Demography (2021) 58(1):321–344 Published online: 18 January 2021 DOI 10.1215/00703370-8937427 © 2021 The Authors This is an open ac­cess ar­ti­cle dis­trib­uted un­der the terms of a Creative Commons license (CC BY-NC-ND 4.0). Cross-Sectional Average Length of Life Childless Ryohei Mogi, Jessica Nisén, and Vladimir Canudas-Romo ABSTRACT Increases in the av­er­age age at first birth and in the pro­por­tion of women remaining child­less have ex­tended the to­tal num­ber of years that women spend child­less dur­ing their re­pro­duc­tive life­time in sev­eral countries. To quan­tify the num­ber of years that re­pro­duc­tive-age women live with­out chil­dren, we in­tro­duce the cross-sec­tional av­er­age length of life child­less (CALC). This mea­sure includes all­ the age-spe­cific first-birth in­for­ma­tion avail­­able for the co­horts pres­ent at time t; it is a pe­riod mea­sure based on co­hort da­ta. Using the Human Fertility Database, CALC is cal­cu­lated for the year 2015 for all­ countries with long enough his­to­ries of fer­til­ity avail­­able. Results show that women in the ma­jor­ity of the stud­ied countries spend, on av­er­age, more than half of their re­pro­duc­tive lives child­less. Furthermore, the dif­fer­ence be­tween CALCs in two countries can be decomposed to give a clear vi­su­al­iz­ a­tion of how each co­hort con­trib­utes to the dif­fer­ence in the du­ra­tion of the length of child­less life in those pop­ u­la­tions. Our il­lus­tra­tion of the de­com­po­si­tion shows that (1) in re­cent years, fe­male co­horts in Japan and Spain at in­creas­ingly youn­ger ages have been con­trib­ut­ing to more years of child­less life com­pared with those in Sweden, (2) the United States con­tin­ues to rep­re­sent an ex­cep­tion among the high-in­come countries with a low ex­pec­ta­tion for child­less life of wom­en, and (3) Hungary ex­pe­ri­enced a strong pe­riod ef­fect of the re­cent Great Recession. These ex­am­ples show that CALC and its de­com­po­si­tion can pro­vide in­sights into first-birth pat­terns. KEYWORDS Childlessness • Fertility mea­sure­ment • Life ta­ble • Decomposition •​ Cohort versus period Introduction The post­pone­ment (Kohler et al. 2002; Sobotka 2004) and, in­creas­ing­ly, for­go­ing (Frejka et al. 2010; Kreyenfeld and Konietzka 2017; Miettinen et al. 2015) of par­ ent­hood have been im­por­tant fam­ily de­mo­graphic trends of the past half-cen­tury in the de­vel­oped world. Since the 1970s, the age at en­ter­ing moth­er­hood has in­creased by an av­er­age of about one year each de­cade in high-in­come countries, with sub­ stan­tial var­i­a­tion in the av­er­age age and its rate of in­crease across countries (Mills et al. 2011). For in­stance, in the United States, the mean age rose from 23.7 years in 1985 to 26.6 in 2016; in Japan, the re­spec­tive in­crease was more sub­stan­tial, from a ELECTRONIC SUPPLEMENTARY MATERIAL The online version of this article (https://doi.org/10.1215/00703370 -8937427) contains supplementary material. Downloaded from http://read.dukeupress.edu/demography/article-pdf/58/1/321/915868/321mogi.pdf by UAB/SALA DE REVISTES user 322 R. Mogi et al. read­ily higher mean age of 26.7 to 30.7 (OECD 2018). In the United States and in east­ern Eu­ro­pean countries, women con­tinue to en­ter moth­er­hood at a par­tic­u­larly young age, whereas women in south­ern Europe and East Asia en­ter par­ent­hood gen­ er­ally later than else­where, with av­er­age ages in the lat­ter countries cur­rently be­ing above 30. The post­pone­ment of par­ent­hood places women at a higher risk of remain- ing child­less (Kneale and Joshi 2008; Schmidt et al. 2012; Toulemon 1996). In sev­ eral high-in­come countries, remaining child­less has been in­ten­si­fy­ing in the co­horts born from the 1940s to early 1970s. For ex­am­ple, remaining child­less is par­tic­u­larly com­mon in Ger­man-speak­ing countries and south­ern Europe, where slightly more than 20% of women in re­cent birth co­horts are child­less at the end of their re­pro­ duc­tive lives (Sobotka 2017). A no­ta­ble ex­cep­tion to this trend is the United States, where child­less­ness in­creased un­til the fe­male co­horts born in the early 1950s but has be­come less com­mon for more re­cent birth co­horts, stay­ing be­low 13% of women born in 1970 (Frejka 2017). Generally, var­ia­ ­tion in child­bear­ing pat­terns across coun- tries is strongly influ­enced by so­cio­po­lit­i­cal, eco­nom­ic, cul­tur­al, and his­tor­i­cal fac­tors (Frejka et al. 2010; Kreyenfeld and Konietzka 2017; Lesthaeghe and Neidert 2006). The most com­mon in­dexes used to de­scribe the cur­rent first-birth pat­terns aim at cap­tur­ing ei­ther the tim­ing or quan­tum of first birth. For ex­am­ple, the mean age at first birth in­di­cates changes in the tim­ing of the first child­bear­ing, and changes in the pro­ por­tion of child­less women in fe­male co­horts at the end of their re­pro­duc­tive pe­riod at age 50 (here­af­ter re­ferred to as the child­less­ness pro­por­tion) ex­clu­sively show quan­ tum changes. In ad­di­tion, age-spe­cific first-birth rates de­scribe the level of first-birth oc­cur­rence at each age, but it is dif­fi­cult to see the quan­tum of child­less­ness based on these rates be­cause the quan­tum is gen­er­ally obtained from the sum of age-spe­cific first-birth rates. Although these in­dexes are un­doubt­edly use­ful to study ei­ther tim­ing or quan­tum changes, a com­pre­hen­sive mea­sure that cap­tures both tim­ing and quan­tum may also add in­sight in the at­tempt to un­der­stand the phe­nom­e­non of child­less­ness. This is es­pe­cially the case in the cur­rent cir­cum­stances, in which both age at en­ter­ing par­ent­hood and life­time child­less­ness have in­creased in sev­eral high-in­come countries. One pos­si­bil­ity is to study the length of the re­pro­duc­tive life that women are with­ out chil­dren at a given time. Mogi and del Mundo (2020) de­fined this length as the expected years with­out chil­dren (EYWC) for women at re­pro­duc­tive ages. EYWC is cal­cu­lated in a life ta­ble frame­work in which at­tri­tion is the oc­cur­rence of first birth, and it is anal­o­gous to life ex­pec­tancy (Andersson et al. 2017; Bongaarts and Feeney 2006). The EYWC of­fers an ad­di­tional view into the de­mo­graphic pat­terns of first births and the phe­nom­e­non of child­less­ness in con­tem­po­rary so­ci­e­ties by em­pha­ siz­ing the amount of time spent child­less at re­pro­duc­tive ages, with­out mak­ing the dis­tinc­tion be­tween ef­fects of tim­ing and quan­tum of the first child­bear­ing on the ex­pec­ta­tion of the length of life with­out chil­dren. In times of chang­ing fer­til­ity be­hav­ior, co­hort in­dexes may pro­vide an out­dated pic­ture of cur­rent fer­til­ity pat­terns be­cause they are based on in­for­ma­tion on pop­u­la­ tions that are no lon­ger at child­bear­ing ages. The study of child­bear­ing in the pe­riod per­spec­tive (us­ing a syn­thetic co­hort ap­proach), on the other hand, faces the typ­i­cal chal­lenge that in­dexes ex­clu­sively based on pe­riod data do not nec­es­sar­ily re­flect the ex­pe­ri­ence of any real birth co­hort (Bongaarts and Sobotka 2012; Luy 2011). In this study, to over­come the chal­lenge of choos­ing be­tween a pe­riod and co­hort in­dex to study first-birth be­hav­ior, we em­ploy an al­ter­na­tive mea­sure: the cross-sec­tional Downloaded from http://read.dukeupress.edu/demography/article-pdf/58/1/321/915868/321mogi.pdf by UAB/SALA DE REVISTES user Cross-Sectional Average Length of Life Childless 323 av­er­age length of life child­less (CALC). This in­dex uses the en­tire age-spe­cific in­for­ ma­tion of first-birth his­to­ries of co­horts cur­rently at child­bear­ing ages. The con­cept of CALC builds di­rectly on an in­dex de­vel­oped and elab­o­rated in mor­tal­ity re­search (Brouard 1986; Guillot 2003): the cross-sec­tional av­er­age length of life (CAL). The aim of this study is to pres­ent how CALC can pro­vide an al­ter­na­tive per­spec­ tive to the study of first-birth be­hav­ior. The value of us­ing CALC is the abil­ity to (1) cap­ture changes in both tim­ing and quan­tum of first child­birth; (2) com­ple­ment the other pe­riod and co­hort existing mea­sures; and (3) de­com­pose the dif­fer­ence in CALCs be­tween pop­u­la­tions into con­tri­bu­tions from dif­fer­ent co­horts and ages. In this study, we cal­cu­late CALC for all­ countries for which es­sen­tial data are avail­­able in the Human Fertility Database (HFD). Further, we il­lus­trate the use of the mea­sure by com­par­i­sons of se­lected rel­a­tively high-in­come countries with dif­fer­ent ob­served first-birth trends and so­cial con­texts for child­bear­ing. Japan and Spain are countries char­ac­ter­ized by late first-birth tim­ing and high lev­els of child­less­ness, as well as par­tic­u­larly chal­leng­ing so­cial con­di­tions for child­bear­ing. By con­trast, the United States and Hungary are char­ac­ter­ized by early first births and low to mod­er­ate lev­els of child­less­ness. The U.S. con­text is char­ac­ter­ized by low pub­lic sup­port and strong cul­tural var­i­a­tion, and Hungary is char­ac­ter­ized by a his­tory of post-com­mu­nism and cur­rent tra­di­tion­al­ism. Sweden is taken as the coun­try of com­par­i­son: there, first-birth tim­ing ranges from av­er­age to late, strong catch­ing-up of births at older ages is typ­i­ cal, and sup­port of child­bear­ing is par­tic­u­larly strong. Data and Methods Data The HFD is used to ob­tain age-spe­cific first-birth rates by co­hort for the se­lected countries. The HFD is an open ac­cess da­ta­base containing 28 countries from Europe, North and South America, and Asia. The data are heavily scru­ti­nized through qual­ity con­trol, and only those countries with com­pre­hen­sive high-qual­ity in­for­ma­tion are in­clud­ed. To com­pare as many countries as pos­si­ble, we se­lected all­ countries in the HFD with suf­fic­ ient his­to­ries of fer­til­ity in­for­ma­tion needed for the cal­cu­la­tion of CALC avail­­able: Belarus, Czechia, Denmark, Estonia, Hungary, Japan, Lithuania, the Netherlands, Spain, Sweden, and the United States. These countries have long- span birth co­hort data avail­­able in the HFD, in­clud­ing women born in 1966–2003 and cov­er­ing ages 12–50, and thus en­abling the cal­cu­la­tion of CALC for all­ these countries in year 2015. We cal­cu­lated CALC and decomposed its dif­fer­ence be­tween two countries by age and co­hort for all­ these HFD countries for which the nec­es­sary data were avail­­able. In cases where co­hort data for this se­lec­tion were miss­ing for some sin­gle years, we used the pe­riod data from the HFD. We conducted sev­eral sen­si­tiv­ity an­a­ly­ses for countries with com­plete co­hort data and with hy­po­thet­i­cal co­hort data cre­ated from pe­riod da­ta. The re­sults for CALC based on these two data sets were very sim­i­lar, thus jus­ti­fy­ing the par­tial sub­sti­tu­tion strat­e­gy. For ex­am­ple, age-spe­cific first-birth rates cal­cu­lated from both com­plete co­hort data and hy­po­thet­ i­cal co­hort data de­rived from pe­riod in­for­ma­tion were very sim­i­lar (see sec­tion A of the online ap­pen­dix). Downloaded from http://read.dukeupress.edu/demography/article-pdf/58/1/321/915868/321mogi.pdf by UAB/SALA DE REVISTES user 324 R. Mogi et al. Selection of Countries to Compare To il­lus­trate the strengths of the meth­od, we high­light com­par­i­sons of four coun- tries—Hungary, Japan, Spain, and the United States—with Sweden. This se­lec­tion of countries is mo­ti­vated by the ob­served di­ver­sity across first-birth trends as well as the dif­fer­ent con­texts for child­bear­ing across those countries. As shown in Figures 1 and 2, Sweden has witnessed the post­pone­ment of first births since the mid-1970s and is cur­rently char­ac­ter­ized by average to late tim­ing of the first child­bear­ing (29.2 years in 2016). The level of ul­ti­mate child­less­ness in Sweden has remained rel­a­tively sta­ble among birth co­horts of women born since the mid-1950s, reaching ap­prox­i­ma­tely 13% among women born in the late 1960s but in­creas­ing slightly in the more re­cent co­horts. Typical of child­bear­ing in Sweden has been a strong catch­ing-up ef­fect of births at older ages, which has hin­dered de­clines in co­hort fer­til­ity de­spite the trend of post­pone­ment (Andersson et al. 2009; Frejka and Calot 2001). Sweden rep­re­sents a so­cio­po­lit­ic­ al con­text sup­port­ive of gen­der equal­ity and child­bear­ing of du­al-earner cou­ples (Ellingsaeter and Leira 2006; Esping-Andersen 2009): pub­lic sup­port for fam­il­ ies is uni­ver­sal and gen­er­ous, en­abling job-protected long pa­ren­tal leaves with high lev­els of in­come re­place­ment, en­cour­ag­ing fa­thers’ ac­tive par­tic­i­pa­tion in childcare, and en­sur­ing high childcare cov­er­age and flex­i­ble childcare ar­range­ments. Indeed, among Eu­ro­pean countries in 2019, Sweden ranked the highest in gen­der equal­ity (Eu­ro­pean Institute for Gender Equality 2020). Swe- den was not strongly hit by the global Great Recession in 2007–2013 (Matysiak et al. 2018). Japan and Spain are countries where the post­pone­ment of par­ent­hood has been vis­i­ble only since the 1980s, and av­er­age age at en­ter­ing moth­er­hood is cur­rently rel­ a­tively high com­pared with other high-in­come countries (Figure 1). In 2016, Spain reached the highest mean age of all­ countries avail­­able in the HFD, at 30.8 years. Lack of fer­til­ity catch­ing-up at older ages has con­trib­uted to strong in­creases in the ul­ti­mate share of women remaining child­less in re­cent birth co­horts, reaching more than 20% and about 28% of women born in the early 1970s in Spain and Japan, re­spec­tively (Figure 2). Japan records highest ultimate childlessness levels globally (Frejka et al. 2010). Spain, and par­tic­u­larly Japan, rep­re­sent countries where in­creases in wom­en’s ed­u­ ca­tion and em­ploy­ment took place rel­a­tively late but rap­idly and where con­di­tions for fam­i­ly-work rec­on­cil­ia­ ­tion for women re­main dif­fi­cult (Brodmann et al. 2007; Esping-Andersen 2009; McDonald 2008). The con­tri­bu­tion of high op­por­tu­nity costs of moth­er­hood to low first-birth rates arises from less de­vel­oped pub­lic sup­port, his­tory of fam­i­ly-based wel­fare pro­duc­tion, less fam­i­ly-friendly la­bor mar­ket prac­ tices, and men’s weaker con­tri­bu­tion to un­paid work at home (Luci-Greulich and Thevenon 2013). Further, Jap­a­nese women of­ten hes­i­tate to make full use of re­cently im­proved paid leave op­por­tu­ni­ties be­cause they fear it will sig­nal low work com­mit­ ment (McDonald 2008). In ad­di­tion, eco­nomic un­cer­tainty is likely to have neg­a­tively af­fected fer­til­ity rates of young adults in Spain and Japan (Matysiak et al. 2018; McDonald 2008). Japan ex­pe­ri­enced eco­nomic hard­ship since 1989 for 20 years, in the so-called Lost 20 Years (Raymo and Shibata 2017). Spain was hard hit by the Great Recession in 2007–2013, and un­em­ploy­ment and eco­nomic un­cer­tainty are con­sid­ered to have Downloaded from http://read.dukeupress.edu/demography/article-pdf/58/1/321/915868/321mogi.pdf by UAB/SALA DE REVISTES user Cross-Sectional Average Length of Life Childless 325 35 Hungary Japan Spain Sweden United States Mean Age at First Birth 30 25 20 1965 1975 1985 1995 2005 2015 Year Fig. 1 Mean age at first birth in selected countries in the period 1960–2017. Source: Authors’ calculations using the Human Fertility Database. con­trib­uted to lower fer­til­ity rates in the sub­se­quent years. Exclusive to Japan is that child­bear­ing re­mains strongly tied to the in­sti­tu­tion of mar­riage: in 2017, only 2.2% of all­ births took place out­side mar­riage (National Institute of Population and Social Security Research 2019); in Spain, this share in­creased from 3.9% in 1980 to 46.8% in 2017 (Instituto Nacional de Estadística [INE] 2019). Further, in the deeply rooted pa­tri­ar­chal cul­ture of Japan, mar­riage for women typ­i­cally means a com­mit­ment to the care of not only their own chil­dren but also their el­derly par­ents. In light of in­creased ca­reer op­por­tu­ni­ties for wom­en, these care ex­pec­ta­tions of women of­fer an­other in­cen­tive to de­lay fam­ily for­ma­tion (Frejka et al. 2010). Postponement of par­ent­hood has been a prominent trend also in the United States and Hungary but to a lesser ex­tent than in Sweden, not to men­tion east Asian and south­ern Eu­ro­pean countries, such as Japan and Spain (Figure 1). In the United States, the mean age at en­ter­ing moth­er­hood has remained lower than in most high- in­come countries for sev­eral de­cades, and the av­er­age age of 27.2 years upon en­ter­ ing moth­er­hood in the United States in 2016 is low among high-in­come countries. Opposite to the trend in most high-in­come countries (Frejka and Sardon 2006; Fre- jka et al. 2010; Sobotka 2017), child­less­ness has, in fact, de­creased in the United States since the early 1950s fe­male co­horts, down to a level of 11% among women born in the early 1970s (Figure 2). In turn, in Hungary, typ­ic­ al of east­ern and cen­tral Eu­ro­pean countries, post­pone­ment gained ground late, first oc­cur­ring mainly af­ter the fall of the Soviet Union in 1990 (Sobotka 2011). The in­crease was rapid in the fol­low­ing eco­nomic cri­sis, but since 2010, it seems to have come to a halt. Also typ­i­cal of the east­ern and cen­tral Eu­ro­pean countries (Frejka and Sardon 2006), child­less­ness in Hungary be­gan to in­crease first in the co­horts born since the early 1960s, but the in­crease since has been quick, reaching 15% of women born in the early 1970s. Downloaded from http://read.dukeupress.edu/demography/article-pdf/58/1/321/915868/321mogi.pdf by UAB/SALA DE REVISTES user 326 R. Mogi et al. 30 Hungary Japan Spain Sweden 25 United States Childlessness Proportion (%) 20 15 10 5 0 1935 1940 1945 1950 1955 1960 1965 1970 Birth Cohort Fig. 2 Childlessness rate at age 44 or older among women born in 1935–1973 in selected countries. Source: Authors’ calculations using the Human Fertility Database. The United States and Hungary rep­re­sent two con­texts very dif­fer­ent from Sweden as well as each oth­er. Gender-equal prac­tices in the fam­ily in the United States can be viewed as more de­vel­oped than those in Japan, Spain, or Hungary but as some­what less de­vel­oped than those in Sweden (Esping-Andersen 2009; Korintus and Stropnik 2009; Tsuya et al. 2000). In the United States, the pub­ lic sup­port for fam­i­lies is weak, wel­fare pro­duc­tion is strongly mar­ket-based, and in­ter­rup­tions in em­ploy­ment among moth­ers are com­par­at­ ively short (Berger and Waldfogel 2004; Esping-Andersen 2009; Waldfogel 2001). Ethnic and re­li­gious di­ver­sity con­trib­utes to strong var­i­a­tion in fer­til­ity be­hav­ior in the United States, and rates of early child­bear­ing are high in the non-White sub­groups of the pop­ u­la­tion (Lesthaeghe and Neidert 2006; Sutton and Mathews 2004). Further, the U.S. fer­til­ity rates were neg­a­tively af­fected by the Great Recession (Comolli 2017; Currie and Schwandt 2014). In Hungary, on the other hand, tra­di­tional gen­der at­ti­tudes and prac­tices re­main largely prev­a­lent, and child­birth typ­i­cally leads to long work in­ter­rup­tions for moth­ ers (Korintus and Stropnik 2009). For ex­am­ple, com­pared with cou­ples in other Eu­ro­ pean countries, cou­ples in Hungary are least likely to share house­hold tasks (Aassve et al. 2014). Public sup­port for long leaves is gen­er­ous, and young chil­dren are al­most ex­clu­sively cared for at home by their moth­ers. Family pol­icy since the so­cial­ist pe­ riod has con­cen­trated on pro­vid­ing fi­nan­cial aid for fam­i­lies in or­der for moth­ers to stay at home, and the de­vel­op­ment of pol­i­cies that would en­able moth­ers to pur­sue em­ploy­ment has been slow. Consequently, the op­por­tu­nity costs of moth­er­hood are high, and these may boost the trend of post­pone­ment as well as for­go­ing moth­er­hood. Hungary, like Spain, is among the Eu­ro­pean countries that were most strongly hit by the Great Recession in 2007–2013, which is also reflected in the de­creas­ing fer­til­ity rates since 2008 (Matysiak et al. 2018). Downloaded from http://read.dukeupress.edu/demography/article-pdf/58/1/321/915868/321mogi.pdf by UAB/SALA DE REVISTES user Cross-Sectional Average Length of Life Childless 327 Methods Expected Years Without Children (EYWC) The con­cept of life ex­pec­tancy has been pre­vi­ously used in fam­ily de­mog­ra­phy re­ search (Andersson et al. 2017; Bongaarts and Feeney 2006, Bumpass and Lu 2000), but it has fur­ther po­ten­tial to shed light on the cur­rent trends in fer­til­i­ty. The in­creases in the av­er­age age at first birth and in­creas­ing share of women remaining child­less in sev­eral countries trans­late into an ex­ten­sion of the av­er­age length of life that women are child­less dur­ing their re­pro­duc­tive years. A life ta­ble for first births fol­lows a co­hort of child­less women from age 12 un­til age 50 where the event of in­ter­est (exit- ing the co­hort) is the oc­cur­rence of first birth. The expected years with­out chil­dren (EYWC) (Mogi and del Mundo 2020) is de­fined for pe­riod and co­hort as 50 (1a) () EYWC p t = ∫ l p x,t dx 12 ( ) and 50 ( ) EYWCc y = ∫ lc x, y dx, 12 ( ) (1b) where EYWC p (t) and EYWCc ( y) are the EYWC for year t (pe­ri­od) and the co­hort born in year y, re­spec­tive­ly. Similarly, l p (x,t) and lc (x, y) are the prob­a­bil­i­ties of remaining child­less at age x cal­cu­lated from pe­riod data in year t and for the co­hort born in year y, re­spec­tive­ly. The EYWC of­fers ad­di­tional in­sight into the de­mo­ graphic pat­terns of first births and the phe­nom­e­non of child­less­ness in con­tem­po­rary so­ci­e­ties by plac­ing em­pha­sis on the amount of time lived with­out chil­dren at re­pro­ duc­tive ages. Furthermore, both per­spec­tives, the pe­riod and co­hort EYWC, are the ba­sis of the new child­less mea­sure, CALC. Cross-Sectional Average Length of Life Childless (CALC) CALC(t) is a pe­riod mea­sure that includes age-spe­cific first-birth his­to­ries of all­ fe­male co­horts at re­pro­duc­tive ages 12–50 at a given time t. Traditional pe­riod in­dexes use a syn­thetic co­hort ap­proach ag­gre­gat­ing val­ues from dif­fer­ent co­horts from dif­fer­ent ages at a given time t, but CALC uti­lizes real co­hort da­ta. The anal­o­gous mea­sure to CALC is the cross-sec­tional av­er­age length of life (CAL), which was de­vel­oped and elab­or­ ated to com­ple­ment pe­riod and co­hort life ex­pec­tan­cies in mor­tal­ity re­search (Brouard 1986; Guillot 2003). Because the event of in­ter­est is the first birth, the an­a­lyt­ i­cal age range cor­re­sponds to the re­pro­duc­tive ages 12–50, with CALC(t) as fol­lows: 50 () ( CALC t = ∫ lc x,t − x dx, 12 ) (2) where lc (x,t − x) is the prob­a­bil­ity of remaining child­less from age 12 to the age attained in year t—that is, x for the co­hort born in year t – x. The lc (x,t − x) se­ries Downloaded from http://read.dukeupress.edu/demography/article-pdf/58/1/321/915868/321mogi.pdf by UAB/SALA DE REVISTES user 328 R. Mogi et al. 50 40 l c (35, 1980) 30 E Y W C c (1966) l c (25, 1990) Age CALC (2015) 20 l c (15, 2000) 12 10 E Y W C p (2015) 0 1966 1975 1985 1995 2005 2015 Year Fig. 3 Illustration of the age-specific fertility rates used in the probabilities of remaining childless for each of the three measures of duration: the expected years without children for the period of 2015, EYWC p (2015) , and for the cohort of 1966, EYWCc (1966), and for the cross-sectional average length of life childless in 2015, CALC(2015). lc (x,t − x) represents the life table probability of remaining childless for women reaching age x at time t who were born in year t − x. are not nec­es­sar­ily mono­ton­i­cally de­creas­ing with age x be­cause each prob­a­bil­ity of remaining child­less cor­re­sponds to a unique co­hort (Guillot 2003). Figure 3 il­lus­trates the dif­fer­ence be­tween CALC from pe­riod and co­hort expected years with­out chil­dren. The pe­riod EYWC p (2015) is an age-ag­gre­gated in­dex of the syn­thetic co­hort con- structed with the prob­a­bil­i­ties of remaining child­less by age in 2015 (the ver­ti­cal line in 2015 in Figure 3). EYWCc (1966) is obtained from the prob­a­bil­ity of remaining child­less by age for the co­hort of 1966 (the di­ag­o­nal thick line in Figure 3). Different from these unique co­hort or pe­riod in­di­ca­tors, CALC includes all­ the co­hort first-birth in­for­ma­tion at a given time. Thus, CALC(2015) uses all­ the prob­a­bil­i­ties of remaining child­less from the 1966 to 2003 birth co­horts, marked by the shaded tri­an­gle area in Figure 3. Each di­ag­o­nal dashed line in Figure 3 cor­re­sponds to the prob­a­bil­ity of remaining child­ less for per­sons who were born in year t – x and reached age x in 2015. For ex­am­ple, CALC(2015) con­tains the 1966 birth co­hort up to age 50, the 1967 birth co­hort up to age 49, and so on un­til the 2003 birth co­hort, which reached age 12 in 2015. Specifically, CALC(2015) = lc (50,1966) + lc (49,1967) +!+ lc (13,2002) + lc (12,2003); cal­cu­la­tion de­tails us­ing dis­crete-time data can be found in sec­tion B of the online ap­pen­dix. Because CALC con­tains all­ avail­­able co­hort first-birth in­for­ma­tion, it cap­tures the fer­til­ity changes over co­horts. Simulations fix­ing the age-spe­cific first-birth rates across dif­fer­ent co­horts helps to un­der­stand the dy­nam­ics of CALC. For ex­am­ple, if all­ U.S. women of re­pro­duc­tive age in 2015 had the age-spe­cific first-birth rates of women of the post-baby boom of 1966, with rel­a­tively early en­try to moth­er­hood (mean first-birth age of 25.13 years) and a mod­er­ate pro­por­tion remaining child­less (13.8%), then CALC in 2015 would be 17.64 years as op­posed to the ob­served 18.67 years. That is, the post­pone­ment (Figures 1 and 2) of fer­til­ity in the lat­er-born U.S. Downloaded from http://read.dukeupress.edu/demography/article-pdf/58/1/321/915868/321mogi.pdf by UAB/SALA DE REVISTES user Cross-Sectional Average Length of Life Childless 329 co­horts had a sub­stan­tial im­pact on the ex­pec­ta­tion of years remaining child­less in 2015. Another ex­am­ple is us­ing the Jap­a­nese fer­til­ity of the 1966 birth co­hort, with late first birth (mean first-birth age of 28.01) and high child­less­ness pro­por­tion (26.2%), for the lat­er-born fe­male co­horts. This would re­sult in a 2015 CALC of 22.91 years, which is about 1.5 years lower than the ob­served value of 24.40 years. Cohorts born later than 1966 post­poned their child­bear­ing more, and a higher share of women remained child­less (Figures 1 and 2), both of which in­creased the over­all num­ber of years remaining child­less in 2015. The CALC mea­sure as de­fined in Eq. (2) con­sid­ers only first birth as the pos­ si­ble exit from the pop­u­la­tion, as­sum­ing that no other com­pet­ing way of at­tri­tion is pres­ent. To test the sen­si­tiv­ity of this as­sump­tion, we in­cor­po­rated in­for­ma­tion on mor­tal­ity for women (al­though prop­erly only mor­tal­ity for child­less women should be in­clud­ed) from the HMD in an ad­justed CALC in­clud­ing both first birth and mor­tal­i­ty. Table A1 in the online ap­pen­dix shows the com­par­i­sons be­tween pe­riod and co­hort EYWC and CALC, and their ad­justed val­ues when mor­tal­ity was in­clud­ed. All three mea­sures show mi­nor de­vi­a­tions from those val­ues presented in Table 1, and thus the rest of the anal­y­sis in­clude only the first birth in the CALC re­sults. Cohort-Decomposition of the Difference Between Two CALCs General meth­ods of de­com­po­si­tion are widely known and used (Gupta 1978; Horiuchi et al. 2008; Kitagawa 1955; Vaupel and Canudas-Romo 2002). Our cur­rent in­ter­est lays in ex­am­in­ing the dif­fer­ences be­tween CALCs of two pop­u­la­tions and partitioning them by age and co­hort. As such, the de­com­po­si­tion method that Canudas-Romo and Guillot (2015) de­vel­oped for the mor­tal­ity mea­sure CAL is employed to de­com­pose the dif­fer­ence anal­o­gously for CALCs. Let the CALC for pop­u­la­tion i be denoted as CALCi (t). The dif­fer­ence in CALC be­tween two pop­u­la­tions A and B is then 50 () () ( ) CALC B t − CALC A t = ∫ lc,B x,t − x − lc,A x,t − x dx, 12 ( ) (3) where lc,i (x,t − x) is the prob­a­bil­ity of remaining child­less at age x for the co­hort born in year t − x, and the in­te­gral cor­re­sponds to co­horts aged 12–50 and pres­ent at time t. The dif­fer­ences in co­hort prob­a­bil­i­ties of remaining child­less, seen on the right side of Eq. (3), al­low iden­ti­fic­ a­tion of the fer­til­ity con­tri­bu­tion of each of the co­horts pres­ent in a given year. Each of the co­horts presented in Eq. (3) can be fur­ther decomposed by their age con­tri­bu­tions to the dif­fer­ence in CALCs. These age con­tri­bu­tions al­low a com­par­i­ son of co­horts in dif­fer­ent pop­u­la­tions and an as­sess­ment of their fer­til­ity tran­si­tions over the life course. Age Decomposition of the Difference Between Two CALCs The de­com­po­si­tion of the change in CALC over a con­tin­u­ous var­i­able is explained first. Equations in con­tin­u­ous terms are el­e­gant and easy to elab­o­rate, but de­mo­graphic data Downloaded from http://read.dukeupress.edu/demography/article-pdf/58/1/321/915868/321mogi.pdf by UAB/SALA DE REVISTES user 330 R. Mogi et al. are dis­crete, so in this sec­tion, we com­bine the con­tin­u­ous for­mu­la­tions with the es­ti­ ma­tion pro­ce­dures for dis­crete da­ta. We as­sume that CALC and all­ the de­rived mea­sures are func­tions of a var­i­able ξ and then ex­am­ine the de­riv­a­tives of CALC with re­spect to this var­i­able to as­sess dif­fer­ences be­tween pop­u­la­tions. The de­riv­a­tive of CALC(t,ξ) with re­spect to the con­tin­u­ous var­ia­ ble ξ is . 50 . ( ) CA LC t,ξ = ∫ l c x,t − x,ξ dx, 12 ( ) (4) where the dot above the var­i­able de­notes its de­riv­a­tive with re­spect to ξ. Similar no­ta­tion for de­riv­a­tives has fa­cil­i­tated the de­vel­op­ment of de­com­po­si­tion meth­ods (Canudas-Romo and Guillot 2015; Vaupel and Canudas-Romo 2002). The rest of the de­com­po­si­tion re­quires the use of age-spe­cific prob­a­bil­i­ties of remaining child­less. We de­note 1 pa (t − x) as the prob­a­bil­ity of remaining child­less from age a to age a + 1 for the co­hort born in year t – x. Thus, we can ex­press the prob­ab­ il­ity of remaining child­less up to age x as the prod­uct of sin­gle age prob­a­bil­i­ties of remaining child­less from 0 to age x as lc (x,t − x,ξ) = 1 p0 (t − x,ξ) 1 p1 (t − x,ξ)! 1 px−1 (t − x,ξ), with the ra­dix of the life ta­ble be­ing equal to 1. The de­riv­a­tive of the prob­a­bil­ity of remaining child­less on the right of Eq. (4) can then be re­writ­ten as . x −1 . ( ) l c x,t − x,ξ = ∑ 1 p a t − x,ξ ∏ 1 pb t − x,ξ . a =12 ( ) b≠ a ( ) Rearranging and substitut­ing in­side the in­te­gral in Eq. (4), we ob­tain . . ( ) ⎡ x −1 1 p a t − x,ξ CA LC t,ξ = ∫ lc x,t − x,ξ ⎢ ∑ 50 ( ) ( ) ⎤⎥ dx. (5) 12 ⎢⎣ a =12 1 pa t − x,ξ ( ) ⎥⎦ In Eq. (5), the rel­a­tive de­riv­a­tives with re­spect to ξ of the prob­a­bil­ity of remaining p a (t − x,ξ) . child­less, denoted as 1 , cor­re­spond to the age a con­tri­bu­tion of the co­hort 1 pa (t − x,ξ) reaching age x at time t—name­ly, lc (x,t − x,ξ)—in the over­all change in CALC. For com­par­i­sons be­tween pop­u­la­tions, we also use Eq. (5). If data for two pop­u­la­ tions are avail­­able—for ex­am­ple, for countries A and B—we can es­ti­mate the rel­ a­tive de­riv­a­tives in Eq. (5) as the log­a­rithm of the ra­tio of prob­a­bil­ity func­tions, . . ⎡ p (t − x, B) ⎤ 1 p a (t − x,ξ) = ln[ 1 pa (t − x,ξ)] ≈ ln ⎢ 1 a ⎥ . Similarly, co­hort sur­vival to 1 pa (t − x,ξ) ⎣ 1 pa (t − x, A) ⎦ age x,  lc (x,t − x,ξ) , is taken as the av­er­age of the two pop­u­la­tions. The age-co­hort con­tri­bu­tion to the dif­fer­ence in CALCs be­tween pop­u­la­tions A and B is es­ti­mated as the el­e­ments in­side the in­te­gral in Eq. (5) as ( ( ) ( ⎡ lc x,t − x, B + lc x,t − x, A ⎤ ⎡ 1 pa t − x, B ⎤ Δage − cohort a,t − x = ⎢ ) ⎥ ln ⎢ ⎥ . (6) ) ( ) ⎢⎣ 2 ⎥⎦ ⎢⎣ 1 pa t − x, A ⎥⎦ ( ) Downloaded from http://read.dukeupress.edu/demography/article-pdf/58/1/321/915868/321mogi.pdf by UAB/SALA DE REVISTES user Cross-Sectional Average Length of Life Childless 331 Then, the over­all dif­fer­ence in CALCs be­tween the two pop­u­la­tions is cal­cu­lated as the sum of all­ the age-co­hort con­tri­bu­tions over co­horts and ages as 50 x () () CALC B t − CALC A t = ∑ ∑ Δage − cohort a,t − x . x =12 a =12 ( ) (7) Equations (3) and (5) in con­tin­u­ous for­mat and Eq. (7) in dis­crete for­mat are dif­ fer­ent from pe­riod or co­hort de­com­po­si­tions, which can be ap­plied to mea­sures such as the expected years with­out chil­dren (EYWC) and can re­veal only cur­rent fer­til­ity con­di­tions or con­di­tions for one spe­cific co­hort, re­spec­tive­ly. Cohort fer­til­ity dif­fers from year to year and from age to age, and CALC con­denses the avail­­able fer­til­ity his­tory of a num­ber of co­horts pres­ent at time t into one mea­sure. It is pre­cisely the de­com­po­si­tion shown in Eq. (7) that al­lows iden­ti­fy­ing age- and co­hort-spe­cific con­ tri­bu­tions to dif­fer­ences in CALCs. Decomposition of the Difference in CALCs The de­com­po­si­tion anal­y­sis il­lus­trates how each birth co­hort con­trib­utes to the dif­fer­ ence in CALCs be­tween the two pop­u­la­tions by age. Figures 4–7 in the Results sec­ tion cor­re­spond to the cu­mu­la­tive age and co­hort con­tri­bu­tions. In other words, the fig­ures de­pict how a co­hort is progressing in their du­ra­tion of life spent child­less at re­pro­duc­tive ages com­pared with the re­spec­tive co­hort in the com­par­i­son pop­u­la­tion. The cu­mu­la­tive con­tri­bu­tions of dif­fer­ent co­horts at time t sum to the to­tal dif­fer­ence in CALC(t). Thus, the in­spec­tion of the cu­mu­la­tive con­tri­bu­tions al­lows study­ing how a co­hort dif­fer­ence be­tween two countries evolves over time and age. Alternatively, the con­tri­bu­tion to the dif­fer­ence of CALCs of two countries can be il­lus­trated with non­cu­mu­la­tive age- and co­hort-spe­cific con­tri­bu­tions, which can be found in Table 2 and cor­re­spond­ing fig­ures shown in sec­tion C of the online ap­pen­dix. Comparisons and decompositions between any two of the studied countries are shown in an inter- active online application (https://rmogi.shinyapps.io/CALC/). Results Table 1 shows CALC in 2015 as well as EYWC for the year of 2015 (pe­ri­od) and the co­hort born in 1966 in 11 countries. These three in­dexes con­tain dif­fer­ent in­for­ ma­tion, as shown in Figure 3; thus, the val­ues and rank­ings dif­fer some­what across in­dex­es. In gen­er­al, the co­hort EYWCs are lower than pe­riod EYWCs. This is a re­flec­tion of on­go­ing child­bear­ing post­pone­ment, which in­creases the ex­pec­ta­tion of child­less life in the pe­riod com­pared with the co­hort per­spec­tive. Further, coun­try rank is strongly cor­re­lated be­tween the mea­sures, but in some cases, the coun­try rank dif­fers sub­stan­tially across the three mea­sures. For in­stance, Hungary ranks much higher in the pe­riod EYWC than the co­hort EYWC, thus il­lus­trat­ing a case where a pe­riod and a co­hort mea­sure give a dif­fer­ent pic­ture be­cause of re­cent de­mo­graphic changes. In Hungary, the pe­riod age at first birth rose rap­idly from an ini­tial low level in the late 1990s and early 2000s (Figure 1). Likewise, the pro­por­tion of women Downloaded from http://read.dukeupress.edu/demography/article-pdf/58/1/321/915868/321mogi.pdf by UAB/SALA DE REVISTES user 332 R. Mogi et al. Table 1 Three mea­sures of child­less­ness du­ra­tion and their rank­ings: CALC in 2015, and EYWC for the year of 2015 (pe­ri­od) and the co­hort of 1966 in 11 countries Country CALC(2015) EYWC p (2015) EYWCc (1966) Japan 23.23 (1) 23.57 (2) 21.78 (1) Spain 22.49 (2) 23.68 (1) 18.97 (3) The Netherlands 20.86 (3) 21.58 (4) 20.49 (2) Hungary 20.38 (4) 22.29 (3) 13.54 (8) Denmark 20.01 (5) 20.98 (5) 18.42 (4) Sweden 19.83 (6) 20.75 (6) 18.02 (5) Czechia 19.15 (7) 20.44 (7) 12.55 (11) Lithuania 18.12 (8) 18.67 (10) 15.56 (7) Estonia 18.00 (9) 20.32 (8) 13.21 (9) United States 17.45 (10) 19.37 (9) 16.57 (6) Belarus 15.97 (11) 17.45 (11) 13.06 (10) Source: Authors’ cal­cu­la­tions based on the Human Fertility Database. remaining child­less remained low in the fe­male co­horts born un­til the early 1960s but has in­creased strongly in the more re­cent co­horts (Figure 2). CALC shows that ex­cept for Belarus, Estonia, Lithuania, and the United States, women in the stud­ied countries spend, on av­er­age, more than half of their re­pro­ duc­tive lives child­less (i.­e., more than 19 years). Women in Japan and Spain re­cord the highest CALCs: in 2015, women in these countries spent an av­er­age of al­most 60% of their re­pro­duc­tive life with­out chil­dren (23.23 and 22.49 years, re­spec­ tive­ly). Czechia, Denmark, Hungary, the Netherlands, and Sweden have a CALC rang­ing be­tween 19.15 and 20.86 years. Sweden is the coun­try with the me­dian expected length across the stud­ied countries. Swed­ish women spend an av­er­age of 19.83 years with­out chil­dren be­tween ages 12 and 50. Women in Belarus, Estonia, Lithuania, and the United States have lower CALCs. The low­est val­ues are found in Belarus and the United States, where women spend (re­spec­tive­ly) 15.97 and 17.45 years child­less. To il­lus­trate the de­com­po­si­tion of CALC, we study which ages and co­horts con­ trib­ute to the dif­fer­ence in CALC be­tween our four se­lected countries (Hungary, Japan, Spain, and the United States) and the coun­try of com­par­i­son (Sweden). Among these four countries, Japan has the highest 2015 CALC (23.23 years), and the United States has the low­est (17.45 years). The dif­fer­ences in CALC of Japan, Spain, Hungary, and the United States com­pared with Sweden are 3.40, 2.66, 0.55, and −2.38 years, re­spec­tive­ly. Thus, the United States is the only one of the four countries with a lower ex­pec­ta­tion of child­less life in 2015 than Sweden. The de­com­po­si­tions of the dif­fer­ences in CALC in 2015 be­tween countries are presented in Figures 4–7. Figure 4 il­lus­trates the de­com­po­si­tion of the 3.40-year CALC dif­fer­ence be­tween Japan and Sweden. A pos­i­tive con­tri­bu­tion from a par­tic­u­lar age and co­hort in Figure 4 means that Jap­a­nese women be­long­ing to that co­hort are con­trib­ut­ing to a higher ex­pec­ta­tion of child­less life up to that age than Swed­ish wom­en. By con­trast, a neg­ a­tive con­tri­bu­tion would cor­re­spond to a con­tri­bu­tion to higher child­less­ness years by a par­tic­u­lar co­hort and up to that spe­cific age in Sweden than Japan. Positive Downloaded from http://read.dukeupress.edu/demography/article-pdf/58/1/321/915868/321mogi.pdf by UAB/SALA DE REVISTES user Cross-Sectional Average Length of Life Childless 333 Japan 1 45 0.1 Cumulative Age and Cohort Contribution 0.01 40 0.001 35 0.0001 30 0 −0.0001 25 −0.001 20 −0.01 −0.1 15 −1 1970 1975 1980 1985 1990 1995 2000 Sweden (1982) (1987) (1992) (1997) (2002) (2007) (2012) Birth Cohort (Year) Fig. 4 Lexis surface for the cumulative age- and cohort-specific contributions to the difference in CALCs between Sweden (19.83) and Japan (23.23) in 2015 for the 1966–2003 birth cohorts. Positive values cor- respond to a higher risk of remaining childless in Japan than in Sweden; negative values represent a lower risk of remaining childless in Japan than in Sweden. Source: Authors’ calculations based on the Human Fertility Database. con­tri­bu­tions are ob­served at all­ ages and birth co­horts, mean­ing that women in Japan across co­horts and ages con­trib­ute to a lon­ger ex­pec­ta­tion of child­less life than women in Sweden in all­ co­horts of child­bear­ing age in 2015. Especially from age 20 on­ward, across co­horts, the dif­fer­ences in the cu­mu­la­tive con­tri­bu­tion to the expected length of life remaining child­less be­tween Sweden and Japan are rel­a­tively large, at 0.01 to 0.1 years. Furthermore, al­though the dif­fer­ence in­creases fur­ther around age 35 among the older co­horts born in the late 1960s, the gap in­creases grad­u­ally at youn­ger ages among the youn­ger co­horts. This fig­ure clearly il­lus­trates that changes in first-birth trend af­ter age 30 are the key to un­der­stand­ing the in­creas­ing dif­fer­ence be­tween Sweden and Japan in the expected length of child­less life (see also Figure A2 in the online ap­pen­dix, which graphs re­sults of non­cu­mu­la­tive age and co­hort con­tri­bu­tions). The 2.66-year CALC dif­fer­ence be­tween Spain and Sweden is presented in Figure 5. As in the case of Japan, there is a clear age pat­tern for this com­par­i­son, with some changes across co­horts. Unlike in the case of Japan, teen­age moth­er­hood is more com­mon in Spain than in Sweden among all­ co­horts, es­pe­cially in the co­horts born af­ter the mid-1980s. Beyond the teens and early 20s, pos­i­tive val­ues are ob­served in all­ co­horts: Span­ish women be­yond these ages have ac­cu­mu­lated a con­tri­bu­tion to a lon­ger child­less life than Swed­ish wom­en. In the co­horts born un­til the mid-1970s, the con­tri­bu­tion to the ex­pec­ta­tion of child­less life re­mains rel­at­ively sta­ble at high ages. In the youn­ger co­horts, how­ev­er, an in­creas­ing dif­fer­ence is ob­serv­able at in­creas­ ingly young ages, sim­il­ar to that found for Japan. This find­ing il­lus­trates that the more re­cent co­horts of women in Spain are post­pon­ing their en­try into moth­er­hood more Downloaded from http://read.dukeupress.edu/demography/article-pdf/58/1/321/915868/321mogi.pdf by UAB/SALA DE REVISTES user 334 R. Mogi et al. Spain 1 45 0.1 Cumulative Age and Cohort Contribution 0.01 40 0.001 35 0.0001 30 0 −0.0001 25 −0.001 20 −0.01 −0.1 15 −1 1970 1975 1980 1985 1990 1995 2000 Sweden (1982) (1987) (1992) (1997) (2002) (2007) (2012) Birth Cohort (Year) Fig. 5 Lexis surface for the cumulative age- and cohort-specific contributions to the difference in CALCs between Sweden (19.83) and Spain (22.49) in 2015 for the 1966–2003 birth cohorts. Positive values cor- respond to a higher risk of remaining childless in Spain than in Sweden; negative values represent a lower risk of remaining childless in Spain than in Sweden. Source: Authors’ calculations based on the Human Fertility Database. than the re­spec­tive co­horts in Sweden, which in­creases Spain’s CALC and wid­ens the dif­fer­ence in CALC com­pared with Sweden. Interestingly, the two low­est-low fer­til­ity countries, Japan and Spain, show sim­i­lar yet dif­fer­ent trends: the sim­i­lar trends are the pos­i­tive con­tri­bu­tions af­ter age 20 in all­ co­horts and in­creas­ing post­pone­ment in youn­ger co­horts. In both countries, con­tri­ bu­tions to lon­ger life spent child­less are in­ten­si­fy­ing in youn­ger fe­male co­horts com­ pared with Sweden. The higher CALCs of Japan and Spain com­pared with Sweden are largely due to low first birth af­ter age 20. The dif­fer­ence be­tween Japan and Spain in com­par­i­son to Sweden is mainly the con­tri­bu­tion from teen ages. Whereas con­tri­ bu­tions at teen ages are neg­a­tive (fa­vor­ing lower CALC) in the case of Spain (Figure 5), the com­par­i­son of Japan with Sweden does not re­veal neg­a­tive con­tri­bu­tions at any ages (Figure 4). The lat­ter find­ing may re­flect the strong link­age of child­birth and mar­riage in Japan, con­trib­ut­ing to low teen­age fer­til­ity rates there. The de­com­po­si­tion of the 0.55-year CALC dif­fer­ence be­tween Hungary and Swe- den, shown in Figure 6, re­veals a pat­tern very dif­fer­ent from those presented ear­li­er. In co­horts born un­til the early 1970s, women in Hungary ac­cu­mu­lated a lower ex­pec­ ta­tion of child­less life than women in Sweden by the end of their re­pro­duc­tive life. However, this pat­tern is re­versed in the co­horts born there­af­ter: by the age they have reached by 2015, these women have ac­cu­mu­lated a pos­i­tive con­tri­bu­tion to the dif­ fer­ence in CALCs. Interestingly, this pos­it­ive con­tri­bu­tion is ev­i­dent only at higher ages, reflecting higher birth rates at youn­ger ages in Hungary than Sweden. The com­ par­i­son of Hungary and Sweden is likely to il­lus­trate a pe­riod ef­fect be­gin­ning with around 2009 be­cause sev­eral co­horts are be­ing af­fected at dif­fer­ent ages at the same Downloaded from http://read.dukeupress.edu/demography/article-pdf/58/1/321/915868/321mogi.pdf by UAB/SALA DE REVISTES user Cross-Sectional Average Length of Life Childless 335 Hungary 1 45 0.1 Cumulative Age and Cohort Contribution 0.01 40 0.001 35 0.0001 30 0 −0.0001 25 −0.001 20 −0.01 −0.1 15 −1 1970 1975 1980 1985 1990 1995 2000 Sweden (1982) (1987) (1992) (1997) (2002) (2007) (2012) Birth Cohort (Year) Fig. 6 Lexis surface for the cumulative age- and cohort-specific contributions to the difference in CALCs between Sweden (19.83) and Hungary (20.38) in 2015 for the 1966–2003 birth cohorts. Positive values correspond to a higher risk of remaining childless in Hungary than in Sweden; negative values represent a lower risk of remaining childless in Hungary than in Sweden. Source: Authors’ calculations based on the Human Fertility Database. point in time. This pat­tern is likely to have been influ­enced by the re­cent Great Reces- sion in 2007–2013, which de­pressed first-birth rates in Hungary com­pared with Swe- den across all­ ages be­yond the early 20s. Finally, Figure 7 shows the cu­mu­la­tive age and co­hort con­tri­bu­tions to the −2.38-year CALC dif­fer­ence be­tween U.S. and Swed­ish women in 2015 (i.­e., reflecting 2.38 fewer years of child­less life in the United States than in Sweden). Figure 7 shows that Swed­ish co­horts born up to the mid-1970s con­trib­ute to higher CALC un­til their late 30s com­ pared with their U.S. coun­ter­parts; at higher ages, though, the cu­mu­la­tive con­tri­bu­tion turns to fa­vor higher child­less life ex­pec­tancy in the United States. In lat­er-born co­horts, this pat­tern re­verses: across co­horts born since the mid-1970s, the con­tri­bu­tion to CALC in 2015 is neg­a­tive at all­ ages. This neg­a­tive con­tri­bu­tion re­sults in Swed­ish women hav­ing a higher CALC in 2015—a pat­tern that even intensifies in the co­horts born in the mid-1980s, with a stron­ger dif­fer­ence fa­vor­ing child­less­ness in Sweden com­pared with the United States for women in their 20s. This in­ten­si­fy­ing pat­tern may re­flect stron­ger post­pone­ment of births in Sweden than in the United States, which dom­i­na­tes the cu­mu­la­tive child­less­ness de­spite the non­cu­mu­la­tive age-spe­cific pat­terns fa­vor­ ing higher child­less­ness at higher ages in the United States com­pared with Sweden. Table 2 il­lus­trates the case of the U.S.-Sweden CALC dif­fer­ence, show­ing the age and co­hort con­tri­bu­tions for se­lected co­horts. The col­umns in Table 2 show both the non­cu­mu­la­tive age and co­hort con­tri­bu­tions as cal­cu­lated in Eq. (6) (denoted as “Δ age con­tri­bu­tion”) as well as the cu­mu­la­tive con­tri­bu­tions as presented in Figure 7 (denoted as “ ∑ Δ age con­tri­bu­tion”) for the co­horts of 1966, 1978, and 1991. The 1966 birth co­hort starts with higher first-birth rates in the United States than Sweden Downloaded from http://read.dukeupress.edu/demography/article-pdf/58/1/321/915868/321mogi.pdf by UAB/SALA DE REVISTES user 336 R. Mogi et al. The US 1 45 0.1 Cumulative Age and Cohort Contribution 0.01 40 0.001 35 0.0001 30 0 −0.0001 25 −0.001 20 −0.01 −0.1 15 −1 1970 1975 1980 1985 1990 1995 2000 Sweden (1982) (1987) (1992) (1997) (2002) (2007) (2012) Birth Cohort (Year) Fig. 7 Lexis surface for the cumulative age- and cohort-specific contributions to the difference in CALCs between Sweden (19.83) and the United States (17.45) in 2015 for the 1966–2003 birth cohorts. Positive values correspond to a higher risk of remaining childless in the United States than in Sweden; negative values represent a lower risk of remaining childless in the United States than in Sweden. Source: Authors’ calculations based on the Human Fertility Database. (neg­a­tive con­tri­bu­tion to the CALC dif­fer­ence) un­til age 23, when the re­verse oc­curs such that first-birth rates are higher in Sweden (pos­i­tive con­tri­bu­tion to CALC dif­ fer­ence). However, for this co­hort, it is not un­til age 37 when the cu­mu­la­tive con­tri­ bu­tion of child­less­ness pro­por­tion switches from higher lev­els in Sweden than in the United States. We in­ter­pret this as the long-last­ing ef­fect of first-birth post­pone­ment and catch­ing up at later age for Swed­ish fe­males. Table 2 also il­lus­trates that youn­ger co­horts of Swed­ish fe­males ac­cu­mu­late more years of child­less life than the older co­horts, as seen in the com­par­i­sons for the 1966, 1978, and 1991 birth co­horts. For ex­am­ple, in the 1978 co­hort, the catch­ing-up pro­ cess at later ages is also ob­served: the cu­mu­la­tive neg­at­ive dif­fer­ence starts de­creas­ ing from age 28 on­ward, but given larger dif­fer­ences ac­cu­mu­lated at ear­lier ages and up to a higher age, the neg­a­tive cu­mu­la­tive dif­fer­ence re­mains up to the ob­served age of 37. In the 1991 co­hort, only ages up to 24 can be ob­served. These ages are char­ac­ter­ized by con­tri­bu­tions to more years of child­less life in Sweden than in the United States, and these con­tri­bu­tions are larger than those in pre­vi­ous co­horts. Taken to­geth­er, the ear­lier first-birth tim­ing of U.S. women more strongly dom­i­na­tes the pic­ture in the lat­er-born co­horts. Discussion This study aims to pres­ent an al­ter­na­tive way of study­ing first-birth trends in a pe­ riod per­spec­tive by us­ing in­for­ma­tion on first-birth his­to­ries of fe­male co­horts in Downloaded from http://read.dukeupress.edu/demography/article-pdf/58/1/321/915868/321mogi.pdf by UAB/SALA DE REVISTES user Table 2 Age-spe­cific con­tri­bu­tions to the dif­fer­ences in CALCs of −2.38 years be­tween the United States (17.45) and Sweden (19.83) up to 2015, and age- and co­hort-spe­cific con­tri­bu­tions of the 1966, 1978, and 1991 birth co­horts All Cohorts 1966 1978 1991 by UAB/SALA DE REVISTES user Δ Age ∑ Δ Age Δ Age ∑ Δ Age Δ Age ∑ Δ Age Age Δ Survival ∑ Δ Survival Contribution Contribution Contribution Contribution Contribution Contribution 12 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 −0.0001 −0.0001 13 −0.0002 −0.0002 −0.0001 −0.0001 −0.0002 −0.0002 −0.0004 −0.0005 14 −0.0011 −0.0013 −0.0005 −0.0006 −0.0009 −0.0011 −0.0018 −0.0023 15 −0.0045 −0.0058 −0.0017 −0.0023 −0.0028 −0.0039 −0.0061 −0.0084 16 −0.0124 −0.0182 −0.0034 −0.0057 −0.0053 −0.0092 −0.0132 −0.0216 17 −0.0269 −0.0451 −0.0051 −0.0108 −0.0077 −0.0169 −0.0212 −0.0428 Cross-Sectional Average Length of Life Childless 18 −0.0520 −0.0971 −0.0064 −0.0172 −0.0097 −0.0266 −0.0292 −0.0720 19 −0.0878 −0.1849 −0.0059 −0.0231 −0.0104 −0.0370 −0.0338 −0.1058 20 −0.1255 −0.3104 −0.0042 −0.0273 −0.0095 −0.0465 −0.0302 −0.1360 21 −0.1577 −0.4681 −0.0024 −0.0297 −0.0075 −0.0540 −0.0219 −0.1579 22 −0.1805 −0.6486 −0.0006 −0.0303 −0.0065 −0.0605 −0.0160 −0.1739 23 −0.1898 −0.8384 0.0010 −0.0293 −0.0048 −0.0653 −0.0096 −0.1835 24 −0.1861 −1.0245 0.0026 −0.0267 −0.0039 −0.0692 −0.0026 −0.1861 Downloaded from http://read.dukeupress.edu/demography/article-pdf/58/1/321/915868/321mogi.pdf 25 −0.1776 −1.2021 0.0033 −0.0234 −0.0025 −0.0717 26 −0.1650 −1.3671 0.0036 −0.0198 −0.0023 −0.0740 27 −0.1562 −1.5233 0.0032 −0.0166 −0.0006 −0.0746 28 −0.1456 −1.6689 0.0021 −0.0145 0.0008 −0.0738 29 −0.1229 −1.7918 0.0017 −0.0128 0.0026 −0.0712 30 −0.1102 −1.9020 0.0021 −0.0107 0.0057 −0.0655 31 −0.0950 −1.9970 0.0015 −0.0092 0.0065 −0.0590 32 −0.0823 −2.0793 0.0016 −0.0076 0.0067 −0.0523 33 −0.0722 −2.1515 0.0019 −0.0057 0.0052 −0.0471 34 −0.0487 −2.2002 0.0017 −0.0040 0.0046 −0.0425 35 −0.0420 −2.2422 0.0021 −0.0019 0.0030 −0.0395 36 −0.0356 −2.2778 0.0015 −0.0004 0.0019 −0.0376 337 37 −0.0362 −2.3140 0.0018 0.0014 0.0014 −0.0362 338 Table 2 (continued) All Cohorts 1966 1978 1991 by UAB/SALA DE REVISTES user Δ Age ∑ Δ Age Δ Age ∑ Δ Age Δ Age ∑ Δ Age Age Δ Survival ∑ Δ Survival Contribution Contribution Contribution Contribution Contribution Contribution 38 −0.0401 −2.3541 0.0021 0.0035 39 −0.0386 −2.3927 0.0020 0.0055 40 −0.0309 −2.4236 0.0013 0.0068 41 −0.0293 −2.4529 0.0009 0.0077 42 −0.0255 −2.4784 0.0004 0.0081 43 −0.0185 −2.4969 0.0006 0.0087 44 −0.0109 −2.5078 0.0001 0.0088 45 −0.0020 −2.5098 0.0001 0.0089 46 0.0047 −2.5051 0.0001 0.0090 47 0.0082 −2.4969 48 0.0039 −2.4930 49 0.0090 −2.4840 Downloaded from http://read.dukeupress.edu/demography/article-pdf/58/1/321/915868/321mogi.pdf Notes: Δ Age con­tri­bu­tion pres­ents the age con­tri­bu­tion to the dif­fer­ence in CALCs be­tween the United States and Sweden, as in Eq. (6); ∑ Δ Age con­tri­bu­tion means the cu­mu­la­tive age con­tri­bu­tion over age to that dif­fer­ence; Δ Survival is the cu­mu­la­tive age-spe­cific con­tri­bu­tion un­til the age attained in 2015, and ∑ Δ Survival is the ad­di­tion of Δ Survival among all­ ages, as in Eq. (7). The de­vi­a­tion be­tween the dif­fer­ence of CALCs (−2.38) and ∑ Δ Survival at age 49 (−2.48) is caused mostly by our ap­prox­i­ma­tion pro­ce­dures in Eqs. (6) and (7). R. Mogi et al. Cross-Sectional Average Length of Life Childless 339 re­pro­duc­tive ages in a given pe­ri­od. We in­tro­duce a new mea­sure, cross-sec­tional av­er­age length of life child­less (CALC), to the study of first-birth be­hav­ior. This mea­sure of­fers three ad­van­tages. First, it is com­pre­hen­sive: whereas tra­di­tional com­ mon in­dexes def­i­nitely carry value in ex­am­in­ing ei­ther tim­ing or quan­tum of first birth, CALC has value in cap­tur­ing changes in both. Additionally, it in­te­grates the life ta­ble con­cept of du­ra­tion in the study of fam­ily de­mog­ra­phy (Andersson et al. 2017; Bongaarts and Feeney 2006, Bumpass and Lu 2000). The level of CALC in­di­cates the to­tal amount of life spent child­less, in­clud­ing women who even­tu­ally have chil­ dren and those who do not. In this way, CALC can com­ple­ment existing in­dexes of tempo and quan­tum of first births, such as mean age at first birth and the pro­por­tion of remaining child­less. Our il­lus­tra­tive re­sults show that re­pro­duc­tive-age women in most of the 11 coun- tries an­a­lyzed spend more than one-half of their re­pro­duc­tive pe­riod remaining child­ less. Exceptions to this among the stud­ied countries avail­­able in the HFD are Belarus, Estonia, Lithuania, and the United States. On the other hand, women in Japan and Spain spend al­most 60% of their re­pro­duc­tive life child­less—on av­er­age, 23.2 and 22.5 years, re­spec­tive­ly, be­tween ages 12 and 50. Clearly, these ex­am­ples il­lus­trate that CALC may re­flect trends in both tim­ing and quan­tum of first births given that Japan and Spain are countries char­ac­ter­ized by late tim­ing of first births and high child­less­ness, whereas an early tim­ing and rel­a­tively high oc­cur­rence of first births are com­mon in Belarus and the United States. The sec­ond and most im­por­tant ad­van­tage of CALC lies in its abil­ity to pro­vide a pe­riod mea­sure that is in­for­ma­tive not only of the child­bear­ing be­hav­ior of a given pe­riod but also of fer­til­ity be­hav­ior of fe­male co­horts who are at re­pro­duc­tive ages in that pe­ri­od. This char­ac­ter­is­tic is ev­i­dent in the de­com­po­si­tion of the dif­fer­ences of the mea­sure be­tween two countries. Because CALC includes all­ age-spe­cific co­hort first-birth in­for­ma­tion, it al­lows the de­com­po­si­tion of the dif­fer­ences in the mea­sure into con­tri­bu­tions from dif­fer­ent co­horts and ages. CALC can thereby re­veal co­hort, pe­ri­od, and age pat­terns—some­thing that con­ven­tional pe­riod in­dexes based solely on data from the cur­rent pe­riod do not al­low. Traditional pe­riod mea­sures of fer­til­ ity are not in­for­ma­tive of how dif­fer­ent co­horts or prior pe­ri­ods con­trib­ute to them. CALC, how­ev­er, al­lows un­der­stand­ing of changes (or lack of them) in first-birth be­hav­ior across dif­fer­ent co­horts, pe­ri­ods, and ages and can pro­vide in­sight into how cur­rent child­bear­ing pat­terns and trends are influ­enced by the re­cent past. In our view, this is the main strength of the mea­sure. As we il­lus­trate, the re­sults of the de­com­po­ si­tion can also be presented in an in­for­ma­tive way vi­su­al­ly. In this study, four coun­try com­par­i­sons serve as ex­am­ples of us­ing the de­com­po­ si­tion of the CALC. Sweden, the coun­try of ref­er­ence, is char­ac­ter­ized by av­er­age to the late tim­ing of first births, mod­er­ate lev­els of child­less­ness, and a strong ten­dency of birth catch­ing-up at higher ages. The com­par­i­son of Sweden with Japan and Spain, which had a par­tic­u­larly high CALC in 2015, re­flects the later tim­ing of moth­er­ hood in these two countries. These two countries dis­play a larger risk of remaining child­less at re­pro­duc­tive ages across all­ co­horts com­pared with Sweden, with the ex­cep­tion that first-birth rates are higher in Spain than Sweden at very young ages. In ad­di­tion, the youn­ger co­horts of Japan and Spain con­trib­ute to more years of child­less life from in­creas­ingly ear­lier ages. This il­lus­trates the in­ten­si­fy­ing phe­nom­e­non of child­less life at re­pro­duc­tive ages in these countries com­pared with Sweden. In Japan, Downloaded from http://read.dukeupress.edu/demography/article-pdf/58/1/321/915868/321mogi.pdf by UAB/SALA DE REVISTES user 340 R. Mogi et al. this may re­flect the in­crease in the nev­er-mar­ried fe­male pop­u­la­tion, which di­rectly in­flu­ences fer­til­ity be­cause of the strong link be­tween child­birth and mar­riage in this con­text (National Institute of Population and Social Security Research 2019). On the other hand, in Spain, the over­all late tran­si­tion to adult­hood (i.­e., leav­ing the pa­ren­tal home and forming a union) may play a key role in the pro­lon­ga­tion of child­less life (Esteve et al. 2020). The de­com­po­si­tion of the dif­fer­ences in CALC be­tween the United States and Sweden re­flects the im­pact of the high first-birth rates at young ages in the United States on one hand, and the catch­ing-up pro­cess of child­bear­ing at later ages in Swe- den on the other hand, among women born from the late 1960s to mid-1970s. In the youn­ger co­horts, born from the mid-1970s, there are in­di­ca­tions that the dif­fer­ence in life spent with­out chil­dren at re­pro­duc­tive ages in the United States com­pared with Sweden is in­ten­si­fy­ing. A fac­tor that helps ex­plain this dif­fer­ence is the sig­nif­ i­cant var­i­a­tion in first-birth be­hav­ior by race/eth­nic­ity in the United States (Sullivan 2005). His­panic White and non-His­panic Af­ri­can Amer­i­can women con­tinue to en­ter moth­er­hood rel­a­tively early in life, whereas a pat­tern of late moth­er­hood is pres­ent among non-His­panic White wom­en. Given these het­ero­ge­neous trends in first-birth be­hav­ior by race/eth­nic­i­ty, women in the United States have, on av­er­age, lived fewer years of child­less life com­pared with women in Sweden, in which first births are con­cen­trated at later ages. In ad­di­tion, the de­com­po­si­tion of the dif­fer­ence in CALC be­tween Hungary and Sweden is par­tic­u­larly in­ter­est­ing be­cause it shows how dif­fer­ ent co­horts are af­fected at dif­fer­ent ages at the same point in time, thereby show­ing a pe­riod as op­posed to a co­hort ef­fect on first births. This pe­riod ef­fect oc­curred in the years fol­low­ing the on­set of the Great Recession in 2007, and it seems prob­a­ble that the re­ces­sion con­trib­uted to more years spent child­less at re­pro­duc­tive ages in Hungary than in Sweden. The third ad­van­tage of CALC is that it bypasses the distorting ef­fect of chang­ing the tim­ing of first child­bear­ing over a pe­ri­od. Previous fer­til­ity re­search has mostly fo­cused on ways to over­come the distorting ef­fect of chang­ing tim­ing when assessing pe­riod quan­tum—for ex­am­ple, pe­riod quan­tum that is be­ing de­pressed when births are be­ing post­poned and may there­fore not be in­for­ma­tive of quan­tum of child­bear­ing of any real birth co­horts (Bongaarts and Feeney 1998, 2006; Bongaarts and Sobotka 2012; Schoen 2004). By in­clud­ing all­ co­hort in­for­ma­tion, CALC cir­cum­vents the confounding ef­fect of the pe­riod tempo ef­fect on the ex­pec­ta­tion of child­less life. In this way, CALC is an al­ter­na­tive so­lu­tion to pe­riod tempo dis­tor­tions in the study of first-birth be­hav­ior. The pro­posed mea­sure, CALC, also has lim­it­a­tions. CALC re­quires a long his­tory of first-birth da­ta, thus restricting its use to countries that can pro­vide data extending back sev­eral de­cades. However, CALC can also be cal­cu­lated as a trun­cated mea­ sure us­ing in­com­plete co­hort first-birth in­for­ma­tion, sim­i­lar to the trun­cated cross- sec­tional av­er­age length of life, a mea­sure de­vel­oped and used in mor­tal­ity re­search (Canudas-Romo and Guillot 2015). The trun­cated ver­sion of CALC could be used par­tic­u­larly in the mid­dle- or low-in­come countries, where de­tailed fer­til­ity data extending back de­cades are less of­ten avail­­able. The age-co­hort de­com­po­si­tion for trun­cated CALC com­par­i­sons be­tween countries with the same length of fer­til­ity se­ries can still be adapted and ap­plied. Additionally, the es­ti­ma­tes of pe­riod EYWC do of­ten not dif­fer sub­stan­tially from the val­ues of CALC. Thus, pe­riod EYWC, Downloaded from http://read.dukeupress.edu/demography/article-pdf/58/1/321/915868/321mogi.pdf by UAB/SALA DE REVISTES user Cross-Sectional Average Length of Life Childless 341 which can be cal­cu­lated us­ing pe­riod data only, can be a prox­i­mate es­ti­mate for CALC if the long his­tor­i­cal data are not avail­­able. Another lim­i­ta­tion is that CALC may be bi­ased by mor­tal­ity and mi­gra­tion at re­pro­ duc­tive ages. Our sen­si­tiv­ity anal­y­sis us­ing the HMD showed that mor­tal­ity at re­pro­ duc­tive ages in­flu­ences CALC only mar­gin­al­ly. However, the age-spe­cific mor­tal­ity data avail­­able were not fur­ther disaggregated by moth­er­hood sta­tus. Because mor­tal­ ity rates of women dif­fer according to their child­bear­ing his­tory (Barclay et al. 2016; Einiö et al. 2016), this is­sue could be fur­ther in­ves­ti­gated us­ing age- and par­it­y-spe­cific mor­tal­ity data where avail­­able. In the high-in­come countries where mor­tal­ity at re­pro­ duc­tive ages is gen­er­ally low, we ex­pect mor­tal­ity to bias the re­sults only mar­gin­al­ly. However, larger bias is pos­si­ble when CALC is ap­plied in mid­dle- and low-in­come countries where ma­ter­nal mor­tal­ity and other young adult fe­male mor­tal­ity is higher (Canudas-Romo et al. 2014). In ad­di­tion, in- and out-mi­gra­tion might po­ten­tially af­ fect CALC. The data set used, the HFD, as­sumes that mi­grants and non­mi­grants do not dif­fer in their birth his­to­ries. Although this as­sump­tion can be questioned, the data needed to fur­ther dis­en­tan­gle the con­tri­bu­tion of mi­gra­tion on CALC are not widely avail­­able. Where pos­si­ble, fu­ture re­search should em­pir­ic­ ally as­sess this is­sue. Further, sim­i­lar to other fer­til­ity mea­sures, CALC does not dis­tin­guish be­tween vol­un­tary and in­vol­un­tary child­less­ness: the cal­cu­la­tion of CALC cov­ers women who re­main ei­ther vol­un­tary and in­vol­un­tary child­less, and women who en­ter moth­er­hood both in­ten­tion­ally and un­in­ten­tion­al­ly. However, ear­lier work has shown that the dis­ tinc­tion be­tween vol­un­tary and in­vol­un­tary child­less­ness is chal­leng­ing (Berrington and Pattaro 2014; Keizer et al. 2007; Kreyenfeld and Konietzka 2017; Kuhnt and Trappe 2016). From a more gen­eral per­spec­tive, CALC can pro­vide use­ful in­for­ma­tion for policymakers in the ar­eas of em­ploy­ment, pen­sions, and pub­lic health. For in­stance, CALC is likely to re­flect em­ploy­ment rates of women at work­ing ages, at least in countries where moth­ers typ­i­cally have lower em­ploy­ment rates than child­less wom­ en. Furthermore, as shown ear­li­er, the de­com­po­si­tion can be used to study which age groups and co­horts con­trib­ute to pop­u­la­tion-level dif­fer­ences in child­less­ness, and it can also re­veal pe­riod ef­fects on fer­til­i­ty. From a pol­icy per­spec­tive, this char­ac­ter­is­ tic of the CALC is valu­able be­cause, for ex­am­ple, spe­cific pop­u­la­tion sub­groups still at re­pro­duc­tive ages may be iden­ti­fied and targeted if con­sid­ered nec­es­sary. How- ever, given that CALC in a given year is influ­enced by the first-birth his­to­ries of all­ co­horts pres­ent, it is chal­leng­ing to an­a­lyt­ic­ ally re­late it to struc­tural fac­tors, such as em­ploy­ment rates, in a re­gres­sion frame­work. Cohorts are influ­enced by such fac­ tors through­out their re­pro­duc­tive lives, and CALC in a given year is a com­pos­ite of dif­fer­ent co­horts’ ex­pe­ri­ences. Comparisons of CALC be­tween countries can and should, how­ev­er, be mo­ti­vated and discussed in the light of con­tex­tual in­for­ma­tion. To con­clude, CALC is a rich tool of de­scrip­tion to dis­en­tan­gle pat­terns and trends of first-time child­bear­ing. CALC is a pe­riod mea­sure of a wom­an’s life spent with­out chil­dren at re­pro­duc­tive ages, which is based on all­ avail­­able first-birth in­for­ma­tion for fe­male co­horts at re­pro­duc­tive ages pres­ent at a given time. This mea­sure fol­ lows the core de­mo­graphic life ta­ble meth­od­ol­ogy ap­plied here to first births. Family de­mog­ra­phy re­search­ers can ben­e­fit from the rich vi­su­al­i­za­tion that the de­com­po­si­ tion of dif­fer­ences be­tween two CALCs pro­vi­des, which high­lights the con­tri­bu­tion to that gap of each of the co­horts at re­pro­duc­tive ages. Moreover, the ap­pli­ca­tion of Downloaded from http://read.dukeupress.edu/demography/article-pdf/58/1/321/915868/321mogi.pdf by UAB/SALA DE REVISTES user 342 R. Mogi et al. the cross-sec­tional length of life to fam­ily de­mo­graphic out­comes is not re­stricted to the study of first births. For in­stance, the ex­pec­ta­tion of time to higher par­i­ties or to fam­ily de­mo­graphic events other than births could be stud­ied with this ap­proach. ■ Acknowledgments The first au­thor is part of the pro­ject GLOBFAM (RTI2018-096730-B-I00) funded by the Span­ish Ministry of Science, Innovation and Universities, National R&D&I Plan. The con­tri­bu­tion of Jessica Nisén was supported by the Academy of Finland (de­ci­sion num­bers 332863 and 320162). We are grate­ful to Marilia Nepomuceno for her ini­tial con­tri­bu­tion. References Aasve, A., Fuochi, G., & Mencarini, L. (2014). 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Ryohei Mogi (cor­re­spond­ing au­thor) rmogi@ced​­.uab​­.es Mogi • Centre d’Estudis Demogràfics, Universitat Autònoma de Barcelona, Carrer de Ca n’Altayó, Edifici E2, Universitat Autònoma de Barcelona, Bellaterra/Barcelona, Spain Nisén • Max Planck Institute for Demographic Research, Rostock, Germany; and INVEST Research Flagship Center, University of Turku, Turku, Finland Canudas-Romo • School of Demography, Aus­tra­lian National University, Canberra, Australia Downloaded from http://read.dukeupress.edu/demography/article-pdf/58/1/321/915868/321mogi.pdf by UAB/SALA DE REVISTES user