which, in case the fertility rates are specific for five age groups, becomes f =

Transcription

1 1. Introduction CLUSTER ANALYSIS ON AGE PATTERNS OF FERTILITY Prem P. Tlwr Peter A. Lchenbruch, Interntionl Fertility Reserch Prorm The e -specific fertility rte curve, in enerl, is bell- shped, unimodl curve which first rises slowly then shrply in the e roup 15-19, ttins its modl vlue somewhere between es then declines, first slowly then steeply, until it pproches zero round the e of 45 yers. Its two dimensions re the re the shpe. The demorphic term for the former dimension, re, is the totl fertility rte or the completed fmily size, ccordin to whether period dt or cohort dt re utilized for computtion of such rtes. The ltter dimension, nmely, the shpe of the curve, my be termed the fertility pttern or the pttern of reproduction or the e pttern of fertility. The shpe of the e- specific fertility rte curve (subsequently referred to s ASFR) hs severl useful pplictions in demorphic nlysis (1,2,3,8). In study of reltionship between different fertility indices, totl fertility rte hs one -to-one reltionship with either crude birth rte or enerl fertility rte for iven e pttern of fertility (shpe of the e- specific fertility rtes). In comprin the levels of fertility t two time periods or between two or more popultions, it my be necessry to control for differences in e ptterns of fertility. Cole Tye (2), in n ttempt to explin differences in the levels of fertility in two ethnic roups in Sinpore, hd to consider fertility ptterns of births. In popultion projections, where totl number of births re needed for the future time periods, it my be essentil to consider chnes in the e pttern of fertility, especilly in popultions where fertility hs shown decline. Despite the need for considerin e pttern of fertility in demorphic nlysis, the only detiled study devoted to this topic is one by the United Ntions (10). They noted the vritions in terms of two bsic chrcteristics of the curves: the pek -fertility e roup the deree of concentrtion round this pek determined some bsic fertility ptterns in which most of the popultions belon. This study ttempts to determine such bsic fertility ptterns by usin more informtion from the individul curves. The ols of this study re: (1) to investite the extent to which fertility ptterns vry mon popultions, (2) to investite whether bsic e ptterns of fertility exist, if so, wht they re. 2. Sources of Dt The United Ntions hs compiled dt on fertility rtes specific for quinquennil e roups for 73 popultions in round the yer This study uses their dt (10). The UN exmined the dt for vrious inccurcies djusted for most of them. For 36 low- fertility 2/ popultions (TFR < 4.12), the numertor of the ASFR ws derived from reistrtion sttistics on births by e of mother. It ws divided by the officil estimtes of e -specific femles popultion to et the fertility rtes. The reistrtion sttistics were reported to be complete in these countries except for Greece where under - reistrtion ws not extensive. The 37 hih - fertility popultions hve TFR bove 4.12 their dt re of stisfctory qulity for the purposes of this study. Dt of doubtful ccurcy were omitted. For the present study the dt on period ASFR's iven in the UN publiction were converted to e- specific reltive fertility rtes (ASRFR) by dividin them by the sum of ll ASFRs (equlizin the re for ech fertility curve). In other words, if dx is the re of ASFR curve between x x +xdx, fx is defined s the pro- portion of the totl fertility chieved in e x x + dx is obtined by mkin totl re under the curve equl to unity. In mthemticl nottions f dx x dx w w the 1 beinnin end es of reproductive life which, in cse the fertility rtes re specific for five e roups, becomes f = This trnsformtion ws done in order to investite primrily vritions in fertility ptterns or e ptterns of fertility. In the further description f will be referred to s e- specific reltive fertility rtes (ASRFR). 3. Vritions in Individul Fertility Ptterns A lre vrition ws observed in the individul e ptterns of fertility. No two popultions hd exctly the sme pttern; thouh there re some similrities in the ptterns mon different popultions. The men reltive fertility rtes for the 73 popultions under study re 8.4, 26.7, 27.4, 19.5, 21.1, re ifferent from the reltive fertility rtes observed in individul popultions. For instnce, the men reltive fertility rte for the e roup 15-19,is 8.4 compred to the individul popultion rtes which rne from 1.0 for Irel to 21.4 for Gbon. The coefficients of vrintion clculted for ech e roup revel tht roups 15-19, hve reltively lrer vritions. The lrer vrition in the erlier prt of the reproductive life cn be ttributed to vritions in the ptterns of mrrie in ptterns incidence of 808

2 illeitimcy mon countries. Tht in th ltter prt is due to severl bioloicl socioculturl fctors which ffect termintion of the reproductive life. The fertility pttern of hih -fertility country is likely to be different from tht of low- fertility country becuse both these roups of countries differ not only in their fmily size norms but lso in the effects of bioloicl socio- culturl fctors. Hence the first prt of the study considered whether ll 73 popultions should be considered tobther or in sub -roups. For such study the popultions were rouped into two sub -roups--those with TFR reter thn 4.12 those with less thn tht. This roupin ws dopted in view of the U.N. observtion (10) tht no other socil or economic indictor seprtes countries so well into two roups s does the TFR level of Step -wise discriminnt nlysis ws dopted to obtin n eqution which best discriminted fertility ptterns of these two roups. A posterior clssifiction ws obtined on the bsis of this eqution. It ws found tht fertility ptterns of hih -fertility countries re different from those for low- fertility countries. The fertility pttern of low- fertility popultion typiclly hs lower reltive fertility rtes t es hiher reltive fertility rtes t es The typicl fertility pttern for low- fertility country is 7, 29, 30, 19, 11, for seven quinquennil e roups s compred to 10, 24, 24, 20, 14, 6 2 for hih -fertility. Since the fertility pttern for hih -fertility country is different thn for low- fertility country, the two roups were considered seprtely to determine bsic e ptterns for ech. 4. Approch to the Problem The United Ntions ttempted to determine bsic e ptterns on the bsis of two chrcteristics of the curve: (1) modl fertility e roup (2) spred round the modl e roup. We will use cluster nlysis to pportion the totl number of e ptterns of fertility into the "best" number of homoeneous disjoint subsets(4). The vere pttern of ech subset is the bsic e pttern for tht roup of countries. Cluster nlysis identifies homoeneous clsses or roups in lrer set of dt (7). The first step in its use is to choose similrity index which provides mesure of closeness between two units. Such n index is clculted for ll pirs of units. In the second step clusterin lorithm is used to form few clusters so tht the units in one cluster re more similr thn those in different clusters. A "stoppin rule" is ultimtely needed to terminte the clusterin process. The third step in the cluster nlysis pplicble in our cse, is to choose the "best" set of clusters if there re more thn one set of clusters formed by tkin more thn one similrity indexes. 5. Cluster Anlysis on Ae Ptterns of Fertility. Similrity Index The indices of similrity were defined by subtrctin distnces between the fertility ptterns of two popultions from the mximum possible distnce. Two types of distnces between popultions i j considered in this study re: = [ D. r(fi - Fj)T(Fi Fj)1 f where T = Trnspose of the mtrix Column vector of reltive fertility rtes for seven quinquennil e roups for country i These distnces were converted to mesures of similrity by subtrctin them from The mximum distnce between 7 two fertility curves. Thus the similrity indices, for our nlysis, were defined s s S D - dij D - Dij The first index considers the discrepncy in fertility pttern curves in ech e roup, while the second considers discrepncy in cumultive reproduction up to different e roups. In our nlysis, we identified these similrity indices s those bsed on simple distnce on cumulted distnce respectively. The clusters formed by these two similrity indices for low- fertility hih- fertility countries re discussed seprtely. b.l Process of Cluster Formtion The cluster formtion process consisted of two prts. The first prt clculted the (n x n) symmetric mtrix S of similrity coefficients between every pir of units. The second prt formed clusters by doptin lomertive lorithm (4). Specificlly, the cluster formtion prorm proceeded in the followin fshion: (1) Initilly, the n units were considered to constitute n clusters. Ech unit ws desinted by the subscript i (í = 1,2 n) (2) In the first step, the (n x n) symmetric similrity mtrix ws scnned for the mximum vlue of Sij, i <j. We denote Mx ll i <j *(Sij) Sij Then S.. becme the level of similrity t which clusters were formed t this step. (3) In cse the coefficient (k = 1, 2 units then rom f vlue of the similrity between units i k n; k j) or between k were tied with selection decided 809

3 whether cluster ws formed by (i,j), (i,k) or (j,k) units. For this exmple, let us ssume tht the units i clustered toether. (4) The roup which clustered (1,1)íh units toether is represented by the ith unit (note: i < j)whose row column elements were replced by those of Min (Sik, S.k); (k = 1,2,...n; # i or j) Elements correspondin to unit j were replced by those of the nth unit. Thus ech cluster ws represented by unit in the next step of the similrity mtrix whose dimension ws reduced by one. (5) The vlue of vrious clusters their constituent units were recorded (6) Steps (2) to (5) were repeted until ll the units were included in one cluster. This lorithm ws used to form two sets of clusters from two indexes of similrity considered here. The two indexes re the simple distnce the cumulted distnce between two units. It ws noted tht the process of clusters formtion ws slower for the low- fertility counties for both these sets. This miht be n indictor of more heteroeneity in the fertility ptterns mon the low - fertility countries becuse of reter control on their reproductive lives. b. 'Stoppin rules' for decision on the optimum number of clusters The clusterin process strted from the ste when ech unit ws n individul cluster to the lst ste when ll units formed one cluster. Some rule ws needed to decide on the optimum number of 'homoeneous' clusters. One possible pproch is to use distribution theory, lthouh fully wre tht it is not pplicble in the usul sttisticl sense. This pproch looks t some sttistics of within - cluster between -cluster vritions t ech step of the clusterin process. Thouh these sttistics re enerlly used to test the sttisticl sinificnce, they were used here to study their chned pttern when the clusterin process psses from one ste to the next. An brupt chne in the expected chnin pttern cn be suestive of stoppin ste. The sttistics bsed on between clusters within clusters vritions considered but not pursued further were (1) those enerlly used in multivrite testin -- lrest root, trce or the likelihood criterion, (2) multivrite outlier test sttistics, becuse of smller number of units per cluster compred to the number of vrites (six correspondin to seven quinquennil e roups). The other sttistics computed t different stes of the clusterin process whose results were used to decide on the optimum number of clusters will be described here. The purpose of look- in t severl sttistics ws to check the consistency in decision bout the stoppin ste since no one sttistic ppered to be better thn the other. Two types of mesures were used: (1) sttistics bsed on unweihted distnce, (2) sttistics bsed on weihted distnce. The word distnce hs been used here thouh nloy with univrite nlysis of vrince is very close. Like nlysis of vrince, totl vrince is bein prtitioned into within- cluster between -clusters. The first type of mesure is F -rtio test in the nlysis of vrince the second mesure is similr to the enerlized D2 iven by Ro(6). The mesure bsed on unweihted distnce t step when there were clusters is defined s where UD - (B/(-1)) 1 W/(n-) W = overll within -cluster vrince = Wk k =1 within -cluster vrince for kth cluster is clculted s Ek(fk)- f-(k)) T(fk)- f(k)) - where f(k)= column vector for reltively fertility rtes for seven e roups for country i which belons to cluster k, f(k) = column vector for men reltive fertility rtes for seven e roup in cluster k. T nk = trnspose of the column vector, Wk number of countries in cluster k B = overll between -cluster vrince clculted s n E nk where f(') n (f(k)-f()) = column vector for overll men reltive fertility rtes for seven e roups. = number of clusters in which n countries hve been rouped with nl,n2...n countries in ech cluster = totl number of countries The mesure bsed on weihted distnce t step when there re clusters is defined s 810

4 cline where WD - 1 ( -1) (p -1) k=1 (k) nk (f f (S)(-?) ) ( f S = (f. (k) (k) (k) (k) ) w T k=1 i-1 n- P = number of independent vrites = 6 in our cse Obviously, the lrer vlues of UD or WD would men lrer distnce betweenlclusters smller distnces within clusters. Thus lrer vlues of these indices re preferred for the choice of stoppin ste., These two mesures were clculted t ech ste of the clusterin process. It my be noticed tht both these mesures consider the number of clusters t ech ste which is essentil since these mesures re clculted t different stes of the clusterin process when there re different numbers of clusters. Tbles 1 2 ive vlues of UD1 WD t different stes of the clusterin process. Tble 1 ives results for the cse when the simple distnce ws tken s the similrity index Tble 2 when the cumulted distnce ws used. In enerl the vlues of UD 1 WD in Tbles 1 2 decline with the dein the number of clusters. But our choice of the "best" number of clusters ws bsed on two desirble properties. First,the number of clusters should not be too lre secondly, the within clusters distnce should not be too lre (or between clusters distnce should be lre)i.e. lrer vlue of the indices were to be preferred. These two considertions suest five clusters for the low - fertility countries from Tble 1. Both UD1 WD suest ret increse in the within clusters vrince compred to between clusters if one considers choice of less thn five clusters (shrp decrese in UD1 WD). The choice hs to be on eiht clusters if the within clusters vrince hs to be further reduced. The sme considertions suest three clusters for the hih -fertility countries in Tble 1. In the cse of Tble 2, UD} suests seven clusters for the developed countries but WD suests six, lthouh it seems tht the vlue of WD remins stble from step 3 to 6 except for rom fluctutions. Hence, the choice in this cse ws seven clusters for the low -fertility countries. For the hih -fertility countries, the vlue of remins stble from steps 4 to 7, while tht of WD is stble from steps 3 to 6 except for the rom fluctutions. Hence, the choice ws three clusters for the hih- fertility countries. In relity, there re only five clusters for the low- fertility countries bsed on Tble 2, since Jpn Irel hve unique fertility ptterns form clusters of their own. The sme rument suests three clusters for the hih -fertility countries in Tble 1. The bsic pttern for this cluster is obtined by verin the fertility ptterns of ll popultions formin tht prticulr cluster. Althouh some countries belon to the sme cluster on the bsis of the two similrity indices considered here, there re other countries which belon to different clusters if different similrity indices re used. c. The Choice of "best set of e ptterns of fertility" The choice of the two similrity indices led to the two sets of e ptterns of fertility both for the low- fertility the hih - fertility countries. The next decision ws to mke choice of the best set out of the two vilble. The first set bsed on simple distnce s mesure of similrity hd five fertility ptterns for the low- fertility three for the hih -fertility countries. The second set (bsed on cumulted distnce s mtter of similrity) hd seven fertility ptterns for the low- fertility three for the hih fertility countries. The choice of the "best" set ws bsed on sttisticl demorphic considertions. It ws desired tht the bsic fertility ptterns should be distinct. In other words, countries within pttern (cluster) should be more homoeneous thn those in different ptterns. Hence the sttisticl considertions were bsed on smller within -clusters distnce or lrer between -clusters distnce. The demorphic considertions required tht the "best" set should cover extreme shpes of the e- specific reltive fertility curves. The followin prrphs discuss the results of vrious sttisticl demorphic techniques utilized for investitions into the choice of the "best" set. A method bsed on rnks of the pir -wise distnces ws lso used for this purpose. Step -wise discriminnt nlysis ws used to study whether different fertility ptterns within set were distinct non -overlppin. In Set I ll low- fertility countries were correctly clssified in the three clusters; only one of the 37 hih -fertility countries ws misclssified; it hd 0.45 posterior probbility of belonin to the correct roup. For Set II, ll hih- fertility countries were correctly clssified, but one of the low- fertility countries ws misclssified with 0.37 probbility of correct clssifiction. If the number of misclssified cses their posterior probbilities re tken s n index of cler -cut distinct fertility ptterns in set, then Set I seems to be slihtly better by this nlysis. A sttisticl test for outliers ws used to 811

5 determine whether ll countries formin cluster hve come from some bsic pttern. It provided test for homoeneity of ll fertility ptterns in cluster. In its ppliction, we hve ssumed tht the bsic distributionl ssumptions re fulfilled by our dt. Wilks' (11) test for multivrite outliers ws pplied to individul clusters within the two sets 4/. In Set I, no outliers were found. In Set II, there ws one outlier. If the number of outliers is tken s n index of heteroeneity of clusters, then this test, too, tends to fvor Set I. Demorphiclly, the fertility ptterns should cover the extreme shpes since they hve effect on the popultion birth rtes the rowth rtes. Stble popultion prmeters were estimted for ech fertility pttern within the different sets under considertion. It is found tht the fertility ptterns in Set I covered the lrer rne of the stble popultion prmeters. The rnk order of distnces method considers ll the (2)pir -wise distnces between countries. The principle underlyin this technique is tht two countries with shorter distnces between them re more likely to belon to the sme cluster. Tht is, pir -wise distnces between units in the sme cluster should be smller thn between those in different clusters. Let there be clusters CI, C2 Cwith ni, n2 number of units sucn tht n nk. The totl number of k =1 pir -wise, within -cluster distnces re iven by N = (2l) + (22) + +(Z) where N is less thn (2) unless = 1. Now if ll (n) distnces re rnked in n rry of descendin order then idelly, the lst N distnces should be from countries which belon to sme clusters. Opertionlly, ll the n ()distnces were rnked in descendin rry; 'S' ws mrked for distnce between those countries both of which were in the sme cluster 'D' for those in the different, then the per cent of S's in the lst N distnces were tken s n index of distinct cler - cut roupin in clusters within set. The idel vlue of this index is 100. This index ws clculted for both the sets; Tble 3 shows the results for the developed the developin countries. Set I shows the lrest percente of S's thouh they re fr from the idel vlue of 100 per cent. Thus this method lso suests Set I s the "best" choice. Different techniques dopted in the investition on the "best" set hve enerlly shown results in fvor of Set I. e. Results: The nlysis discussed bove shows tht the bsic e ptterns of fertility, i.e. percent of the totl fertility rtes ccounted for by different quinquennil e roups, for the developin countries re: Pttern I= UPI : 19.7, 24.1, 21.2, 16.6, 11.5, 4.8, 2.2 Pttern II= UP2: 12.0, 25.9, 24.1, 18.4, 12.6, 5.4, 1.6 Pttern III = UP3: 5.9, 21.7, 25.8, 22.2, 15.7, 7.0, 1.6 those 6/ for the developed countries re iven s follows: Pttern I = DPI: 13.1, 37.9, 26.6, 13.6, 6.6, 2.0, 0.2 Pttern II = DP2: 6.5, 30.9, 30.4, 18.8, 9.7, 3.3, 0.3 Pttern III = DP3: 3.4, 23.4, 31.5, 22.6, 13.6, 5.0, 0.6 The subscripts 1,2 3 hve been ssined with view of suestin typoloy to the bsic fertility ptterns. The ptterns with subscripts l's (UP1 DP1) hd lower men, medin modl es t reproduction compred to the other subscripts in the sme roup(u or D). The results obtined by cluster nlysis technique were confirmed when individul fertility ptterns were rphiclly represented by () Prmeters of mthemticl curve fitted to the fertility ptterns(5), (b) First two fctors in scores in fctor nlysis on six -vrite fertility ptterns Different bsic e ptterns of fertility occupied different non -overlppin sub -spces thus reinforced the fertility ptterns obtined in our erlier nlysis. 6. Summry Ae- specific fertility rte dt for the period round 1960 from 36 low- fertility 37 hih - fertility countries were converted to e- specific reltive fertility rtes by equlizin the re for ech fertility curve. This ws done primrily to investite vritions in fertility ptterns. It ws found tht the fertility ptterns for the hih -fertility countries were different from those for the low- fertility countries. Thus these two roups of countries were considered seprtely in the investition of the bsic e ptterns. A cluster nlysis technique ws utilized to determine "resonble" number of ptterns. It ws found tht both the low- fertility countries, excludin Jpn Irel, the hih- fertility countries could be rouped into three bsic ptterns ech. Their bsic ptterns, i.e. percent of the totl fertility rtes ccounted for by different quinquennil e roups, re iven in the pper. 812

6 TABLE 1 Some Descriptive Sttistics Clculted t Different Steps, to Determine the Stoppin Ste, of the Clusterin Process with Simple Distnce s the Similrity Index, for the Low -Fertility the Hih -Fertility Countries Low -Fertility Countries Hih -Fertility Countries of Unweihted Weihted of Step Clusters Step Clusters Unweihted Weihted See text for the definitions of these nottions. TABLE 2 Some Descriptive Sttistics Clculted t Different Steps, to Determine the Stoppin Ste, of the Clusterin Process with Cumulted Distnce As the Similrity Index, for the Low- Fertility the Hih -Fertility Countries Low -Fertility Countries Hih Fertility Countries Step of Clusters Unweihted (UDI) Weihted Step of Clusters Unweihted (UD1) Weihted See text for the definitions of these nottions. 813

7 TABLE 3 of S's And D's in the Lst N of the (2) Pir-Wise Rnked (Descendin) Distnces by Sets of Clusters, for the Low -Fertility Hih Fertility Countries. Country Type Set No. Low -fertility Percent S of totl N Hih -fertility Percent S of totl N N N FOOTNOTES 1/ This is prt of the first uthor's Ph.D. disserttion, which oes beyond wht hs been reported here. Vrious fctors responsible for differentil fertility ptterns hve been studied. Mny demorphic pplictions hve lso been shown. 2/ The U.N. divides ll 73 popultions into two roups, those with GRR >2.0 those with GRR < 2.0. We trnslted these into totl fertility rtes (TFR). 3/ The cluster nlysis prorm roups n units into few clusters the totl vrince into two components: between - clusters vrition within -clusters vrition. Ech successive step of the prorm tends to increse the within - clusters component (pooled within vrince) thus reduces the one correspondin to the between -clusters vrition. 4/ A cluster must hve seven or more countries for this test to be pplicble. 5/ "U" sts for underdeveloped popultions. Similrly "D: will be for the developed ones. 6/ Jpn Irel re omitted, since both of them hve unique fertility ptterns form clusters of their own. Jpn's unique fertility pttern is explined by very hih incidence of bortion. A very hih e t mrrie very hih incidence of spinsterhood in Irel is responsible for its unique fertility pttern. REFERENCES 1. Cole, A.J.(1965) "Birth rtes, deth rtes rtes of rowth in humn popultion ". Public Helth Popultion Chne. Eds. Mindel C. Sheps Jenne C. Ridley. University of Pittsburh Press, Pittsburh, Popultion. Milbnk Memoril Fund Qurterly 39(4), Cole, A.J. Zelnik, M. (1963). New Estimtes of Fertility Popultion in the United Sttes. Princeton University Press. Princeton. 4. Gower, J.C.(1967) "A Comprison of Some Methods Cluster Anlysis ", Biometrics 23(4). 5. Johnson, N.L. (1949). Systems of Frequency Curves Generted by Methods of Trnsltion. Biometrik 36(1 &2), Ro, C.K. (1952). Advnced Sttisticl Methods in Biometric Reserch. John Wiley & Sons, New York. 7. Sokl, R.R., Sneth P.H. (1963). Principles of Numericl Txonomy. W.H. Freemn Compny, Sn Frncisco London. 8. Tlwr, P.O.(1970). Ae Ptterns of Fertility. Institute of Sttistics Mimeo Series No. 656, University of North Crolin, Chpel Hill, North Crolin. 9. Tlwr, P.P. (1971). "Vritions in Fertility Ptterns Amon Popultions ". Presented in All -Indi Seminr on Demorphy Sttistics, Demorphic Reserch Center, Burs Hindu University, Vrnsi, April 23-25, United Ntions (1965). Popultion Bulletin of the United Ntions No. 7, 1963: Conditions Trends of Fertility in the World. New York 11. Wilks, S.S. (1963). "Multivrite Sttisticl Outliers ". Snkhy Series A, 25(1). 2. Cole, A.J. Tye, C.Y.(1961). The Sinifiof Afe Ptterns of Fertility in Hih Fertility 814