SciELO - Scientific Electronic Library Online

 
vol.21 issue2The biological and social determinants of illness: a study of iron-deficiencyHaenszel's standardized coefficient of lost years of life: a comparison with the standardized coefficient of general mortality with regard to its use as a health level indicator for populations author indexsubject indexarticles search
Home Pagealphabetic serial listing  

Services on Demand

Journal

Article

Indicators

Related links

Share


Revista de Saúde Pública

Print version ISSN 0034-8910On-line version ISSN 1518-8787

Rev. Saúde Pública vol.21 no.2 São Paulo Apr. 1987

http://dx.doi.org/10.1590/S0034-89101987000200004 

ORIGINAL ARTICLE

 

Swaroop and Uemura's proportional mortality ratio. The need for periodic revision of the definition1

 

 

 

Elias Rodrigues de Paiva; Yára Juliano; Neil Ferreira Novo; Walter Leser

Departamento de Medicina Preventiva da Escola Paulista de Medicina. Rua Botucatu, 740 - 04023 - São Paulo, SP - Brasil

 

 


ABSTRACT

Using reliable data from 34 countries for the years 1950, 1960, 1970 and 1980 it was observed that the proportional mortality ratio for 50 years of age and above, proposed by Swaroop & Uemura, did not provide the best discriminatory power between more and less developed countries in any of the years studied. In 1970 and 1980, the greatest discriminatory power was obtained by using the proportional mortality ratio for 75 years of age and above. The displacement of deaths to upper age groups over a certain period of time was better translated by variations in the 75 years and above than in the 50 years and above proportional mortality ratio. It is also useful to complement this information by computing the percentage of deaths at 65 years of age and above. It is suggested that the classes proposed by Swaroop & Uemura should be reformulated using new classes based on the proportional mortality ratio for 75 years and above, with the following limits: 0 |—20; 20 |—40; 40 |—50; 50 |—55 and 55 and above, with the possibility of subdividing the last group, if necessary, in the future.

Uniterms: Health status indicators. Mortality.


 

 

INTRODUCTION

The majority of health level indicators is based on mortality data. One of these indicators, the Proportional Mortality Ratio (PMR), was proposed by Swaroop and Uemura13 in 1957. They compared two groups of countries which they classified as "developed" and "under-developed", calculating, by the linear discriminating function technique, Mahalanobis's10 (1936) generalized quadratic distance values (D2) with reference to the percentages of deaths counted as from each one of the limits of the normal age groups. They thus discovered that the greatest value of D2 was obtained when the percentage of deaths corresponded to the group of 50 and above. Neither isolatedly nor in combination did other indicators (Coefficient of Infant Mortality, Life Expectancy and Coefficient of Crude General Mortality) provide higher values.

Among the innumerable advantages advanced by these authors, in favor of their indicator, one may emphasize that which relates to the "availability of data, related to a large number of countries, on a regular basis."

It is well known that, after the passage of practically thirty years since the date of the proposal of this indicator, considerable improvements in the health level of the majority of countries have become apparent - and this has been clearly reflected, in the mortality rates.

Among the many factors which must have contributed to this improvement, basic sanitation, the expansion of the use of insecticides with residual action, and of chemotherapeutical and antibiotic substances, as well as the greater extension of the services of medico-sanitary care, and in a general way the progress with regard to the living standards of the populations, may be recorded.

It is probable that some adverse factors may also be infered, mainly in Third-World countries, but a general assessment should show a favorable result.

It seemed to us therefore that there was a sufficient foundation for the formulation of certain hypotheses as to the consequences, in terms of the proportional mortality by age, of this evolution of the health level:

I - the values of the PMR for successive periods, as from 1950, for each country, have shown a tendency to increase;

II - the age group defined by Swaroop and Uemura, that of 50 years of age and above, has not continued to be that which has shown the greatest power of discimination - which, in successive periods, has moved to older age groups;

III - the ordering of the countries into classes, proposed by Swaroop and Uemura13, based on that age-group, has ceased to describe, significantly, their relative positions, in terms of health levels.

Beyong this, the confirmation of hypothesis I would mean the progressive transfer of deaths from the age groups of less than 50 years of age to those with 50 years of age and obove. In those countries in which, at the beginning of a period, the level of health was low, expressed as a low PMR value, the degree of improvement achieved during the period under study would correspond to the volume of such transfers. If the value of the contingent of deaths below 50 years of age at the beginning of the period was high, notable improvements would express themselves in substantial increases of the PMR.

In contradistinction, in countries with high health levels, the number of deaths at less than 50 years of age is already small at the beginning of the period, thus diminishing the possibility of such transfers, during the period under study, to that group which defines the PMR. As a result, even though there were to occur a notable improvement in the health levels, as occasionally evidenced by other indicators, the repercussion in the increase of the PMR value would be very small. It is to be admitted that, in such a case, the transfers will have occurred within the 50 or above age group, with a displacement of proportional mortality to those higher age groups included within it.

Based on this reasoning, it seemed reasonable to formulate hypothesis IV, dependent on the confirmation of hypothesis I:

IV - if the age subgroups resulting from the division of that proposed by Swaroop and Uemura13 are considered, the displacement of proportional mortality towards greater age, from one period to another, will not be adequately translated by the variations of the PMR as proposed by Swaroop and Uemura - however, they would be so translated were another, higher, age-limit adopted for the definiton of this rate.

It was decided that proportional mortality by age for a group of countries from which trustworthy data were available, at ten-year intervals from 1950 up to 1980, should be studied with a view to putting this hypothesis to the test.

 

MATERIAL AND METHODS

Basic Data

Bearing in mind the hypotheses formulated, the plan of work proposed the study of mortality data, by age group, for countries from which data were available for 1950, 1960, 1970 and 1980. It was allowed that, should data for these dates not be available, use would be made of those with not more than three years of difference, with the one exception made in relation to 1950-in which case, for four countries, it was necessary to accept distances of four years.

Thus, so that a country should be included in the study, it was necessary that data for all the four periods should be available and that the distribution of deaths by age should have been made in accord with groups equal to those adopted by Swaroop and Uemura. In this way it was possible to obtain trustworthy data for 34 countries from editions of the Demographic Yearbook1-5 (1950, 1951, 1960, 1970 and 1980). As will be seen, when the study of the last period only was considered, i.e. 1980, this number rose to 66.

The studies to be undertaken sought to:

1 - observe the evolution of the PMR values, as from 1950, for each country;

2 - calculate the power of discrimination, as measured by Mahalanobis10 generalized quadratic distance (D2), of the accumulated percentages of deaths up to the upper limit of each age group, according to the model employed by Swaroop and Uemura13;

3 - discover the behavior, for each period, of the distribution of the countries classified in the way proposed by Swaroop and Uemura13;

4 - examine, in view of the results obtained, the convenience of modifying the definitions, both of the Proportional Mortality Ratio as also of the classifications based on its value.

Formation of the Groups

For the formation of the groups to be compared by the use of the calculation of the linear discriminating function, it was preferable to use a more objetive criterion than that used by Swaroop and Uemura13, who classified countries as "developed" and "underdeveloped" by what they knew of them a priori. Recourse was had to a health level indicator which, in terms of the variable which it represents, permitted the ordering of the countries studied, and which, for its calculation, involved proportional mortality values by age-groups; the indicator in question is that of Guedes7 (GI) (1972).

In order to determine the size of the groups to be compared, the criterion proposed by Kelley9 (1939) for the selection of groups for analysis, by means of items in educational tests, was adopted; according to that author the greatest discrimination between groups is attained when each of them, at both extremes of the ordering of the overall result of the test, includes 27% of the total number of observations. In the present case this percentage corresponds, by approximation to the whole number, to 9 countries.

Thus the group of countries designated "less developed" includes those which, according to the ranking of GI, occupy positions 1 to 9, while those considered "more developed" occupy positions 26 to 34.

The composition of the groups varied, from period to period, by function of the differences in evolution, in terms of GI, shown by the countries concerned.

The groups of countries which were compared in each period are presented in Table 1, which also shows the years to which the data relate.

Another study, beyond that related to the 34 countries focussed on in the four periods and including 66 countries, was carried out for 1980. On the basis of the above-mentioned criterion, in this case, each group included 18 countries.

Statistical Methods

Linear discriminant function

The discriminatory power between the groups provided by the differences between the accumulated percentages of the proportional mortality, up to the upper limit of each age group, was measured by the value of Mahalanobis's10 generalized quadratic distance (D2).

Designating the groups as A and B, we get:

where Z is a linear discriminant function which, in the case of a single variable, assumes the form: Z = bX. For the calculations of this function the model presented by Goulden6 (1951) was followd.

Non-parametric methods of analysis

Bearing in mind the nature of the data to be analysed in the further stages of the study, involving comparisons of distribution in non-independent samples, the impossibility of accepting that its distributions should satisfy the requirements demanded for the use of parametric methods, even if recourse were had to transformation of the variables, became evident.

The following methods were, therefore, used:

a) Friedman's test (in Holander and Wolfe8, 1973), for the comparison of non-independent distributions;

b) multiple comparisons based on the sum of the test ranks of Friedman (in Holander and Wolfe8, 1973), effected when the statistical value calculated reached the level of significance.

Adaptation of Bartlett's method for three groups for the estimation of equations of regression with two variables subject to error (in Sokal and Rohlf12, 1969)

In view of the fact that the two variables involved in the equations of regression which it was the intention to estimate were expressed, one as a percentage and the other in differences between percentages, the use of the averages of the two groups which occupy extreme positions could raise doubts of a theoretical nature. Beyond this, the frequent occurrence of values differing greatly from others, substantially affecting the value of the average, constituted a weighty reason for the replacement of this statistic by the average. This was the only modification introduced by the authors into Bartlett's method to adapt it to the nature of the data used. It should be, further, emphasized that it was not intended to carry out tests of significance with regard to the equations estimated, their use being limited to the calculation of differences between values observed and those estimated on the basis of them.

Level of rejection of the nullity hypothesis

The rejection of the hypothesis of nullity was decided on in tests of significance when the probability of the casual occurrence of the value of the statistic calculated was equal to, or less than, 0.05. In this case the values were marked with an asterisk.

 

RESULTS

The observation of the PMR values for each of the countries, in the four periods, presented in Table 2, demonstrates the validity of hypothesis I; as a matter of fact only in one instance, that of Guatemala in 1950 and 1960 was there a reduction in the value for the PMR.

 

 

The calculation of Mahalanobis's generalised quadratic distance (D2) was carried out, with a view to testing hypothesis II, according to the model adopted by Swaroop and Uemura, for each percentage of accumulated deaths up to increasing age limits, contrasting the groups of countries "more" and "less" developed, for each of the four periods studied.

The results obtained are presented in Table 3, a reading" of which shows the validity of hypothesis II. As a matter of fact the highest value found for D2 did not correspond, for any of the periods, to the accumulated percentage of deaths at below fifty years of age indicated by Swaroop and Uemura. The grouping of ages for which the highest value of D2 was found was not the same in the four periods, including deaths below 60 years of age in 1950, with less than 70 in 1970 and with less than 75 in 1970 and 1980.

 

 

Table 3 allows, further, verification of the fact that the power of discrimination showed, as a general rule, progressive decreases, from decade to decade. Thus was verified the supposition that the difference between the groups of countries "more" and "less" developed was progressively reduced over the period of 30 years.

As a result of the growth of the PMR, as shown in Table 2, the situation which Table 4 expresses with clarity appeared. The distribution of countries according to the classes defined by Swaroop and Uemura13 has changed clearly and progressively, with an increasingly higher concentration, with each passing decade, in the class of 75% and above. This position suffered no appreciable modification even when a further 32 countries were added, for 1980. These results corroborate hypothesis III.

 

 

Once these results had been achieved it became necessary, with the hypothesis IV, in mind, to find a variable capable of measuring the displacements of mortality to subgroups of increasing age, within the group of 50 or more years of age.

For the sake of greater simplicity, from this point on, the Proportional Mortality Ratio will be designated by R. Thus Rj will signify this rate when the accumulated percentage of deaths is calculated from j years of age. will correspond, respectively, to the rates in the final and initial years of a period.

The difference , that is, the variation of the PMR, as between the beginning and end of a period, will represent the displacement of deaths; when positive, the displacement will have occurred from the group of ages below j to that of ages equal to or greater than j; when negative, as happened in some 1% of cases, the direction of the displacement will be the opposite. For even greater simplicity, we will make

 

Friedman's test

Multiple comparisons

Difference between the sum of the ranks

 

The percentage of deaths, in year I, at less than j years of age, is given, evidently, by , and may decrease or increase up to year F, according as be, respectively, positive or negative.

If, by , is designated the relative decrease or increase of these percentages of deaths with less than j years of age, during the period from I to F, the result, in terms of percentages for the value of the year I; is:

With the measure of the relative displacements for ages equal to or greater than j thus defined, the study of hypothesis IV became possible, by means of the calculation of the equations of regression between and , allowing the definition of a new variable represented by the module of the difference between the observed values of between the observed values of and the estimates made on the besis of the equation. In each period considered, and for each Dj, this difference can be expressed, in a simplified form, by

Thus, in each period, for each value of j, in pj, there will be as many equations of regression as there are values attributed to j in Dj. Each of these equations will give a distribution of values for the above-defined variable.

The comparison of these distributions will then permit the verification as to which of the Dj, for the period in question, will provide the best forecast of the "values of pj, that is to say, of the relative displacement of deaths for ages less than j towards ages equal to or greater than j.

The ages selected for the calculation of the were 50, 65 and 75 years — the first because it was that defined by Swaroop and Uemura, and the others, finishing in 5, not only because they correspond to the age distribution which appears in the World Health Statistics Annual edited by World Health Organization, but also because, by their use, the source of error represented by the rounding off of the age to the final 0 is reduced. Further because, as can be seen from Table 3, the greatest value for D2 corresponds to one of them or to another age to which it approximates.

In the case of , ages ending in 0 or 5, from 50 up to 80 inclusive, were used. They were studied for the periods previously considered, namely: 1970 - 1980, 1960 - 1980, 1950 - 1980, 1960 - 1970,1950 - 1970 and 1950 - 1960.

The values of , of 100 - Rj and of , calculated for each country, constituted the basic elements for the studies to be elaborated from this point on. Because of limitations of space it is impossible to present the tables of these values.

The equations of regression from to were calculated by means of the adaptation of Bartlett's (in Sokal and Rholf12) three-group method.

Then were calculated, for each period, the values of , which, again for reasons of space, are not presented2. As can be seen, in each period, for one and the same value of j, in pj there are three distributions of the values of , corresponding to the equations which include D75, D65 and D50. Friedman's test was used for the comparison of these distributions, and the values for the sums of the positions (SDj) related to each distribution, appear in the same table.

Values for the statistic S are presented in Table 5. They have been calculated for each of Friedman's tests (Hollander and Wolfer8), as also were the differences obtained in the multiple comparisons. It should be remembered that, when the statistic S did not reach the critical level, the values of the differences were presented only as points of reference, without there being any pretension to relate them to the critical value of the significant minimum difference.

An examination of Table 5 allows the verification that:

a) in the cases of the distributions related to p70, p75 and p80, transposing displacements of deaths from lower ages to ages equal to or greater than those shown, the statistic S, in the 18 tests, passed the critical value, even for the level of significance given by a = 0.01; in the multiple comparisons the sums of the positions related to the equations involving D50 were, also without exception, significantly greater than those given by the equations in which D75 was involved, that is to say, the shifts in the former were greater than those in the latter; the sums of the ranks, relating to D50, were also always greater than those given by D65, with a difference of significance, for a rate of experimental error equal to 0.05, in three cases; in another three, significance was reached when a = 0.10; the sum of the ranks related to the equations with D65 was always, with the exception of one single case, greater than that obtained with D75, and the level of significance was reached 12 times for the experimental rate of error equal to 0.05, and in another two instances for the rate of 0.10;

b) from p50 up to p65 the statistic S reached the level of significance in 10 of the 24 comparisons; in five of them the smallest sum of the ranks corresponded to the equations with D65 and, in the five others, to the equations with D75. In the 14 comparisons in which S showed a lower than critical value, in six of them the smallest sum corresponded to D75 and in eight to D65. In the multiple comparisons with a rate of experimental error equal to 0.05 the sum of the positions related to D75 was, three times, significantly less than that related to D65, and six times less than that corresponding to D50 and five times that with D65 was less than that with D50. With a rate of experimental error equal to 0.10 the sum of the ranks with D65 was twice significantly less than that obtained with D75 and once than that related to D50; once the sum of the ranks with D75 was less than that with D65 and one other time less than that related to D50.

These results, taken together, demonstrate that the relative displacements of deaths, from lower ages to those equal to or higher than 70, 75 and 80, were better interpreted by the variations, during any given period, of the proportional mortality rate which represents the percentage of deaths at 75 years of age or above. When the displacements of deaths referred to 50, 55, 60 and 65 years of age, both rates deserve to be taken into consideration, those including deaths at 65 or more and at 75 or above.

Thus the results obtained by the method used show that the variations, within any one period, of the proportional mortality ratio, as proposed by Swaroop and Uemura13, do not allow such exact forecasts as those provided by the ratios related to 65 or 75 years of age, of the relative displacements of deaths to any of the age-classifications focussed on in this study. Thus the validity of hypothesis IV, may be admitted.

 

PROPOSITIONS ARISING FROM THE ANALYSIS OF THE RESULTS

The results presented in Table 3 made two important points, which must be taken into consideration, evident:

1st) in none of the periods studied was the maximum value of D2 that which referred to the accumulated percentage of deaths with less than 50 years of age, wich means to say that it was always greater for the percentage of deaths at 50 or more years of age;

2nd) the age group which corresponded to this maximum value varied during the period studied.

According to the criterion adopted by Swaroop and Uemura13, the PMR should be represented by mortality at 60 years or above in 1950, at 70 years or over in 1960 and at 75 years or over in 1970 and 1980. It is evident, however, that comparisons through time only make sense when the value of the PMR refers to the same age group.

There then arises the problem of the choice of the new value for the definition of the PMR. Immediately reasons come to mind against the choice of the values 60 and 70. It is true that in the age groups adopted by the World Health Organization for the distribution of deaths which appears in the World Health Statistics Annual, edited by the WHO, the classifications have, as their upper limits, the mean value between the tens, with the single exception of that which refers to deaths in the first year of life. As the Yearbock is one of the most widely used sources of data on vital statistics, the estimates for age-groups with limit or limits ending in zero involve the need for interpolations, naturally subject to errors of greater or lesser weight. The use of such limits could be justified by reference, as a data source, to the Demographic Yearbook, in which the distribution of deaths is presented by classes with five year intervals; as has already been mentioned, this was the reason which led to the use of this source in the light of the methodology envisaged for this study. It is evident, however, that in this way data which may be presented solely in the WHO publication14 are left aside.

For these reasons, there remain as options the values of 65 and 75 for the age which may be used in the definition of the PMR, and the following advantages may be indicated with regard to the choice of the latter:

1st - this has been the age-group with the greatest discriminatory power in the two most recent periods; in another, 1960, this power still showed itself to be quite high, approximating to that corresponding to 65 years of age and above;

2nd - the group 75 years or above is that which permits the best forecast, in terms of its variations, from the beginning to the end of a period, of the relative magnitude of the displacements of deaths to higher age-groups, especially above 70 years of age.

However, in view of the fact that the displacement of deaths to classes with the lower limits equal to 50, 55, 60 or 65 years, had a good relationship with the variations of the PMR defined for deaths at 65 years of age or above, at times better than that refering to 75 years or above, the consideration of this indicator as a means to complement the evaluation of the evolution of the level of health of a population, during a particular period, is advisable.

Another relevant aspect worthy of examination is related to the corroboration, as demonstrated in the appreciation of the results, that the ordering in wide classes, with the limits proposed by Swaroop and Uemura, would no longer permit the description of the countries' relative positions, in terms of health level, when the group of 50 years of age or above was adopted for the definition of the PMR.

According to the new definition suggested, the distribution of the countries into the same classes would undergo modification, as may be seen in Table 6.

 

 

An examination of Table 6 enables one to verify immediately, also in terms of the new definition of the PMR, the progress, in terms of health levels, presented by the countries studied. There is however a concentration in the classification 25 50 and two classifications remain empty, up to 1970, and one in 1980. In this last year, as much for the 34 as for the 66 countries, the highest value observed was 55.0, still far removed from the upper limit of the class 50 75.

Thus, the classification proposed by Swaroop and Uemura13, beyond not giving, as has already been seen, a meaningful differentiation of the countries when the PMR is defined in terms of 50 years of age or above, is also not the most adequate for this classification according to the variable which results from the new definition of the PMR, that is to say, the percentage of deaths at 75 or more years of age.

Bearing in mind the past and present situations, as well as with a view to the possibility of future development, the proposition of new classification limits as set out in Table 7 seems worthy of consideration. In this table are presented the distributions which would result, for the four periods, if these new classes were adopted.

 

 

It may be foreseen that, with the continued advances in the level of health, the classification 0 20 will tend to empty out, while the classification 55 , empty up to 1970, will become increasingly thickly populated. It is even possible that it will be convenient to fix an upper limit for it, making it equal to 55 60, thus creating a new classification for 60 .

 

CONCLUSIONS

The results obtained and analysed, related to the countries studied in years approximating to 1950, 1960, 1970 and 1980, lead to the formulation of the following conclusions.

1 - In the contrast between countries more or less developed, involving accumulated percentages of deaths, the maximum value of Mahalanobis's generalized quadratic distance found for each one of the periods studied always corresponded to accumulated percentages which reached higher age levels than those refered to by Swaroop and Uemura.

2 - With the progressive improvement of the health level of the countries studied, its distribution within the classifications defined by Swaroop and Uemura, according to the values of the Proportional Mortality Ratio will no longer provide adequate differentiation, as the large majority are concentrated in the last of these classifications.

3 - In the light of the advantages which it offers, as much in terms of Mahalanobis's generalized quadratic distance, in recent years, as also in the availability of sources of data and of the possibility it gives for the interpretation of the significance of variations from one period to another, the Proportional Mortality Ratio related to the age-group of 75 years of age or above ought to be adopted in substitution to that proposed by Swaroop and Uemura.

4 - The variations of the Proportional Mortality Ratio, during a particular period, as defined by the percentage of deaths at 65 years of age or above, complement the information provided by the variations in this same Ratio based on the deaths at 75 or more years of age.

5 - Taking into consideration the values presented, in the four periods, by the various countries studied, the reformulation of the classification proposed by Swaroop and Uemura becomes necessary in terms of the values of the Proportional Mortality Ratio related to deaths at 75 years of age or above, with the following limits: 0 20, 20 40, 40 50, 50 55 and 55 or above, with the possible necessary future division of this last and the emptying out of the first.

 

REFERENCES

1. DEMOGRAPHIC YEARBOOK: 1949-50. (United Nations) New York, 1950.

2. DEMOGRAPHIC YEARBOOK: 1951. (United Nations) New York, 1951.

3. DEMOGRAPHIC YEARBOOK: 1960. (United Nations) New York, 1960.

4. DEMOGRAPHIC YEARBOOK: 1970. (United Nations) New York, 1971.

5. DEMOGRAPHIC YEARBOOK: 1981. (United Nations) New York, 1983.

6. GOULDEN, C.H. The discriminant function. In: Goul- den, C.H. Methods of statistical analysis. 2nd ed. New York, Wiley, 1952. p. 378-93.

7. GUEDES, J.S. Contribuição para o estudo da evolução do nível de saúde do Estado de São Paulo: análise das regiões administrativas (1950-1970). São Paulo, 1972. [Tese de Doutorado - Faculdade de Saúde Pública da USP].

8. HOLLANDER, M. & WOLFE, D.A. Nonparametric statistical methods. New York, John Wiley & Sons, 1973.

9. KELLEY, T.L. The selection of upper and lower groups for the validation of test items. J.educ. Psychol., 30: 17-24, 1939.

10. MAHALANOBIS, P.C Apud SWAROOP, S. & UEMURA, K.13

11. ORGANIZACIÓN MUNDIAL DE LA SALUD. Grupo de Estudio de Medición del Nivel de Salud, Ginebra, 1955. Informe. Ginebra, 1957. (Ser.Inf. tecn., 137).

12. SOKAL, R.R. & RHOLF, F.J. Biometry: the principles and practice of statistics in biological research. San Francisco, Freeman, 1969.

13. SWAROOP, S. & UEMURA, K. Proportional mortality of 50 years and above: a suggested indicator of the component "health, including demographic conditions" in the measurement of levels of living. Bull. Wld Hlth Org., 17: 439-81,1957.

 

 

Received in August 5th, 1986.
Accepted in November 12th, 1986

 

 

1 Dissertation sinopsis presented at the "Escola Paulista de Medicina", by Elias Rodrigues Paiva, with the same title, in 1981.
2 The tables will be supplied on request by the authors.

1. DEMOGRAPHIC YEARBOOK: 1949-50. (United Nations) New York, 1950.        [ Links ]

2. DEMOGRAPHIC YEARBOOK: 1951. (United Nations) New York, 1951.        [ Links ]

3. DEMOGRAPHIC YEARBOOK: 1960. (United Nations) New York, 1960.        [ Links ]

4. DEMOGRAPHIC YEARBOOK: 1970. (United Nations) New York, 1971.        [ Links ]

5. DEMOGRAPHIC YEARBOOK: 1981. (United Nations) New York, 1983.        [ Links ]

6. GOULDEN, C.H. The discriminant function. In: Goulden, C.H. Methods of statistical analysis. 2nd ed. New York, Wiley, 1952. p. 378-93.        [ Links ]

7. GUEDES, J.S. Contribuição para o estudo da evolução do nível de saúde do Estado de São Paulo: análise das regiões administrativas (1950-1970). São Paulo, 1972. [Tese de Doutorado - Faculdade de Saúde Pública da USP].        [ Links ]

8. HOLLANDER, M. & WOLFE, D.A. Nonparametric statistical methods. New York, John Wiley & Sons, 1973.        [ Links ]

9. KELLEY, T.L. The selection of upper and lower groups for the validation of test items. J.educ. Psychol., 30: 17-24, 1939.        [ Links ]

10. MAHALANOBIS, P.C Apud SWAROOP, S. & UEMURA, K.13

11. ORGANIZACIÓN MUNDIAL DE LA SALUD. Grupo de Estudio de Medición del Nivel de Salud, Ginebra, 1955. Informe. Ginebra, 1957. (Ser.Inf. tecn., 137).        [ Links ]

12. SOKAL, R.R. & RHOLF, F.J. Biometry: the principles and practice of statistics in biological research. San Francisco, Freeman, 1969.        [ Links ]

13. SWAROOP, S. & UEMURA, K. Proportional mortality of 50 years and above: a suggested indicator of the component "health, including demographic conditions" in the measurement of levels of living. Bull. Wld Hlth Org., 17: 439-81,1957.        [ Links ]

Creative Commons License All the contents of this journal, except where otherwise noted, is licensed under a Creative Commons Attribution License