A Re-look at the Methods for Decomposing the Difference Between Two Life Expectancies at Birth

Transcription

1 Sankhyā : The Indian Journal of Statistics 008, Volume 70-B, Part, pp c 008, Indian Statistical Institute A Re-look at the Methods for Decomposing the Difference Between Two Life Epectancies at Birth Krishna Murthy Ponnapalli International Institute for Population Sciences, (Deemed University), Deonar, Mumbai, India Abstract Earlier Chandra Sekar (1949) and Arriaga (1984) followed by several others, suggested different formulas for decomposing a difference between two life epectancies. The present paper at first using the Keyfitz (1968) equation for decomposition, attempts to demonstrate how one can derive different s defined by Arriaga (from several s defined by Chandra Sekar). Then it attempts to show how one can derive Preston et al. s (001) decomposition formula in a scientific way; and also attempts to show how one can etend the same to give a symmetric formula that gives eactly the same result as that of the United Nations (1985) formula. At the end, a meaningful decomposition and interpretation method is suggested to the United Nations formula following Preston et al. s approach. It is demonstrated that the present paper finally succeeds in suggesting a new look to the Preston s and United Nations formulas. AMS (000) subject classification. Primary 9D5, 6P05. Keywords and phrases. Life epectancy at birth, decomposition, Chandra Sekar method, Arriaga method, Preston method, United Nations Decomposition Formula, Keyfitz equation 1 Introduction A review of eisting literature on decomposing a difference between two life epectancies over time or cross-sectionally shows that a number of methods were suggested by several researchers. Perhaps, no other method in the field of demography has attracted as much attention as that of decomposing the difference between two life epectancies. While Chandra Sekar (1949, 1986), (now known as Chandrasekaran), defined four concepts of main, operative, -interaction deferred and -interaction forwarded to eplain a change in mortality conditions

2 84 Krishna Murthy Ponnapalli in a specific age group during a given time period, Arriaga (1984) suggested several different s, namely direct, indirect, eclusive, interaction, other, and total as a result of the difference between two life epectancies. Lopez and Ruzicka (1977), on the other hand, preferred to decompose the difference between two life epectancies into only two s - eclusive and interaction. Pollard (198, 1988) approached the problem by continuous analysis and preferred to suggest a symmetric formula for decomposition; however, he discussed several other formulae in his article. The United Nations (198, 1985) suggested a symmetric formula by discrete approach but did not give a clue about how they derived their formulae. Andreev (198) and Pressat (1985) also suggested a symmetric formula. Canudas Romo (003), Vaupel and Canudas Romo (00, 003), and several other researchers also made attempts to suggest formulae to decompose the difference between two life epectancies over time or cross-sectionally. Recently, Murthy (005) has shown successfully that different decomposition methods, discrete as well as continuous, in their symmetrical form will give eactly the same results as those of United Nations, Pollard, Andreev and Pressat. He further suggests using the symmetric formulae only of the different methods for obvious reasons. Preston et al. (001) in their classic tet book Demography: Measuring and Modeling Population Processes tried to re-represent Arriaga s (1984) method by means of direct and other terms, but did not give any details of the derivation of the formulae, perhaps being a tet book. Their other s term includes the indirect and interaction terms as suggested by Arriaga (1984). Preston eplains that a change in the death rate between ages and + n has a direct on n L ; but, it also has an indirect on all the n L above age + n because of the change in the number of survivors at age + n. (Preston et al., 001: 64). It seems Preston et al. (001) might have simply added two different formulae suggested by Arriaga (1984) to arrive at their present formulae 3.11 given in page no. 64 of Preston et al. (001). However it will be interesting if one can show how to arrive at the formulae suggested by Preston et al. (001) in a mathematical way without much complication. Further it will be more interesting if one can etend the formulae given by Preston et al. (001) to arrive at the formulae suggested by United Nations (1985). Then we may say another methodology has been added to the band wagon of formulae already suggested by several authors as already mentioned, instead telling

3 A re-look at the methods for decomposing the difference between two life epectancies at birth 85 that Preston et al. formulae has been derived from the already eisting Arriaga (1984) methodology. Fortunately the present researcher finds a solution to both the above problems raised by himself by simply substituting in a systematic way a simple formula suggested by Keyfitz (1968) in the various formulae suggested by Chandra Sekar in his famous work (1949), which was earlier used by the present researcher (Murthy, 005) to show all methods give the same results if modified further in a particular fashion. Interestingly the present study as a silent component of the present work also shows that Arriaga (1984) all formulae can also be derived in a scientific way from the Chandra Sekar (1949) formulae without much effort. It is realized that Arriaga (1984) did not derive his formulae but simply presented them with a strong logical argument. Thus indirectly this study also gives full mathematical proof and support for the brilliant work carried out by Arriaga (1984) further etensively used by several demographers and other researchers throughout the world over the years. Thus, the three specific objectives of the present study are: 1. To derive different s defined by Arriaga from several s defined by Chandra Sekar in a methodical way, using the Keyfitz equation for decomposition.. To show that Keyfitz s (1968) formula helps to understand better the formulae given by Preston et al., (001); and to show that it is an outcome of the objective (1) above. 3. To show that Preston et al. s (001) formulae, when etended, equals the United Nations (1985) method; and to show that it is also an outcome of the objective (1) above. However, to start with, and to avoid any confusion afterwards, we state here that the main thrust of the study is to show how one can derive and also etend the Preston et al. (001) formulae to arrive at the United Nations (1985) symmetric formulae. To maintain the flow of the methodology for deriving various formulae suggested by Murthy (005) as etensions of Arriaga (1984) method earlier, in the present study also, a systematic way was followed and thus results of the Preston et al. (001) and United Nations formula (1985) modifications were shown as outcomes of the methodology somewhere in the middle of the research paper. Hope the readers will bear this in mind while going through the present paper and not to be confused.

4 86 Krishna Murthy Ponnapalli Thus said, the remaining portion of the study is organized into three sections analysis, discussion, and conclusion. In the analysis section, I derive several new formulae by substituting Keyfitz s (1968) formula in the Chandra Sekar s (1949) formulae and it is also shown that each of the results led to the derivation of different s proposed by Arriaga (1984). An application of the new formulae was illustrated by using the life table data on the United States and the results are presented in Tables 1 to 3. The input data is further reproduced in Appendi Table 1 and Appendi Table, for convenience. Further, following Murthy (005), the results of the analysis are presented in Table 1, Table and Table 3 as Approach I, Approach II and Approach III for easy understanding. In the analysis section, an attempt has also been made to compare the present methodology and results with those of Carlson (006). Results of Table 3 are meant for comparison with those of Carlson (006:3-33). The various life table notations used in the present study are as follows: nq = Probability of dying between age and + n; np = probability of surviving between age and + n = 1- n q l 1 = number of persons alive at eact age, in the initial time period 1 l = number of persons alive at eact age, in the latter time period nl 1 = number of person-years lived between ages and + n, in the initial time period 1 nl = number of person-years lived between ages and + n, in the latter time period T 1 = number of person-years lived above eact age, in the initial time period 1 T = number of person-years lived above eact age, in the latter time period e 1 = epectation of life at eact age, in the initial time period 1 e = epectation of life at eact age, in the latter time period na = Average number of years survived in the age interval to + n by those dying during the age interval n = length of the age interval

5 A re-look at the methods for decomposing the difference between two life epectancies at birth 87 Analysis According to Keyfitz (1968) one may epress e by the equation: e = n a n q + (n + e ) n p (.1) Namboodiri and Suchindran (1987:41) state, One could use this result (.1) above to decompose into meaningful parts the difference between the epectation of life at age from two life tables. When further simplified, (.1) equals to: e = n L l + T l (.) Following Namboodiri and Suchindran (1987), when 1 and refer to two time periods, the difference between the epectation of life at age from the two time periods may be written as: ( e e 1 nl = l n L 1 ) ( T l 1 + l T 1 ) l 1 (.3) Thus said, the above equation (.3) may now be used in the present study to obtain various s defined by Arriaga from the various s defined by Chandra Sekar (1949) in a neat fashion. This can also be used to eplain that Preston et al s (001) formula is nothing but one of the outcomes of the above eercise..1 Obtaining Arriaga s Direct and Indirect Effects from the Main Effect of Chandra Sekar (1949). According to Arriaga (1984:87), while direct is the on life epectancy due to the change in life years within a particular age group as a consequence of the mortality change in that age group, indirect is the number of life years added to a given life epectancy because the mortality change within (and only within) a specific age group will produce a change in the number of survivors at the end of the age interval. Chandra Sekar (1949) defines the main term as the difference between e 1 0 and the epectation of life at birth which would have resulted if the mortality conditions had changed only in the age group under consideration to the etent it had and the mortality conditions in the other age groups had remained unchanged.

6 88 Krishna Murthy Ponnapalli Murthy (005) has shown that the main of Chandra Sekar (1949) equals to the eclusive or the sum of the direct and indirect s as defined by Arriaga. Here, it can be demonstrated how one can obtain the direct and indirect s of Arriaga by simple substitution of equation (.3) in the main term defined by Chandra Sekar (1949) as follows: Chandra Sekar (1949) defines the main by a formula as = ( ) ( ) 1 l 1 {l ( e e 1 ( ) l e e 1 )} l 1 0 l Substituting (.3) in (.4), and simplifying further gives the equation: l 1 l 1 0 ( nl l n L 1 ) l 1 + T 1 ( l 1 l ) l 0 l l 1 1 (.4) (.5) It is realized that while the first term in equation (.5) above represents the direct, the second term in equation (.5) above represents indirect, and equation (.5) as a whole represents the eclusive term of Arriaga (1984). Thus it seems a re-representation of Chandra Sekar s (1949) main formula given in equation (.4) by a different format as shown in equation (.5), not only allows us to arrive at the direct, indirect and eclusive terms as defined by Arriaga (1984) but also allows the researcher to give a quite meaning and justification for the formulae developed by Chandra Sekar way back in To state further, all the formulae suggested by Chandra Sekar (1949) for decomposition purpose thus are re-represented in the present work by the present researcher as a combination of different s as suggested by Arriaga, in the following few sections in an orderly manner and tried to bring to the fore the interrelationships between the scholarly works independently carried out by two great scholars. The results of the application of the above formula (.5) to the data sets under consideration are shown in columns, 3, and 4 of Table 1, Table and Table 3 as Approach-I (due to Arriaga, 1984). Table shows how the results of the analysis will be changed when the age groups considered differ from the ones given in Table 1. It is noted that, when age groups reported are re-considered into broad age groups, as in Table for eample, one has also to re-consider n L in the above formula as equal to (T T ), to get the epected results. Table 3, however, intends here to show that Carlson s (006) formulae are also a result of one of the present eercises.

7 A re-look at the methods for decomposing the difference between two life epectancies at birth 89 Table 1. Decomposition results obtained by using the new formulae: US, females, Age Direct Indirect Eclusive Other Interaction Total Effect Effect Effect Effect Effect Effect (1) () (3) (4)=()+(3) (5) (6)=(5)-(3) (7)=(4)+(6) = ()+(5) = ()+(3)+(6) Approach-I (Due to Arriaga) TOTAL Approach-II (Due to the present researcher) TOTAL

8 90 Krishna Murthy Ponnapalli Table 1. Decomposition results obtained by using the new formulae: US, females, (CONTD.) Age Direct Indirect Eclusive Other Interaction Total Effect Effect Effect Effect Effect Effect (1) () (3) (4)=()+(3) (5) (6)=(5)-(3) (7)=(4)+(6) = ()+(5) = ()+(3)+(6) Approach-III (Due to the present researcher) TOTAL Source: Appendi Table 1, note applicable.. Obtaining Arriaga s direct and indirect s from the operative of Chandra Sekar (1949). It is also possible to obtain Arriaga s (1984) direct and indirect s from the Chandra Sekar s (1949:36) operative, which is defined by Chandra Sekar as the difference between e 0 and the epectation of life at birth which would have resulted if the mortality conditions had remained unchanged (or inoperative) in the specified age group and the mortality conditions in all other age groups had changed to the etent they had. Similar to Approach-I above, by substituting (.3) in operative formula defined by Chandra Sekar (1949) the direct, indirect terms can be obtained as follows: Chandra Sekar (1949) defines the operative by a formula as = ( ) ( ) 1 l {l ( 1 e e 1 ( ) l 1 e e 1 )} (.6) l 0 l 1

9 A re-look at the methods for decomposing the difference between two life epectancies at birth 91 Table. Decomposition results obtained by using the new formulae: US, females, Approach-I (Due to Arriaga) Age Direct Indirect Eclusive Other Interaction Total Effect Effect Effect Effect Effect Effect (1) () (3) (4)=()+(3) (5) (6)=(5)-(3) (7)=(4)+6) = ()+(5) = ()+(3)+(6) Total Approach-II (Due to the present researcher) Total Approach-III (Due to the present researcher) Total Source: Appendi Table 1, note applicable Note: While applying the new formulae for different age groups here, as required, nl is considered as (T T )

10 9 Krishna Murthy Ponnapalli Substituting (.3) in (.6), we get: ( l l 0 ) ( nl l n L 1 ) l 1 + T ( l 1 l l 1 ) 0 l 1 l (.7) While the first term in equation (.7) represents the direct, the second term represents the indirect. Thus, equation (.7) as a whole represents the eclusive. The direct and indirect s thus obtained from the operative of Chandra Sekar, are not the same as the ones obtained in Approach-I, but equal to the one obtained by Murthy (005) as Arriaga s Approach-II (due to the present researcher) or Approach-II (due to Lopez and Ruzicka). The total of direct and indirect s obtained in Approach- II here are equal to the eclusive term defined by Lopez and Ruzicka (1977) as the s of mortality differentials between the two time periods within specified age intervals. For convenience, following Murthy (005), the results of the application of the above formula (.7) to the data sets under consideration are shown in columns, 3, and 4 of Table 1, Table and Table 3 as Approach-II (due to the present researcher)..3. Obtaining Arriaga s Direct and Indirect Effects from Averaging Main and Operative Effects of Chandra Sekar (1949). Through Approach-I and Approach-II, this study has shown how the direct and indirect s of Arriaga (1984) can be obtained from the main and operative of Chandra Sekar (1949). Following Chandrasekaran (1986), the present author has shown (see Murthy, 005:146) that one may also obtain the eclusive of Arriaga by averaging the main and operative s of Chandra Sekar. Similarly, it is possible to obtain the direct, indirect and eclusive s of Approach-III of Arriaga in Murthy (005:15) by averaging the formulae (.5) and (.7). Thus, averaging the main and operative terms leads us to the equation: ( ) ( 1 l 1 + l l 1 nl 0 l 0 l ) ( nl1 + 1 T 1 ( l 1 ) ( l l T l 1 0 l l 1 l 0 1 l l1 l 1 l )) (.8)

11 A re-look at the methods for decomposing the difference between two life epectancies at birth 93 Age s (1) Direct () Table 3. Decomposition results obtained by using new formulae: US, White Male-Black Male Approach-I (Due to Arriaga) Indirect (3) Eclusive (4) = () + (3) Other (5) Interaction (6) = (5) (3) Total (7) = (4)+(6) = ()+(5) =()+(3)+(6) TOTAL Thus, equation (.8) may also be seen simply as a resultant of averaging equations (.5) and (.7). Again, while the first term in the above formula (.8) is the direct term, the second term is seen as the indirect term, and formula (.8) as a whole is equal to the eclusive term in Approach-III of Arriaga s method or Approach-III of Lopez and Ruzicka s method as given in Murthy (005). It is also the same as Approach-III in Pollard s method as presented by Murthy (005:169). This formula is also equal to the formula (8) given in Carlson (006:4). Carlson further states, Averaging l i values avoids taking one population or the other as a standard or baseline. As stated here, Carlson (006:4) also clearly indicates the first term in his equation (.8) as direct component (an average of l i values times the difference in n e i ) and the second term as indirect component (an average of n e i values times the difference in l i ).

12 94 Krishna Murthy Ponnapalli Age s (1) Table 3. Decomposition results obtained by using new formulae: US, White Male-Black Male (contd.) Approach-II (Due to the present researcher) Direct () Indirect (3) Eclusive (4) = () + (3) Other (5) Interaction (6) = (5) (3) Total (7) = (4)+(6) = ()+(5) =()+(3)+(6) TOTAL The results of the application of the above formula (.8) to the data sets under consideration are shown in columns, 3, and 4 of Table 1, Table and Table 3 as Approach-III (due to the present researcher)..4. Obtaining Arriaga s direct and other s from the interaction forwarded by Chandra Sekar (1949). Arriaga (1984) defined the other as the one resulting from the years of life to be added because the additional survivors (CS) at age + i will continue living under the new mortality level after mortality changed. Simply, by substituting formula (.3), for the difference between two life epectancies of two time periods, in the formula for interaction forwarded by Chandra Sekar (1949), we may obtain the direct and other s as eplained below. The term interaction is defined later on in Section.8 in a detailed way as epressed by different researchers.

13 A re-look at the methods for decomposing the difference between two life epectancies at birth 95 Age s (1) Table 3. Decomposition results obtained by using new formulae: US, White Male-Black Male (contd.) Approach-III (Due to the present researcher) Direct () Indirect (3) Eclusive (4) = () + (3) Other (5) Interaction (6) = (5) (3) Total (7) = (4)+(6) = ()+(5) =()+(3)+(6) TOTAL Source: Appendi Table, note applicable Note: Life Epectancy at birth Black Male = ; Life Epectancy at birth White Male = ; Life Epectancy at birth difference between White Male Black Male = = Chandra Sekar (1949) defined the -interaction forwarded as the, which would result if all interactions are assigned to the youngest age group involved in its production, and represents the same in a formula as: Effect-interaction forwarded = ( ) 1 {l ( 1 e e 1 ( ) l 1 e e 1 )} l 1 0 (.9) Substituting (.3) in (.9), we will get:

14 96 Krishna Murthy Ponnapalli l 1 l 1 0 ( n L l ) nl1 l 1 + T ( ) l 1 l 1 0 l l1 n l (.10) A comparison of the first term in (.10) with that of the first term in equation (.5) seems to show that they are nothing but the same, and so epress the direct as defined by Arriaga (1984:88), once more. The second term in the above equation (.10) is noticed to be the other as specified by Arriaga (1984:89). A comparison of the formula (.10) above, with that of the formula (3.11) from Preston et al., (001: 64) which is etracted and given below as (.10a) also indicates that both are eactly the same. l 1 l 1 0 ( nl l n L 1 ) l 1 + T ( ) l 1 l 1 0 l l1 n l (.10a) According to Preston et al., (001:64) the first term in their equation (given above) refers to the direct of a change in mortality rates between ages and + n. In other words, the first term gives the that a change in the number of years lived between ages and + n. produces on the life epectancy at birth. The second term is the sum of the indirect and interaction s, i.e., the contribution resulting from the person-years to be added because additional survivors at age + n are eposed to new mortality condition (see Preston et al., 001:64). Thus, Preston s formula is shown here to be the resultant of substituting formula (.3) in the interaction forwarded given by Chandra Sekar (1949). However, we may also realize that as Andreev (198) also suggested similar formulae the above equation derived by Preston may also be the result of substituting formula (.3) in the equations suggested by Andreev. However, Chandra Sekar s (1949) method, perhaps the most attractive research done by him, does not seem to get the epected attention among the researchers due to some unknown reasons. The only drawback in Chandra Sekar s (1949) method when compared to Andreev (198), Pressat (1985) and Pollard (198) is that while Andreev, Pressat and Pollard succeeded in suggesting a formula for symmetrical decomposition of e 0 e1 0, it was not the case with Chandra Sekar. Realizing his oversight in his earlier paper, Chandra Sekar or Chandrasekaran (1986) tried to suggest averaging of his -interaction deferred and -interaction forwarded as a new solution to compare his results with those of the United Nations (198, 1985) and Pollard (198).

15 A re-look at the methods for decomposing the difference between two life epectancies at birth 97 Fortunately, a comparison of Chandra Sekar s (1949) paper with that of Arriaga (1984) made the present author (Murthy, 005), average the main and operative terms defined by Chandra Sekar (1949) to arrive at the eclusive and total interaction terms as obtained from Chandra Sekar s method (1949, 1986); this also led to the final conclusion in Murthy (005) that one should always use the symmetric formulae in order to reduce the of total interaction to the difference in the life epectancy at birth given at two time periods. The present eercise also indirectly demonstrates the strength of Chandra Sekar s (1949) method when compared to other methods suggested in succeeding years by several researchers. The results of the application of the above formula (.10) to the data sets under consideration are shown in columns, 5, and 7 of Table 1, Table and Table 3 as Approach-I (due to Arriaga). As formula (.5) and formula (.10) have the same base, we prefer to represent the results of the present analysis also under Approach-I. However Preston et al s (001) main interest is to show the difference between two life epectancies at birth as the resultant of direct and other s as defined by Arriaga (1984). However, it is of interest to know how one can etend the formulae suggested by Preston et al., (001) to arrive at the formulae suggested by the United Nations (1985). As a result, this study attempts to substitute formulae (.3) in the -interaction deferred of Chandra Sekar and to arrive at the United Nations formula (1985), which does not have any available proof for its derivation, as eplained in the following paragraphs..5 Obtaining Arriaga s direct and other s from the -interaction deferred of Chandra Sekar (1949). Chandra Sekar (1949) defines the interaction deferred as the, which would result if all interactions are assigned to the oldest age group involved in its production and represents the same in a formula as: Effect-interaction deferred = ( ) 1 {l ( e e 1 ( ) l e e 1 )} l 0 (.11) Substituting (.3) in (.11), we obtain: l l 0 ( nl l n L 1 ) l 1 + T 1 ( l ) l 0 l 1 l l 1 (.1)

16 98 Krishna Murthy Ponnapalli A comparison of the first term in (.1) with that of the first term in equation (.7) seems to show that they are the same and so epresses the direct as defined in Arriaga s Approach-II (due to the present researcher) or Approach-II (due to Lopez and Ruzicka) as given in Murthy (005). As eplained by Preston et al., (001) in the above equation (.1), the second term refers to the other term as defined by Arriaga (1984) and equals to the indirect plus interaction s terms of Arriaga (1984). According to Preston et al., (001: 64), it gives the contribution resulting from the person-years to be added because additional survivors at age + n are eposed to new mortality conditions. The results of the application of the above formula (.1) to the data sets under consideration are shown in columns, 5, and 7 of panel two of Table 1, Table and Table 3 as Approach-II (due to the present researcher). As formula (.7) and formula (.1) have the same base, we prefer to represent the results of the present analysis also under Approach-II. Now, we may try to see how to obtain the United Nations formulae or the symmetrical formula for decomposition of the difference between two life epectancies at birth given at two time periods, by simply averaging formulae (.10) and (.1), as follows..6. Obtaining Arriaga s direct and other s by averaging -interaction forwarded and -interaction deferred of Chandra Sekar (1949). As a matter of fact, Chandrasekaran (1986), himself, suggested averaging his -interaction forwarded and interaction deferred to arrive at the United Nations (1985) formula. When averaging the -interaction forwarded and -interaction deferred terms given in (.9) and (.11), we will get the formula: ( ) ( 1 l 1 + l l 1 nl 0 l 0 l 1 ( l 1 l l l 0 ) ( ) ( nl1 + 1 l 1 l 1 + l l 1 T ) 0 l T 1 0 l l ) ( 1 T ) (.13) l T 1 l 1 When rewritten the above formula (.13) can be seen to be same as the United Nations (1985) formula given below: ( ( e e 0 e 1 0) = e 1 ( ) l + l 1 ) ( e e 1 ) ( l + l 1 ) (.13a)

17 A re-look at the methods for decomposing the difference between two life epectancies at birth 99 When further simplified, equation (.13) equals to: ( ) ( 1 l 1 + l l 1 nl 0 l 0 l ) ( nl1 + 1 T 1 l 1 l 0 ( l l 1 ) ( l l 1 + T l 1 l 1 l 0 )) l1 l (.14) While the first term in the above equation (.14) is recognised to be the direct term; the second term is observed to be the other s term as defined by Arriaga (1984) and further eplained by Preston et al. (001:64). However, it is seen to be the result of averaging two non-symmetrical formula arrived at by two different approaches and thus epected to give the results that may be obtained from the formulae given by the United Nations (198, 1985), Pollard (198), Andreev (198), Pressat (1985) and the Approach III (due to the present researcher) for Arriaga s approach, Lopez and Ruzicka s approach and also Pollard s approach as eplained by Murthy (005). Thus, equations (.13) and (.14) may be observed as the symmetric formula, and nothing but the epression of the United Nations formula in a different notation that involves three life table functions namely, l, T and n L. I may say, equation (.13) or (.14) is the epected equivalent of the United Nations or Pollard s or Andreev s or Pressat s formula obtained by epanding the formula given by Preston et al., (001:64). Thus we may consider equation (.14) as the final formula suggested by the present researcher by revisiting Preston et al. (001) method. Also, one may consider this equation (.14) as a modified version of United Nations (1985) formula as, unlike United Nations Method, it now meaningfully decomposes total into direct and other (indirect + interaction) as suggested by Preston et al. (001). The results of the application of the above formula (.14) to the data sets under consideration are shown in columns, 5, and 7 of panel three of Table 1, Table and Table 3 as Approach-III (due to the present researcher). As formula (.8) and formula (.14) have the same first term, we prefer to represent the results of the present analysis also under Approach-III..7. A Comparison with Carlson s (006) Results. Carlson s (006) study is an attempt to show that the origin and destination approaches of decomposing a difference in life epectancies are orthogonally related to each other and also produce an origin-destination decomposition matri that gives Andreev s decomposition results, when summed in one direction. In the language of Carlson (006), we may say that Approach-I is equal to Origin

18 300 Krishna Murthy Ponnapalli decomposition, and that Approach-II is equal to the destination decomposition coined by Carlson. Averaging origin and destination lead to our Approach-III. Interestingly, equation (.13) presented in our study eactly equals to the one (.9a) given by Carlson (006:6). As rightly said here, Carlson (006:6) also recognizes the first bracketed term in equation 9a given by him as the direct specified by Arriaga (1984), stating epect that Arriaga s method did not average the l values, privileging one of them as a baseline. Carlson also recognizes this term as identical to the direct specified in equation 8 of his work (p.4) for destination-decomposition. Similar to Carlson, I may state the direct term given in equation (.8) as the first component, equal to the direct as given in equation (.14). The second term in equation (.14) can be said to be the same as equation (9b) in Carlson s study and it is nothing but the other term defined by Arriaga or Preston et al., (001:64). Equation (8) of Carlson then may be equal to the equation (.8) given here. It may be recognized that the second term in equation (8) of Carlson is equal to the indirect term as defined in equation (.8). As aimed in the present paper, Carlson s (006) final section 4 is also aimed at connecting the origin-decomposition and destination-decomposition methods of a difference in life epectancies, into one method that gives the direct and other terms as defined by Arriaga (1984). Our eample in Table 3 depicts the results of decomposition for the race difference in life epectancy between black men and white men in the United Nations in 000. A comparison of our results in Table 3 with those of Appendi Table A results in Carlson s (006) study further clearly indicates that both strived to produce the total difference in life epectancy as a result of the contribution by different age groups as eplained by Andreev or the United Nations or Pollard. As Carlson (006:7) eplained, out of 0.67 differences in life epectancy due to mortality differences between black and white male infants, 0.08 years comes from the direct, the remaining 0.59 years comes from the other s that includes indirect and interaction s as eplained by Preston et al., (001) or Arriaga (1984). Perhaps, indirectly, Carlson (006) also strives for giving the conclusion that one has to use the symmetrical formulae all the time. Table 3, Approach III results presented in our analysis may be looked into the same as given in Appendi Table A, further depicted in Figure 3 of Carlson (006:3-33 & 8). The Total term column (7) given in Table 3 of the present study may be realized as the Andreev s values given as last column in Appendi Table A, p.33 of Carlson (006). Similarly, the direct terms given in column () of Table 3 may be read diagonally from the Appendi Table A

19 A re-look at the methods for decomposing the difference between two life epectancies at birth 301 starting from 0 age to age 100. Appendi Table A of Carlson (006) also presents the Other terms (given in column 5 of Table 3 in the present study) but they have to be arrived at by simple adding of the values that are in between the direct and total terms of each row of each age group. Thus it is interesting to compare the results of Table 3, Approach III with that of Carlson (006)..8. Obtaining the Interaction Effect. In any analysis of decomposition, interaction /s plays an important role. Interaction s may be easy to understand but difficult to obtain. While its impact, when compared to other s, in certain situations may be highly considerable; its impact in certain other situations may be highly negligible, as has already eplained by Murthy (005). However negligible its impact may be, it is of interest for the researcher to know how much impact is due to the interaction, when compared to other s under consideration. In the contet of our present analysis, according to Arriaga (1984:89) the term interaction refers to the that which cannot be allocated to any particular age group alone, but to the change in mortality at all ages and according to him it can be deduced as the difference between the other and the indirect. Like Arriaga, Chandra Sekar (1949:41) too opines that interactions result from the change in mortality conditions of two or more age groups and that they are not wholly attributable to any single age group. Lopez and Ruzicka (1977), Pollard (198) also defined interaction in a manner similar to Arriaga and Chandra Sekar, in their respective studies. In Chandra Sekar s method, while the main ignores the of interaction, the operative takes it into account. On the other hand, by definition, while -interaction deferred results when the interactions were assigned to the oldest age group involved in its production, -interaction forwarded results when all interactions were assigned to the youngest age groups involved in its production. Further, his method enables one to compute the sum of the interaction of all orders as the difference between the epectation of life at birth recorded by the life tables and the sum of all the main s. Similarly, Chandra Sekar s method enables one to compute the sum of the interaction of all orders as the difference between the epectation of life at birth recorded by the life tables and the sum of all the operative s. However, Chandra Sekar did not pronounce the above statement in his study. As a matter of fact, he maintains that the difference

20 30 Krishna Murthy Ponnapalli between the epectation of life at birth recorded equals to the sum of interaction deferred for all age groups that which again equals the sum of -interaction forwarded for all age groups (Chandrasekaran, 1986:6). The present investigation reveals that while the main and the interaction forwarded formulae have the same base l0 1, the operative and the interaction deferred formulae have the same base l0. As such, probably one may arrive at the total interaction s by age group, by taking the difference between the main and interaction forwarded terms as already hinted by Chandra Sekar (1949:45-46), but for getting the sum of the interactions by all age groups, Chandra Sekar (1949:46) however states, the real difficulty is the lack of a logical basis for apportioning this interaction between the various age groups. Disapproving Chandra Sekar s (1949) doubt, Lopez and Ruzicka (1977) have already indicated in their study how to obtain the interaction even from the operative and interaction-deferred as eplained in detail in Murthy (005: 156). Murthy and Gandhi (004) also suggested a formula for interaction to be derived from the difference between the total and the eclusive, where the eclusive here is seen as the average of main and operative as defined by Chandra Sekar (1949); the total is observed to be the average of the -interaction forwarded and interaction-deferred. A formula for the interaction thus obtained is specified in Murthy (005:146). The intention of the net three sections is to show how to derive the above three types of interaction s from the four s well defined by Chandra Sekar (1949), by means of substituting equation (.3) that we have arrived at here from the Keyfitz (1968) formula further given in equation (.1) Obtaining Arriaga s Interaction Effect from Main Effect and Effectinteraction forwarded of Chandra Sekar (1949). Substituting the main term given in equation (.5) from the -interaction forwarded term given in equation (.10) of the present study; and further simplified, we obtain the interaction as: ( T T 1 ) ( l 1 l l 1 0 l l 1 ) (.15) The results of the application of the above formula (.15) to the data sets under consideration are presented in column 6 of panel one of Table 1, Table and Table 3 as Approach-I (due to Arriaga).

21 A re-look at the methods for decomposing the difference between two life epectancies at birth Obtaining Arriaga s Interaction Effect from Operative Effect and Effect-interaction deferred of Chandra Sekar (1949). Substituting the operative term given in equation (.7) from the -interaction deferred term given in equation (.1) of the present study; and further simplified, we may arrive at the equation: ( T T 1 ) ( l l 1 ) l l 0 l 1 (.16) The above equation (.16) is seen to be the same, as given by Lopez and Ruzicka (1977) as part of their equation that decomposes the difference between two life epectancies at birth into the eclusive and interaction (see Murthy, 005:154). The results of the application of the above formula (.16) to the data sets under consideration are presented in column 6 of panel one of Table 1, Table and Table 3 as Approach-II (due to the present researcher) Obtaining Arriaga s Interaction Effect from Total Effect and Eclusive Effect of Chandra Sekar (1949) as defined by Murthy (005). Substituting the eclusive term given in equation (.8) from the total term given in equation (.14) of the present study, and further simplifying, we may arrive at the equation: T ( T 1 ) ) ( l 1 0 +l 0 l 1 +l l + l +l1 l 1 ( l 1 + l ) (.17) The interaction term given in equation (.17) above is seen to be the same as the one given as equation (1.8) in Murthy (005:146). It is also seen to be the average of the terms given in equation (.15) and (.16). The results of the application of the above formula (.17) to the data sets under consideration are presented in column 6 of panel one of Table 1, Table and Table 3 as Approach-III (due to the present researcher). 3 Discussion In the present paper as a first step an attempt has been made to derive various formulae suggested by Arriaga, Preston and United Nations from

22 304 Krishna Murthy Ponnapalli the Chandra Sekar (1949) formulae, by means of a simple modification in the formulae suggested by Candra Sekar using Keyfitz equation. Secondly, however it is not the main aim of the present paper, it is shown how one may rewrite the Chandra Sekar formulae in a meaningful way in terms of direct or indirect or other terms as defined by Arriaga, by handling the Chandra Sekar s various formula in a systematic way. Thirdly, as Murthy (005) states, a comparison of the results presented in the three panels of Table 1 emphasizes the necessity for considering, all the time, the symmetric formulae of the method when compared to the nonsymmetric formulae, as the contribution of the interaction to the total difference between the two life epectancies at birth under consideration is seen to be negligible in the case of symmetric formulae when compared to non-symmetric formulae. The results in Table as well as Table 3 also indicate the same in a clear-cut manner. Fourthly, a comparison of the results of Table with those of Table 1 clearly indicates that while the difference between the two life epectancies at birth when decomposed by the same method but by using different age combinations are seen to be the same, the s decomposed into direct, indirect, etc. cannot be seen to be the same. The total of direct, indirect, eclusive, other and interaction s obtained by using a large number of age groups, as reported generally, (See Table 1), when compared to the ones obtained by using abridged age groups (made by the researcher with specific aims) (See Table ) are observed to be comparatively low; however, the method applied in obtaining the results is the same. It lead to the conclusion that, it is all due to the age combinations considered in decomposition analysis. A comparison of the results (eclusive, interaction and total ) given in Table 1 with that of the Pollard s results given in Appendi Table of Murthy (005, P.170) shows that both are almost the same, as in both cases the author considered the smaller age intervals of 5 while computing the s, as suggested by Pollard (1988, P.75), co-incidentally. Probably estimation error involved in considering different age combinations may be responsible for the observed differences. Probably, computation of direct, indirect and other s terms may get affected much by the weighting procedure followed when compared to the computation of eclusive, interaction and total terms as can be seen from the results of Table 1 and Table of this paper. Thus, on the part of the researcher, there seems to be a great need to apply caution while considering different combinations of age groups.

23 A re-look at the methods for decomposing the difference between two life epectancies at birth 305 Lastly, the researcher should also realize that, for the open-ended age group we may apply the same formula as that for the other age groups, but the formula will differ slightly, as there would not be term in the last age group. This is to say that the open-ended age group will have a direct, but will not have the indirect, interaction or other s terms (Arriaga, 1984:89). It is customary in many of the research papers to represent the open-ended age group by a formula separately; however, this paper does not attempt to do so. 4 Conclusion In the present paper as a first step I tried to show how one can derive different s defined by Arriaga, and its etension as suggested by Murthy (005) from several s defined by Chandra Sekar in a systematic way by means of substituting the Keyfitz equation for decomposition as suggested by Namboodiri and Suchandran. While one of the outcomes of the above eercise is shown to be the Preston et al., formula given for Arriaga s method, the final outcome is shown to be an etension for the Preston et al., formula for Arriaga s method, that is surprisingly seen to be nothing but the symmetric formula suggested by Carlson. This final formula is noticed to be equal to the symmetric formulae suggested by Pollard, Andreev, Pressat, United Nations, and Murthy. I again suggest use of symmetric formulae. To reduce the estimation error when considering different age groups, I suggest use of all the given ages (single years in case of complete life tables, five year age groups in case of abridged life tables), instead of abbreviated ages of ten years or so. To emphasize the above point, as rightly said by Pollard (1988, P.75) in his conclusions, if the age intervals are made smaller and smaller, one may epect almost the same results when using either discrete or continuous approach. This particular outcome of the paper, however not mentioned as the main aim of the paper, to be considered as important as it has several implications on the results of the analysis. Thus, while a direct outcome of the present paper is to give a new look into the different procedures earlier suggested by Chandra Sekar, Arriaga, Preston and United Nations, an indirect outcome is that the paper cautions the researcher to be cautious while considering different age groups for the analysis, as results of the analysis vary drastically from considering broad age groups to the smaller age groups such as 5 year age groups. To conclude further, as shown in the present study and by Murthy (005) different formulae suggested by several researchers can be etended to give