Foreword The compilation of mortality tables using death flows by sex and age and the population rate resulting from the census figures classified in the same manner, is traditionally performed by the National Statistics Institute. At present, users can now access mortality tables for the population in Spain calculated using the deaths occurred in 1998 and 1999 and of the population forecast for 1998. As the methodology used does not differ from the previous one, the biometric functions obtained are comparable. Carmen Alcaide Guindo INE President Index Foreword 3 Historical background 7 Methodology employed 9 1 Introduction 11 2 Obtaining death probability series 11 3 Obtaining derivative series 12 4 Obtaining prospective series 13 5 Synthesis of the smoothing procedure employed 14 Base information used 15 Results obtained 17 1 Evolution of mortality during the 1980s 19 2 Mortality tables for 1998 20 3 Spanish mortality in the European Union 21 Charts and graphs 23 Complete mortality tables 19981999 35 Glossary of symbols 37 6 Complete mortality tables 1998-199938 Historical background the same (therefore, they do not refer to the studies corresponding to abbreviated tables, or those related to smaller population spheres, like Autonomous Communities and provinces). In 1945, the National Statistics Institute published mortality tables regarding the Spanish population, by sex, calculated using the deaths by age in 1930 and 1931 and the population in the 1930 Census. By doing so, the INE filled the gap that had existed in Spanish statistical research until said year. Moving on, the calculation of the forecast values for the biometric functions of the Spanish population is a relatively recent project, which was implemented to obtain figures for future survivors considering the population forecasts calculated using the 1970 Population Census, and the last forecasts, compiled using the 1991 Population Census. Six years later, with the results from the 1940 Census, tables were created for said year, and the figures from the Censuses of 1900, 1910 and 1920 were also used (after processing them appropriately) as the denominator for deaths by sex and age of the Vital Statistics for the corresponding years, so as to establish the biometric functions that were published alongside those from 1930 and 1940. This provided series for the 1900-1940 period, which allowed the INE to analyse the evolution of general mortality, by sex, in the Spanish population. Finally, the functions of the mortality tables for the Spanish population 1998-1999, analysed in this publication, have been calculated taking forecast population figures, and not observed data (like census information), as the denominators.1. The advantage of this procedure is that it continuously reflects modifications regarding the intensity of the mortality rate, each time new data on deaths is provided. Nevertheless, the tables obtained should be revised to ensure that the populations used to calculate the data are corrected, either in terms of the number of persons by sex and age resulting from a subsequent comprehensive population count, or as a consequence of the errors in the hypotheses devised regarding demographic components when establishing the corresponding forecasts. The INE used the data from the 1950 Census, to compile the tables for said year, following a process identical to the procedure used as the base for the first five tables, consequently, completing the information provided in the abbreviated tables referred to said year, which had been disseminated two years before. The population rates obtained in the 1960 and 1970 Censuses and the figures concerning deaths occurred during those decades, allowed the calculation of new mortality tables for the Spanish population. The following page includes the evolution of life expectancy at birth, for the total population and by sex, throughout the century. Using the aforementioned tables for 1970, the National Statistics Institute has published mortality tables every five years, using the populations by sex and age from the municipal recounts, as well as the census figures. The methodology used enables the comparison between the results for 1975, 1980, 1985 and 1990, always considering the changes in the concepts used which, in any case, are indicated in the publications (as occurs with the concept live birth and still birth which appears in the year 1975). This has resulted in the dissemination of comparable figures since –although the methodology varies as regards the smoothing procedures, which adapt to each situation– the definitions for the basic biometric functions calculated for the whole of the population are 1 Forecast population figures were already used to calculated the tables for 1994-1995 and 1996-1997. Methodology employed 1 Introduction Mortality tables are compiled to measure the incidence of this phenomenon on the population under study, regardless of the structure by age. The type of table used is created after performing a transversal analysis of the mortality, examining how said phenomenon affects the population classified by age or age groups, at a certain moment in time. Given the evolution usually experimented by mortality, that does not present any brusque modifications, these tables appear as an acceptable description of the phenomenon for short periods of time, close to the moment they refer to. To calculate the functions of a complete mortality table, it is necessary to have information on the deceased and the population classified both by ages and referred to the same time period. Since the figures for deaths classified by ages are quite small (except for the oldest age groups), not only in provinces and Autonomous Communities but also on a national level, recount errors and possible disruptions, which could exceptionally affect mortality in a certain year, have a notable bearing on this information. Consequently, it is necessary to eliminate these anomalies since if they were to remain in the data, they would present an incorrect image of the phenomenon under study. This elimination is performed during the initial stage. To calculate the mortality table for a certain moment in time and for each age group, it is necessary to consider average deaths corresponding to a specific number of years (generally from two to four), focusing on that particular moment. In a second stage, it is necessary to eliminate the disruptions, both in terms of the number of deaths and the population, caused by errors when stating the age, and produce an increase of the values observed for certain ages to the detriment of the contiguous ones, distorting the series of death probabilities on the mortality table. This problem is usually avoided applying a smoothing procedure to the original data. 2 Obtaining death probability series Death probability at age x, qX, is defined as the probability a person from a specific generation, exactly x years old, has of dying before reaching age x+1. Therefore, it is necessary to consider possible death cases, in other words, persons who could die, as well as real events, that is, persons of that age and generation who have actually died. Possible cases are persons who are x years old, calculated as the sum of the inhabitants who are that age at the end of the year and half of the persons deceased aged x during the year in question, since it is supposed that deaths are distributed uniformly throughout the year in question. Accepting the hypothesis that the deaths of persons from a certain generation aged x occur half in one year and half in the following, the death probability would be expressed by: 1 / 2 (Dzx + z+ 1 Dx ) qx = z z Px + 1 / 2 (Dx) where: Dzx represents deaths occurred in year z aged x. Dzx1 represents deaths occurred in year z+1 aged x. Pxz population on December 31st of year z aged x. The previous expression has been used to calculate all qx corresponding to all ages ranging from two to ninety years old, both inclusive. Since the deaths of babies under one year old mainly occur during the first weeks of life, it is not possible to apply this hypothesis uniformly throughout the year. Therefore, for this age, the death probability has been calculated using: z q0 = z+ 1 D0, g(z) + D0, g(z) z z P0 + D0, g(z) where: Dz0 ,g(z) deaths occurred in year z, 0 years old, from the generation born that year. D z1 0 ,g(z) deaths occurred in year z+1, 0 years old, from the generation born the previous year. 3 Obtaining derivative series The death probability series can provide the mortality tables functions described hereunder. P0z population on December 31st of year z PROBABILITY OF LIFE OR SURVIVAL AT AGE x, aged 0. Consequently, for babies aged one year old, q1, has been calculated using: z q1 = z+ 1 D1, g(z-1) + D1, g(z-1) z z P1 + D1, g(z-1) px The probability of survival between two exact ages. Therefore, for each age x, px = 1 - qx where: D1z,g(z1) deaths in year z, aged 1, from generation z-1. D1z,g1(z1) deaths in year z+1, aged 1, from SURVIVORS AGED x YEARS OLD, lx Number of persons aged x among the initial l 0 on the mortality table. Therefore, for each age x, generation z-1. lx = lx-1px-1 P1z population on December 31st of year z Surveys usually work with l0 = 100,000 aged 1. The low number of deaths registered for persons who are over ninety years old and the greater repercussion of errors when stating the age, lead to distortions in the death probability series for the aforementioned ages. Therefore, the latter have been estimated adjusting a third grade parabola, by least squares, based on the qx calculated using the previous expression, for X = 90, 91, 92, 93 and 94. The following conditions were established in order to perform said adjustment: a) The cubic parabola passes through point q90, which implies the continuity of the qx adjusted and those calculated for ages under 90 years old, b) value q110 = 1, truncating the parabola as from this point, which means that, a priori, there are no survivors over one hundred and ten years old, and c) the cubic parabola has a tangent parallel to the x axis at point x = 110, which implies an accelerated increase of mortality as from the point of inflection of the cubic parabola, given high mortalities for ages around one hundred and ten years old. THEORETICAL DEATHS AGED x YEARS OLD, dx Deaths occurred between two exact ages x and x+1, obtained from the mortality table. L0 a0 l 0 a1l1 , where a0 a1 1 in which z+ 1 D0, g(z) z+ 1 z+ 1 D0, g(z) + D0, g(z+ 1) Therefore, for each age x, a0 = d x l x q x l x l x 1 where Dz0,g1(z) represents the deaths of children under LIFE EXPECTANCY AT AGE x, ex Average number of years each person aged exactly x is expected to live, for survivors that reach said age, under the supposition that the years lived by all persons are the same for all of them. Considering the hypothesis that all persons who die at a certain age live, on average, half the year in which they die, life expectancy is calculated as ex = 1 2 + 1 lx For x = 99 and x = 100 L99 = e99 l99 - e100 l100 L100 = e100 l100 where L100 are survivors aged 100 years old and older. l PROBABILITY OF SURVIVAL AT x YEARS OLD, TX i i= x+ 1 with representing the oldest age, for which there are supposedly no survivors. 4 one year old occurred in year z+1 among those born in generation g(z). Obtaining prospective series As well as the previous classical biometric series or functions, it was considered of major importance to include the two prospective series specified hereunder. The probability of survival for ages x and x+1 for persons aged x years old. This is easily obtained from the former using Tx = and, for the population aged 99 years old and older, the probability of reaching 100 years old or over is T99 = SURVIVORS AGED x YEARS OLD, LX Represents the number of survivors on the mortality table who are x years old. The estimate of this function has been performed implementing this next formula (see Introduction to the Mathematics of Population. Keyfitz. Addison-Wesley): Lx = 13 24 (lx + lx+ 1) - 1 24 for x = 1, 2, ..., 98. For the remaining ages (lx-1+ lx+ 2) Lx+ 1 Lx L100 L99 + L100 5 Synthesis of the procedure employed smoothing Both population stocks obtained from population censuses and register renewals, and data on deaths obtained from the Vital Statistics, sometimes, contain mistakes due to flaws that appear when interviewees state their age. This increases the values of some ages to the detriment of those corresponding to similar ages, which causes distortions in the death probability series calculated. In order to avoid this problem, it is necessary to implement a smoothing procedure for the original data before employing them. The smoothing procedure employed for the original data was the Variate Difference Method. The National Statistics Institute had used said method to compile the former comprehensive mortality tables. A complete explanation of the application, with vast bibliography, can be found in the book by G. Tintner, The Variate Difference Method, 1940, in the Cowles Commission collection. The following paragraphs explain the foundations of the procedure briefly. The basic hypothesis for the application of the method is that the series observed is the additive superimposition of two other series, one of which expresses the correct value or the value expected for each age x, and the other the random distortion that alters the observed value. In this case, the latter would be the sum of all the causes and circumstances that lead to persons stating an incorrect age. Therefore, the model is: yx = ux + ex where for each age x: yx is the observed value. ux is the expected or correct value. ex is the error or random distortion. In this application, values ux theoretically follow a slow trend, without sharp zigzags, and random errors are supposedly unrelated. This hypothesis could be smoothed given the noncorrelation of the random errors. A second essential hypothesis, that has allowed the implementation of the Variate Difference Method, consists in supposing that the expected value ux is simply a grade n polynomial, when n is an unknown value. The Variate Difference Method determines the exact value of n. Subsequently, after obtaining n, a polynomial for said degree is adjusted to the data observed yx. In this respect, it is necessary to mention the existence of a close relationship between the moving average method and the variate difference method. Specifically, M..G. Kendall (A Theorem in Trend Analysis, Biometrika, vol. 48, 1961. Advanced Theory of Statistics) has proven that the moving average method calculations result from the application of the variate difference method based on a lineal combination of some of the successive terms of the observed values yx. More precisely, all moving average formulae result in an adjustment of 2K + 1 successive terms of a grade p - 1 polynomial, with 2K - p + 1 numbers bi (with p - K < i < K), so that k ûx = y x - p bi y x+ i i= p-k where: p difference of order p. ûx the estimated value of ux. bi coefficients of the Sheppard smoothing formula. If the expected value ux follows a grade n polynomial, it is a case of determining the latter. For this, an iterative process is implemented to calculate the successive finite differences. There will evidently be a point in the process when the expected value u x will disappear, on cancelling the grade n polynomial. That is to say, the corresponding difference will be constant, thus cancelling subsequent differences. Nevertheless, since calculations are performed with the observed values yx, it is necessary to know the moment at which the expected value has supposedly been cancelled in this process of successive finite differences, with only a residue remaining from the existence of random errors ex. This question can be answered using the following consideration: if there is a time series that only contains a random element, the variations of the successive series of finite differences are equal, after correcting them by multiplying a binomial coefficient given that the series, which is random, is not ordered in time. Consequently, the variation of the first and second differences is the same as in the original series. problem to be resolved. Nevertheless, the number of values included in each average should be taken respecting the length of the main cycle that is to be cancelled. In this case, moving averages have been taken considering each series of five consecutive observed values yx. The aforementioned provides a criterion to determine when the expected value ux has disappeared. If a certain difference k is calculated with variation equal to that of difference k + 1, and equal to that of K+2, etc., it is possible to say that the expected value u x has been cancelled, taking K-th difference. Nevertheless, the equation between two variations is never reached, since there is always a random variation residue. Yet since the table uses a probability method it is proven that the only necessary element is that the difference between the variation of two successive series of finite differences is smaller than three time the standard error of the lowest difference. To apply this aspect to the compilation of the mortality tables, the series of the expected values always disappears in the first or second differences. This implies a constant application of moving averages when smoothing original series. 6 The deaths used to calculate each table (males, females and total) have been obtained as an average of the figures registered by age in the Vital Statistics for 1998 and 1999. The irregularities in these original death figures, caused by the possible errors regarding the classification by age, have been cancelled using the smoothing procedure explained in the previous section. After determining the degree of the polynomial to be adjusted, it is merely a case of applying the corresponding weighted average to the coefficients of Sheppard's smoothing formula. The moving average type is determined as follows: if the non-random element or expected value ux is more or less cancelled in the first or second difference, the table uses n = 1 or a moving average that is equivalent to adjusting a straight line to a certain number (not determined by the method) of consecutive observed values yx. If the expected value is cancelled in the third or fourth finite differences, we will obtain n = 2, and select a moving average equivalent to adjusting a second grade parabola to a certain number of consecutive observed values. If the nonrandom element is cancelled in the fifth or sixth differences, n = 3, we use a weighted average equivalent to adjusting a third grade (cubic) polynomial to a selected number of consecutive observed values, etc. If the nonrandom element is cancelled in the k-th finite difference, n = k/2, when k is even, or n = (k +1)/2, when k is uneven. As aforementioned, moving averages are implemented on a specific number of consecutive observed values, which have been centred appropriately. Nevertheless, this number is undetermined. The criterion is open to the experience and specific nature of the Base information used The simple populations by sex and age on December 31st 1998 correspond to the revision of the population forecasts calculated using the results from the 1991 Population Census1, except for the population aged 100 years old and over. As regards the group aged 100 years old and older, of each sex, the figure used is obtained by lineal interpolation between populations aged 100 years old and older from the 1991 Population Census of 1991 and the 1996 Registry Renewal. The figures used appear, alongside the biometric functions of the mortality tables calculated for 1998, in the first two columns. 1 Forecasts for the Spanish population calculated using the 1991 Population Census. Assessment and revision. INE 2001 Results obtained 1 and 7 represent the evolution, from 1980, 1985 and 1990, of the number of survivors, for each sex, at the different ages, for an initial cohort of 100,000 persons in each group. Evolution of mortality during the 1980s The secular evolution of mortality in Spain has been characterised by the decrease of death rates and probabilities corresponding to each age. It is important to note that, opposed to the trajectory observed during the previous period, the survival curve for males corresponding to 1990 decreases rapidly for ages around 20 years old, reaching values under those registered previously and does not increase again until the section corresponding to persons over 60 years old. As regards females, the curve for said year coincides almost completely with that of 1985, without the favourable evolution of the previous quinquennium. The decrease noticed in Spanish mortality during the 1970s continued, albeit less intensely, during the first five years of the 1980s. Nevertheless, in following years, this favourable evolution has been accompanied by changing trends that have appeared among certain young persons (mainly males between 18 and 35 years old); an unfortunate novelty compared to previous years. Graphs 8 and 9 represent the series of theoretical deaths in the mortality tables for males and females for 1980, 1985 and 1990. The upper part of graphs 1 and 2 presents the evolution of mortality quotients by sex and age, between 1980, 1985 and 1990. The evolution of the mortality calendar in the 1980s results in a deceleration of the rhythm of increase of life expectancy, more marked for males than for females. In the early ages, under 15 years old, the trend is still favourable as regards general mortality. With reference to infant mortality specifically, the risk of death of children under 1 year old was 7.8 per thousand in 1990, 8.5 for boys and 7.1 for girls. Life expectancy at birth for Spain’s population in 1990 reached 76.9 years old, 73.4 years for males and 80.5 for females (i.e. women live about seven years longer). Consequently, male excess mortality continues on the rise. On considering the subsequent ages up to 40 years old, the recent mortality trend is unfavourable, with a significant increase between the ages of 18 and 35 years old. The graphs corresponding to males and females show that this situation is more frequent among the former, whilst the rates for females show a very slight decrease. On examining the origin of the increase of deaths of young persons, the three main causes of death are traffic accidents, AIDS and drug addiction. The evolution of the difference between life expectancy for females and males at each age, between years 1980 and 1990, is represented in graph 10, showing the upward trend of the corresponding line. In brief, chart 1 presents life expectancy by age, in years 1980, 1985 and 1990, for each sex and every five years for ages. For people over 40 years old, the increase as regards mortality in 1985 and 1990 is very similar to that of the previous quinquennium. Even when the previous graphs show that the evolution of mortality is more favourable for females than for males, graph 5 represents the quotients between the respective mortality risks by age, for 1980, 1990, 1994 and 1998. All values represented are higher than one. For persons over fifty years old, there is an upward trend of the corresponding curve over time. The quotients of mortality by age provide the distribution of survivors on the table. Graphs 6 2 Mortality tables for 1998 Mortality quotients obtained for 1998 have been represented, alongside those for 1990 and 1994, in the second part of graphs 1 and 2, respectively, for males and females. The results in the mortality tables compiled for 1998 suppose a continuation of the tendency this phenomenon has experienced between both quinquennia of the previous decade, with new reductions of the mortality rates by age, except for persons in the interval between 18 and 40 years old, which present a peculiar trajectory that will be commented in the following paragraphs. The general comment in the previous paragraph is applicable to young persons, up to 15 years old, with very similar mortality quotients for 1990 and 1998. Thus, in the aforementioned interval, 18 to 40 years old, death probabilities up to 25 years old reach their maximum values in 1990, as from when there is a favourable evolution until 1995. On the contrary, between 25 and 40 years old, the trajectory of the risk of death is unfavourable, reaching maximum values in 1995, as appears in the first part of graphs 3 and 4, which present, respectively, the quotients for males and females, from 15 to 44 years old, between 1990 and 1996. As regards infant mortality specifically, the risk of death for children under 1 year old is still evolving favourably: in 1998 the rate was 5.1 per thousand for boys and 4.3 per thousand for girls. The values for death probability obtained for the following years, in other words, 1996, 1997 and 1998, show a decreasing tendency. Consequently, this means an improvement of mortality in all young persons under study. Comparing the levels obtained in 1998 with those from 1985 shows that, for persons under 29 years old, current risks of mortality are inferior. As of said age, although the risks are higher, they tend towards those for said year, as appears in the second part of graphs 3 and 4. Although the increase of mortality at young persons during the period from 1985-1990 was due to three causes, traffic accidents, AIDS and drug addiction, the subsequent aforementioned evolution can be explained by the different repercussion each cause has had in society. Thus, mortality quotients for traffic accidents presented a general reduction during the 1990-1996 period, which was most significant n persons between 15 and 30 years old. Mortality due to AIDS increased death probability between 25 and 40 years old during the 1990-1995 period, showing a gradual decrease afterwards. As regards the third cause, drug addiction, this element is the least relevant; risks of dying due to this cause are very low and, after the increase observed between 1985 and 1990, it is possible to say that they have been stable since 1992. Moving on to consider other age intervals, the reductions noticed in the death probability in persons over 42 years old, between 1990 and 1998, are, once again, decreasing, to the point of being almost imperceptible. Nevertheless, progress was observed in persons aged between 50 and 60 years of age. The different evolution of the mortality for males and females means that the line for quotients between the death probabilities for males and females, at different ages, as appears in graph 5, for 1998, experiences a slight descent compared to 1990, until 47 years old, and starts to increase as of that age. The evolution of the survivors in the mortality tables for 1980, 1985 and 1990, represented for males and females, respectively, in graphs 6 and 7, is completed with the line corresponding to the tables for 1998, which is favourable if compared to years 1990 and 1994, both for the former and the latter. The series of theoretical deaths in the mortality tables for 1980, 1985, 1990, 1994 and 1998 have been represented in graphs 8 and 9, for males and females respectively. As well as the comments in the previous section regarding the evolution of the mortality calendar, and the deceleration of the increasing rhythm of life expectancy during the 1980s, it is necessary to add that the increase of the same, registered between 1990 and 1996 up to 40 years of age approximately, tends to recuperate it. Chart 1 is completed with life expectancy in 1998, for each sex and different ages. Life expectancy at birth of the population in Spain in the year 1998 reaches 78.7 years old, 75.3 years old for males and 82.2 years old for females (in other words, females live 6.9 years longer). The improved mortality registered between 1990 and 1998 in the group from 15 to 40 years old, has contributed to the evident recuperation of life expectancy at birth, positioned at 1.8 years for males and 1.7 years for females. Graph 10 shows the evolution of the difference between life expectancy for females and males, at each age, during the 1980-1998 period. 3 Spanish mortality in the European Union This section assesses the situation of general mortality in Spain comparing it to the other countries in the European Union (EU). Life expectancy at birth has been used as the indicator. The increase regarding the years people live in said countries and the highest values obtained are compared to the data for Spain. Life expectancy at birth for males and females corresponding to dates near years 1980, 1985, 1990 and 1998, in EU countries, appears in Chart 2. In 1980, life expectancy for males in Spain appeared in the third place, with a difference of just 0.2 and 0.3 years with Holland and Sweden, respectively. Since this indicator increased almost a full year over the five-year period between 1980 and 1985, Spain was the second country compared to the other EU countries. Nevertheless, the slow growth rhythm registered in the last quinquennium of the 1980s made Spain dropping back to the fifth place, behind Sweden, Greece, Holland and Italy. Between 1990 and 1998, the notable recuperation observed as regards life expectancy at birth for males is 0.7 years above the EU average, this figure is 0.2 years higher than Spain in Greece and Italy and 1.6 years higher in Sweden. As regards females, the deceleration of the rhythm of growth of life expectancy during the 1980s is not as pronounced as in the case of the males. In 1980, Spanish female's life expectancy was only below Sweden and Holland in 0.2 and 0.7 years, respectively. In 1990, it was only below France and Holland, both with 80.9 years life expectancy at birth, i.e. 0.4 years higher than Spain. The increase of life expectancy at birth for females between 1990 and 1998, as occurs with males, is important. As a consequence, Spain is positioned 1.3 points above the EU average, only below France, 0.1 years higher, in the last year. Glossary of symbols Q(X)= Risk or probability of death between the exact ages of X and X+1. L(X)= Survivors aged 100,000 initial persons. exactly X among D(X) = Theoretical deaths occurred between two specific ages X and X+1. E(X) = Life expectancy at a specific age X. LL(X) = Survivors aged X years old. T(X) = Probability of survival among persons aged X and X+1.