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-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