Methods for age-adjustment of rates

zy zyx zyxwvu STATISTICS IN MEDICINE, VOL. 2,455 466 (1983) METHODS FOR AGE-ADJUSTMENT OF RATES HAZEL INSKIP, VALERIE BERAL AND PATRICIA FRASER Epidemioloyical Monitoriny Unit. London School OJ Hyyiene and Tropical Medicine, Krppel Street (Cower Street) London W C I E 7HT, U.K. AND zyxwv zyxwvu JOHN HASKEY* Medical Statistics Division. Ofice of Population Censuses and Surveys. St Catherines House. 10 Kingsway. London WCZB 6 J P . U.K. SUMMARY Different age structures in two populations complicate any comparison of their levels of mortality. Many methods exist which provide death rates or mortality indices adjusted for age and other factors. Such summary measures inevitably lose information, but they are useful for the initial examination of large quantities of data and for the presentation of results. This paper reviews a number of techniquesavailable for producing age-adjusted death rates or mortality indices, emphasizingtheir historical development. Formulae are given for their calculation. The appropriate context for using each method, and its associated disadvantages are described. K E Y WORDS Standardization Age-adjustment Age-specific rate Mortality index INTRODUCTION As early as 1662, John Graunt successfully used quantitative methods in an analysis of the numbers of deaths recorded in the unpublished weekly Bills of Mortality.' He considered the proportion of all deaths due to certain specific causes to assess the relative impact of those diseases. Few developments in his techniques occurred until William Farr began work in the Office of the Registrar General for England and Wales in 1839. By that time, age at death was recorded routinely on death certificates and in the Registrar General's report for 1841 the importance of knowing the size of population at risk in each age group was emphasized.* Using this information, age-specific death rates were calculated for the first time. These rates were then compared with those for the preceding three years to show how mortality in the different age-groups had changed. To this day, comparison of age-specific rates is still the most comprehensive and reliable way of comparing mortality in different populations. Unfortunately, when many age-groups and populations are to be examined, this method become so cumbersome as to be unhelpful, and since 1841 a number of methods have been developed to summarize population mortality in a single measure. These measures take account of the different age structures in the populations and some can be used to adjust for other factors such as social class or area of residence. In most of the procedures, z zyxwvut * Present address: Cabinet Office, Central Statistical Office, Great George Street. London SWlP 3AQ. U.K 0277-67 15/83/040455-12$01.20 0 1983 by John Wiley & Sons, Ltd. Received 30 December 1981 Revised 5 April 1983 456 zyxwvutsrq zyxwvu zyxwvu HAZEL INSKIP ET A L . comparisons are made between the study population and a reference or standard population giving rise to the term 'standardization' for such techniques. This paper reviews some of the methods that have been suggested for summarizing a set of mortality data. These summary measures take the form of a rate adjusted for age, or an index (or ratio) expressing the mortality in one population relative to another. The formulae for most of the rates and indices discussed below appear in Tables I and 11. With few exceptions the rates can be expressed in the form Zwimi and the indices as Z ( w i m i / M i ) / Z w where i mi and M i are the death rates in the ith age-group in the study population and standard population, respectively, and wi is the weight assigned to the ith age-group by the particular method of standardization. An index expressed in this way is sometimes known as an 'average of ratio^'.^ Some of the indices are more zyxwvuts Table I. Age-adjusted death rates Weight wi Title Crude death rate Indirect method Direct method Comparative mortality rate Life-table death rate (Brownlee) Equivalent average death rate (Yule) Cumulative rate (Day) Formula Pi - P MPi ZMiPi ZMpimi - M d -ZpiMi ZpiMi zyxwvut Z Pi mi P Pi - ~ P i('+) 1 p. P. :C(:+:)m, zyxw zyxwvu Z ni Znimi __ Z ni "i Znimi "i - p, = population in age group i in index population; p = X p i P, = population in age group i in standard population; P = ZP, d, = deaths in age group i in index population; d = Zd D, = deaths in age group i in standard population: D = EDi 4 zyxwvut m,= - = death rate in age group i in index population P, D M , = -1 = death rate in age group i in standard population PI m = d/p M = Dfp n, = number of years in age group i li = number of people alive at age i as determined from a life table e: = expectaiion of life at birth as determined from a life table zyxw zyxw zy zyxw zy 457 METHODS FOR AGE-ADJUSTMENT O F RATES Table 11. Mortality indices Title Weight wi Standardized mortality ratio (SMR) MiPi Comparative mortality figure (CMF) Di Formula z mi Pi D Zmi Pi Fisher’s ‘Ideal Index’ J & X T Comparative mortality index (CMI) Znimi ___ Zn, M i Yule’s index zyx Znimi/Mi Yerushalmy’s relative mortality index Zn, Pi mi Liddell’s relative mortality index CPM. Relative risk index zyxwvut zyxwvuts Zciai - d ZCiai ZCiai Proportional mortality ratio (PMR) Mortality odds ratio (MOR) Kerridge’s inverse method Zmipi(70-hi) Z Mipi(70-hi) Person-years of life lost pi. P i . mi, Mi, 4 , D i , p. P, d, D,ni as in Table I hi = midpoint of the ith age interval ai = number of deaths from all causes in the ith age group zyxwvuts ci, Ci =proportion of all deaths in the ith age group due to the cause under consideration, in the index and standard populations respectively. Thus ci = 4 / a i a; =number of deaths from all causes other than the specific cause of interest, in the ith age group. Thus a; = ai-4 c; = di/a; Notes: ( I ) For the person years of life lost index the sums are only evaluated for positive values of 7@h, (2) For the PMRand MOR.4.refersonly todeathsfrom thespecificcauseofinterest. Forall theother indices4canrefer to a specific cause or to all causes combined. easily represented in the alternative form of ‘ratio of averages’ ( Z w i m i ) / ( Z w i M ibut ) since they can also be presented in the ‘average of ratios’ form, for consistency they are given as such throughout Table 11. Formulae for the standard errors associated with each method are not given here but have been discussed by Chiang4 and key fit^.^ A general formula for the standard error of an age- 458 zyxwvutsrqp zyxwvu zyxwv HAZEL INSKIP ET AL. adjusted death rate applicable to many of the methods described below is also given by Chiang.4 INDIRECT AND DIRECT STANDARDIZATION Reference to a ‘standard population’ first appeared in the Registrar General’s report for 1853.2 The standard population chosen was a set of ‘healthy’countieswhere the crude death rate was less than 17 per 1O00. The crude death rate in a study population was compared with that in the standard population, without taking the age structures of the populations into account. In a letter to the Registrar General four years later, William Farr combined the use of age-specific death rates with the concept of the set of ‘healthy’ counties as a standard population.’ He applied the age-specific death rates in the standard population to populations of other counties and thus obtained the first indirectly standardized death rates. Indirect standardization and its corresponding index, the Standardized Mortality Ratio (SMR), were the only methods of age-adjustment used until 1883 when the direct method, attributed to Ogle, was introduced in the Registrar General’s report.’ In this method of standardization the agespecific death rates in the study population are applied to the numbers in the appropriate agegroups in the standard population. The 1884 report contained an application of the direct method to cancer mortality, this being the first time that such techniques were applied to a specific cause of death.’ The Comparative Mortality Figure (CMF), the index obtained by the direct method, also came into use at that time. Direct and indirect Standardization are, to this day, the most commonly used techniques for summarizing rates and comparing populations. Both methods have advantages and disadvantages and in a given situation one may be more appropriate than the othef. To compare a number of study populations, each standardized against the same standard population, the direct method is advocated because it preserves consistency between the populations. Thus, if each age-s#ecific rate in study population A is greater than the corresponding rate in study population B, then the CMF and directly-standardizedrate in A will be greater than in B, irrespectiveof the standard population employed. This important property has led to the widespread use of the direct method in the United States, notably by the National Cancer Institute in studying the geographical distribution of various cancers.6 The International Agency for Research on Cancer has also used the direct method most effectively for making international comparisons.’ Consistency is not necessarily preserved by indirect standardization, which can, in certain extreme situations, give misleading results. If, however, the age-specific death rates in the study population are unknown or not available, the direct method cannot be employed. The indirect method requires knowledge only of the population at risk in each age-group and the total number of deaths in the study population, and so can sometimes be used when the direct method cannot. In addition the indirect method has the advantage of a low standard error. The maximum likelihood estimate, which inversely weights the age-specific mortality ratios by an estimate of their variance, provides an index with minimum variance. The SMR is the first approximation to this estimate, thus giving it a small standard error. Small numbers of events in the study population leading to unstable death rates may result in large errors of estimation with direct standardization. When studying geographical and occupational variations in mortality within Britain, the number of deaths in each category can be small and for this reason, indirect rather than direct standardization has been widely employed in these analy~es.’.~ Both forms of standardization can also be used in the analysis of mortality data by birth cohort. Cohort rates are used instead of period rates but otherwise the methods are the same as for the CMF and SMR. The resulting indices have been termed Standardized Cohort Mortality Ratios.”. zyxw METHODS FOR AGE-ADJUSTMENT OF RATES zyxw zy 459 VARIANTS OF DIRECT AND INDIRECT STANDARDIZATION zyxwvut Fisher’s ideal index The geometric mean of the SMR and CMF was proposed by Fisher” in 1922 as an ‘Ideal Index’, since it satisfied two criteria in which economists were interested at that time. Only one of these criteria is relevant in dealing with medical data, namely that the index for population A with population B as standard should be the inverse of the index for B standardized against A. Fisher’s index does not possess the consistency property of direct standardization, but the inverse property is useful when comparing two populations, neither of which obviously constitutes the ‘standard‘. Such a situation arises when sex ratios within a population are examined. If rates for males are standardized using the female population as the standard then the resulting value of the ideal index is the reciprocal of that obtained when the data for the two sexes are interchanged. If a method is used which does not possess this inverse property then some other standard has to be used, possibly the combined population, against which rates for either sex can be standardized. Fisher’s index has been little used, except by economists, mainly because it is somewhat unwieldy. Nowadays, with the availability of computers, this is a minor problem and the ideal index could have a wider use. The index, being a combination of the SMR and CMF, also has a certain intuitive appeal when a difficult decision arises as to which is the more appropriate method of standardization. Comparative mortality index I Another index possessing the inverse property was introduced in the Registrar General’s report for 1941.” This was the Comparative Mortality Index (CMI)which was used until 1958. At the time of its introduction there had been considerable discussion about the most appropriate standard population to be used for direct standardization. The 1901 census population of England and Wales had become an established standard and there was reluctance to change, as the previous CMFs would have to be recalculated. However, the direct method‘s lack of the inverse property, had led to marked inconsistencies between CMFs calculated using the 1901 standard and those employing a standard population from more recent years. For example, the CMFs calculated using the 1901 standard showed a decline in mortality between 1901 and 1939 of 50 per cent, but when the 1939 population was used as the standard the decline was only 38 per cent. Similarly a comparison of 1939 with 1938 using 1901 as standard, showed a slight decline in mortality by contrast to the increase in mortality indicated by CMFs calculated using the average 1938-1939 population as standard. The problem concerning the inverse property was resolved by the introduction of the CMI, and the difficulty over the choice of standard population was reduced since the ratio of two CMIs (calculated using the same standard) is approximately the CMI of one population standardized against the other. The standardized rate analogous to the CMI is the average of the crude and directly Standardizedrates. The method has the advantage of being easily calculated and its standard error can be readily obtained. The possession of the inverse property makes it useful in the same situations as Fisher’s ideal index. Person-years at risk For the analysis of longitudinal or follow-up studies, the use of another variant of the SMR has become widespread. The number of person-years at risk contributed by each member of the study population to each age group and calendar year period is calculated. The total person-years at risk in each category for the entire study population is then obtained and rates from the standard zyxwvu zyxwvu zyxwv 460 HAZEL INSKIP ET ,415. population applied to obtain the total number of expected deaths. This procedure is discussed in detail by in a description of the man years computer language, one o f a number of computer programs available for performing these calculations. The observed and expected deaths can then be compared and their ratio forms a mortality index which can be interpreted in a similar way to the SMR. An investigation of lung cancer in asbestos workers by Doll in 1955 was one of the earliest studies to employ this technique, and the method of calculation of expected deaths is clearly explained in his r e p ~ r t . In ' ~ 1956 a study by Doll and used a similar method but, having obtained the person-years at risk, rates in the study population were calculated and applied to a standard population. This is analogous to direct standardization but it is a feasible method only when the study population is large enough to provide stable rates in each age group and calendar year period. ALTERNATIVE METHODS OF AGE-ADJUSTMENT Life-table death rate In 1922, Brownlee challenged the established methods of standardization on the grounds that when the birth rate exceeded the death rate the standardized methods underestimated the risk of dying. Instead he advocated the use of the life-table death rate, calculated as the inverse of the expectation of life obtained from a life table.' 'Life tables assume a stationary population where the number of deaths equals the number of births. The life-table death rate is not strictly a method of standardization because no standard population is used. The age structure of the study population, however, is taken into account and such rates can be used in the comparison of different populations. In a discussion of the value of life tables held at the Statistical Society, Brownlee's ideas were strongly disputed.'* An attempt to adjust for population growth was considered artificial and, since life table populations are entirely hypothetical, the consensus was that such a form ofage-adjustment was impractical. Yule" commented that ifall theage-specificdeath rates in the population were to double the life table death rate would not reflect this, whereas a standardized rate would. This was a marked disadvantage of the life table death rate at that time and it is not surprising that it has rarely been used. The method might, however, be viewed more favourably now that the population of England and Wales is relatively stable, the crude death and birth rates being approximately equal. With the wider use of life tables, the life table death rate could easily be derived and might prove a useful measure. Equivalent average death rate Following his criticisms of Brownlee's suggestions, Yule presented another method of standardization to the Royal Statistical Society in 1934.3 He suggested that two criteria should be met by any method of standardization. The first of these was a variant of the consistency property, and the second was that the ratio of the indices obtained for populations B and C standardized against A should equal the index for B standardized against C. Although the direct method of standardization satisfies the first criterion, neither the direct nor indirect methods satisfy the second. Yule's idea was to average the age-specific death rates weighted according to the width of that age-group in years. The resulting Equivalent Average Death Rate (EADR) is independent of any standard population although the index, being the ratio of two such rates, obviously requires a second population. This method satisfies both of Yule's criteria but has two inherent difficulties. The first is that an arbitrary upper age limit has to be imposed, although in examining occupational mortality for which Yule proposed this index, the problem does not arise. The second criticism of the method METHODS FOR AGE-ADJUSTMENT OF RATES zyxw z 46 1 zyx is that, in giving equal weight to each age, it in effect over-weights the oldest age groups in which the population is smallest but the death rate is highest. Another way ofconsidering this method is as a form of direct standardization using a standard population with equal numbers at each age. Viewed in this way, the unreality of the index is more apparent as such a standard is totally hypothetical.The overwhelmingadvantage of this method, however, lies in its simplicity. If the agespecific death rates are available then the EADR can be calculated rapidly. Yerushalmy’s relative mortality index Another method which used the widths of the age groups as weights was suggested by Yerushalmy in 1951.19 His relative mortality index is a weighted average of the ratios of the age-specific death rates in the index and standard populations. It gives equal weight to equal proportional increases or decreases in mortality in the study population compared to the standard in each age group. One consequence of this is that the mortality of the younger age groups receives greater emphasis, although it is smaller in magnitude. The method thus avoids the problem of giving more weight to deaths at older ages and theconcept of an average of the relative differences in mortality at different ages is not difficult. This index, being independent of the population size in each group, can give, however, a misleading impression of the importance of a disease. Liddell’s relative mortality index A similar idea was proposed by Liddell in 1960” but instead of using a uniform standard population distribution giving weights according to the widths of each age group, he used the size of the standard population. This formulation overcomes the objection to Yerushalmy’s index, but simultaneously destroys the desirable property of equal relative differences in mortality in the different age groups having equal influence on the index. Liddell proposed his index for use in measuring occupational variations in mortality. Relatively larger numbers of errors occur in the recording of occupation at older ages so that a method which gives greater weight to data for these age groups would exaggerate the bias. Liddell’s method, on the other hand, by giving more emphasis to the younger age groups is of particular use in reducing the influence of this recording bias. The cumulative rate Recently an alternative to an age-standardized rate for examining cancer incidence has been proposed by Day.’ This measure, known as the cumulative rate, is the sum of the age-specific incidence rates over each year of age between 0 and 74 expressed as a percentage. Effectively this measure is the same as the numerator of Yule’s equivalent average death rate but the interpretation of the two methods differs. Yule’s method provides a rate adjusted for age which can be used for comparison with similarly standardized rates for other populations, whereas the cumulative rate is a close approximation to the cumulative risk of developing the disease between birth and age 74, assuming that there is no other cause of death. The cumulative rate thus has an intuitive appeal and can also be combined with an estimate of relative risk from other studies to obtain cumulative risk for a subgroup of the population exposed to the risk factor of interest. zyxwvutsr Relative risk index Lilienfeld and Pyne proposed an index of this name in 1979.’’ The weights (wi) used are identical to those proposed by Cochran for weighting differences in proportions when pooling data from a number of 2 x 2 tables.” For the relative risk index, the weights are applied, as in other zyxwvu zyxwv zyxwv 462 HAZEL INSKIP ET ,415. zyxwvutsr standardization methods, to the ratios of the death rates between the study and standard populations in each group(rni/Mi). Ratios of this form, when obtained from epidemiological studies in which death rates can be calculated, are commonly called relative risks, giving rise to the name of the index, which combines the relative risks of each age group. The advantages of this index lie in its similarities to methods for relative risk used in epidemiology and to Cochran’s methods for 2 x 2 tables. One disadvantage of the method, however, is that the weighting procedure does not have an intuitive meaning. METHODS FOR USE WITH INCOMPLETE DATA Proportional mortality ratios Sometimes the data available are too limited to enable any of the methods discussed so far to be used. Such a situation arises, for example, when the size or structure of the base population in which deaths occur is unknown or not clearly defined. John Graunt,’ in his early analyses of the Bills of Mortality was in this position, and nowadays certain studies of occupational mortality can be as well. The occupation and employment status recorded on the death certificate, as stated by a relative or friend, especially for retired people, may differ from that given by the deceased at the previous census. Thus unbiased death rates by occupation or social class are difficult to estimate. Using the proportion of all deaths due to the specific cause of interest in place of the death rate is one way of overcoming this difficulty. Doll used a method of this type in his study of cancers of the ~ lung and nose in nickel workers where the size of the population at risk was ~ n k n o w n . ’The mortality index formed in this way is known as a proportional mortality ratio (PMR). The danger inherent in examining proportional mortality is that a deficit of deaths from one cause may give the erroneous impression of an excess from another. Kupper et aLZ4have shown, however, that the PMR is approximately the ratio of the cause-specific SMR to the all-cause SMR. They called this ratio a relative SMR (RSMR)and demonstrated that using the PMR it is possible to obtain confidence limits for the RSMR. In occupational studies the healthy worker effect often reduces the all-cause SMR calculated using the national population as standard. This phenomenon is due to healthy individuals being selected into and remaining in certain occupations while unhealthy people gravitate to others or become ~nemployed.’~ By contrast, the RSMR and PMR do not suffer from this disadvantage. Thus PMRs, if interpreted with care, can be useful indices in occupational and other similar studies. They have been used for example in the last Decennial Supplement on Occupational M ~ r t a l i t y . ~ Mortality odds ratio An alternative to the PMR, termed a Mortality Odds Ratio (MOR)was proposed by Miettinen and WangZ6Their criticism of the PMR was that it is often interpreted as a cause-specific SMR and to do this involves certain assumptions that are rarely valid. The MOR considers the ratio of deaths from the cause of interest to deaths from other causes. This ratio is equivalent to the odds ratio used in casecontrol studies, the analysis of which is well-do~umented.’~* 2 8 When the proportion of deaths due to the cause of interest is small and the number of deaths from all causes is large, as can occur in occupational studies using data from national statistics, the MOR and PMR hardly differ. In such situations, the PMR is usually preferred because of ease of calculation. Inverse standardization When the age structure of the index population is unknown but the total number at risk is available a method suggested by Doering and Forbes in 1939 can be used.29Their suggestion went largely METHODS FOR AGE-ADJUSTMENT OF RATES 463 unrecognized and it was developed independently in 1958 by Kerridge who called it inverse ~tandardization.’~ This method used the number of deaths in the index population divided by the death rate in the standard to estimate the population at risk in each age group. The total of these estimated populations at risk gives an estimate of the total number at risk. This estimate, divided by the known total number at risk, gives the mortality index. Kerridge showed that it was unbiased, and gave a formula for its standard error. Since its standard error is usually greater than that of the SMR, Kerridge recommended its use only when other techniques cannot be employed. YEARS O F LIFE LOST z zy The majority of the methods discussed above give greater weight to deaths occurring in the older age groups, where the death rates are highest. Since the death of a young person causes greater concern than that of someone who has already lived for many years, a number of methods using weights based on the number of years lost (measured up to a specific age) have been suggested.”-34 The basic form of years of life lost (YLL) can be expressed as Zdiyi where diis the number of deaths in the ith age group and y iis the number of years of life lost by an individual who dies in the ith age group. Two basic forms of y iare widely used. The first, and simplest, is to consider the number of years of life lost prior to a specificage, say 70. Deaths above this age are ignored and yiis calculated as the difference between 70 and the mid-point of the ith age interval. The upper age limit is chosen as appropriate to the study concerned, for example in studies of occupational mortality, the retirement age is usually chosen. The biblical figure of three score years and ten for a lifespan is also commonly used. Logan, in a study on changes in years of life lost, chose the age above which only ten per cent of the population survive.’’ As well as selecting an upper age, a lower limit can be imposed. Thus, infant mortality may be ignored for certain purposes, the argument being that an infant’whodies is often ‘replaced’by another ~ h i l d . ~ ~ D e a t hages s a t below 15 may be disregarded in occupational studies. The second form of y i requires the use of life tables for a standard population. Again there are numerous varieties, the simplest being the difference between the expectation of life at birth and the mid-point of the ith age interval. The expectation of life for people in the ith age group is often used as is the average number of years to be lived prior to a specific upper age. The total number of years of life lost can only be used for ranking causes of death within a population. Creating a rate by dividing by a suitable denominator provides a statistical measure which is more readily interpreted. Denominators which have been suggested are the total number of individuals in the population, the potential total years of life in the population and the total number of deaths. To make valid comparisons between groups of rates based on numbers of years of life lost, the different age structures of the populations must be taken into account. As has been discussed previously, standardized rates are required. Haenszel was the first to propose a suitable method, analogous to direct standardization of death rates.” Although he described his method differently, his procedure is to apply the direct method of standardization using the rate based on YLL in the index population instead of the death rate. In a similar way the indirect method can also be used by replacing death rates in the standard population with rates of YLL. Although rates of YLL are more easily interpreted than total YLL, a mortality ratio based on YLL often facilitates comparisons between populations. There are several such ratios which can be computed in the same way as the SMR and CMF, rates of YLL being used instead of the usual death rates. A value of the index greater than one implies that the study population has a higher level of mortality in terms of years lost than the standard population. The index discussed by 464 zyxwvutsrqp zyxwvu zyxwv zyxwvu HAZEL INSKIP ET A L K1einmat-1~~ is the only form of YLL index presented in Table 11, but as discussed above, a number of other possibilities exist. YLL rates and indices are particularly appropriate for investigating causes of death which occur at young ages. For example, using these methods, accidents are identified as a leading cause of mortality. When used in conjunction with the methods using death rates described above, the technique of YLL offers an additional interpretation of the data which can be particularly useful for health planning. The disadvantage of the methods is that giving greatest weight to age groups in which the number of deaths is smallest, leads to larger standard errors than for many other methods of standardization. DISCUSSION AND CONCLUSIONS It has long been recognized that valid comparisons between populations can only be madeif the confounding effect of age, and other factors, is taken into account. The many methods described in this paper enable such adjustments to be made. The choice of method depends on the way in which comparisons are to be made and on the data available. If data are incomplete, inverse standardization or PMRs may be appropriate, whereas if the specific cause of interest affects people at younger ages a YLL method may be useful. Each method has disadvantages which must be examined for their effect upon the results in each particular case. The consistency or inverse properties may be of importance in one particular situation, whereas a low standard error may be more important in another. Further detailed discussions of the application of some of the standardized rates and indices described in this paper have been given by Woolsey,” K i l p a t r i ~ k ,3~9 ~Kitaga~a?’.~’ . Shryock and Siege14’ and A l d e r ~ o n . ~ ~ It is important to recognize that any summary measure can give misleading results if there are large variations in the ratios of the age-specific rates between the populations. In such circumstances the values of the standardized rates or ratios depend more on the weighting factors used in the particular method of adjustment than upon the rates in the index and standard populations. None of the methods mentioned in this paper can improve upon direct comparisons of the age-specific death rates. The techniques do, however, offer a means of examining large amounts of data with relative ease and provide summary measures which have an immediate interpretation. They are useful for identifying areas of concern requiring further study and for presentation of data in a form which can be readily appreciated. ACKNOWLEWEMENTS We are grateful for the help given by Helen Edwards in typing the many drafts of the manuscript and to the referees, friends and colleagues for helpful comments and suggestions. This work was supported in part by a grant from the Medical Research Council. zyxwvut REFERENCES 1. Graunt, J. Natural and Political Observations made upon the Bills of Mortality, London, 1662. Republished by the Johns Hopkins Press, Baltimore, 1939. 2. General Register Office.Annual Report ojthe Registrar Generalfor England and Wales, 1841,1853,1857, 1883, 1884. HMSO, London. 3. Yule, G. 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