant (= Bi). In tho susceptibles the forces of mortality will therefore be F=i and F~i respectively. O

Now the total survival from all causes Pi exp(-F(=i+Bi)Ti) + (1-Pi) oxp (-(=i+Bi)Ti) is estimated by (Ai-Di)/Ai and the relative mortality Cl) =i/Bi by (Di-Li)/Li (2) Substituting from (I) into (2) and writing Q = exp(-=iTiDi/(Di-Li)) (3) this gives the equation ' pi~F +.(1-Pi)~ = (Ai-Vi)/Ai 14) It can easily be seen that this equation has one real i root in the range (0,I) which can be obtained-iteratlvely ~J .without difficulty. The estimated survival rate in the normal population due to all causes except the one to be eliminated is thus "given by (Di-Li)/Di exp(-=iTi) = Q. (5) Similarly exp(-~iTi) = QLi/Di (6) ' It is now possible to define a sequence of recursive equations to show how the life-table estimate is built up~ Let, at the beginning Of the interval, values of Pi' Ai' Di' Li' Hi and Si be known, where the first four terms have been defined previously, and Hi and Ji are defined as follows. Hi number of living "reincarnated" susceptibles, i.e. the additional number of susceptibles that would have been alive had tho cause of death of interest been eliminated. " Ji number of living "reincarnated" normals. It follows that C:P 0

AiPi Ai (l-Pi) Ai+Hi+Ji A1 is the true number of suzceptlblcs alive. is the true number of normals alive is the adjusted life-table estimate~ We now estimate exp(-miTi) mud exp(-SiTi) from equations (5) and (6). The total number of susceptlbles dying in the interval (DSi) is ~iven by the expression DSi ~ AiPi(I-QF) and the total number of normals dying (DNi) by w DNi = Ai(l-Pi)(l-Q) It follows that Pi+l' the proportion susceptible at the beginning of the next interval (or end of this one), is given by Pi+l = (AiPi-DSl)/(Ai-Di) = (AiPi-DSi)/Ai+l (7) Of the living relncarnated susceptibles Hi, a proportion l-exp(-F ~Ti) will "die" in the interval due to causes other than the one eliminated. However, there will also be an increase in their numbers due to the difference between the actual force of mortality occurring and that that would have occurred had the cause of death been eliminated. It follows that Hi+1 = (Hi+AlPi) exp(-FmiTi) - AlPI exp(ZF~=i+Bi)Ti) (8) Similarly Ji=l = (Ji+Ai(l-Pi)) exp (-=iTi) - Ai(I-Pi) exP(-(=i+Si)Ti) Given that, at the beginning of the first interval H1 and Jl ame defined as O, and given, as noted above the user supplies estimates of Pl sad F, the above equations supply all that is needed for calculation of the adjusted llfe-table. CID O C~ "I

APPENDIX C - Estimation of distribution of England and Wales male population ,by smokinghabit and ave Table 6 gives the estimated distribution of the England and Wales male population by smoking habit and age. This was derived from Lee (1976), assuming such data for the United Kingdom as a whole could be applied to England and Wales, as follows. Firstly, Table IIM was used to extract the following data on percentage of smokers of each type o2 product by age for i975 by averaging the figures given for 1971 and 1975. Age group 35-59 60+ I, All smokers of manufactured cigarettes 49.5% 42.0% - 2. Hand-rolled cigarettes only 5,0% 5,5% 3. Pipe only 4.5% 9.5% 4. Cigars only 4.0% 2.5% 5. Ex-smoEers 19.5% 26.0% 6. Have never smoked 15.5% II.0~ Total 96.5% 98.0% The reason the figures do not add up to 100% is that no data is available for two smoking habit groups; 7. hand-rolled cigarettes and (pipe and/or cigars) and 8. pipe and cigars (but no cigarettes). Arbitrarily assigning the residue equally (1% each) to both categories for 35-59 year olds and 2% to category 7 m~d 1½% to category 8 for 60+ year olds, one could then compute the distribution by the iour smoking habit groups of interest as follows. Current smokers of cigarettes (categories i, 2 and 7) Current smokers o2 pipes and/or cigars only (categories 3, 4 and 8) Ex-smokers Never smoked Age group 35-59 60+ 55.5% 49.5% 9.5% 13, S~ 19.5% 26.0% 15.5% 11.0% Total i00.0~ I00.0~ 0 0 C'- t_~1

@ • . - .. It should be noted that the smoking habit group, current smokers of cigarettes, was defined so as to include both manufactured and hand- rolled cigarette smokers. This was to bring the data in line wlth that from the three prospective studies, in all of which'such a definition had been used. Next we wished to check whether the percentages derived could be takau as validly applying to the whole of the age r~lge they contain. Lee does not give information directly on the distribution of the population by smoking habits and flve year age ranges but some information can he gained on this indirectly from Table 14M which gives the annual consumption of manufactured cigarettes per adult by 5 year ago groups and Table 17M which gives the annual consumption per smoker. Averaging the figures given for each year from 1971-75 in Table 14M and dlvidin~ by the corresponding averages for Table 17M gives estimates of the proportion of the male population smoking manufactured cigarettes as follows. Age group Percentage of manufactured cigarette smokers 35-39 50.6% 40-44 50.5% 45-49 54.0% 50-54 50.7% 55-59 45.4% 60-64 45.4% 65-69 44.5% 70-74 37.9% 75-79 30.8% 80+ 25,3% It can be seen that, within the 35-59 age range, the'percentage of manufactured cigarette smokers is fairly constant within each 5 year age group. For this age range therefore we propose to assume that the dlstribution of the population by the 4 sm0klng~ablt groups given above applies to each 5 year age group within it, ignoring variations which are essentially minor. ~l~.k [~k~i~i J For the 60+ age range, however, it is quite clear that the per- centage of manufactured cigarette smokers, and hence presumably that of all cigarette smokers, decreases steadily with ~a~. Within this age range we therefore, estimated the percentage of all cigarette smokers separately for each 5 year age group by multiplying the above figures by the ratio 49.5/42.0 observed for the ratio ~ of all cigarette smokers/ % 0£ manufactured cigarette smokers for the GO+ age group as a whole. Q C~

Rounding the answers to the nearest }% this gives Estimated percentage of Age ~roup current cigarette smokers 60-64 53.5% 65-69 52.5% , ~m 70-74 44.5% • "/~'t • 75-79 36.5% - ~o 80+ 30.0% Next, as 66% of the 80+ age group are aged 80-84, and as it is likely that a much higher proportion than this were actually inter- viewed, it was decided to take the estimate of 30.0% given in the table above as applying to the 80-84 age group. Estimates of the percentage of current cigarette smokers for ages 85-89 and 90+ were made by extrapolation, assuming the proportional reduction in per- centages of current cigarette smokers observed over the age range 65-69 to 80-84 persisted. This gave figures of 25.0~ for 85-89 end 21.0% for 90-94 year olds. Finally it was assumed that the reductions observed in the per- centage of cigarette smokers were balanced by corresponding increases in ~he percentage of ex-smokers, with percentages of never smokers and pipe and/or cigars only remaining constant throughout the 60+ age range. This resulted in the figures presented in Table 6. It should be noted that, for our purposes, quite marked inaccuracies in the estimates of the relative proportion of ex-smohers, never smokers and pipe and/or cigar smokers at older ages wil~ make little dlf£erence to our conclusions as the relative risk observed in these three groups is not very different (see Table 4). The essential is that the pro- portion of current cigarette smokers should be reasonably accurate, and-it seems likely this is so. + o O Q~ Cr~