LINEARIZED GROWTH CURVE AND AGE AT MENARCHE

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SUT Jo晒al of Hathemat ics．（Foエmerly TRU Hathematics） Volune 26， Nurnber 1 （1990）．， 89−99

LIN］EARI ZED GROWTH CURVE AND AGE AT MENAR（HE

MOtosaburo MASUYAMA （Received January i6， 1990； Revised Hay 7， 1990） ABSTRAC「・α・r find垣9・th・t血divi蝕1i・ed g・・Wh v・1・city・f h・ight is individuality−free and two paral皿eters in linearized growth curve in th・ sec・nd phase are hi餉・y・・g・ti鴨・y・・r・e・・t・d・…ec・pfi皿・d with American femaユe data． The coefficient b． i皿 the definition of t proper age in the second phase， but not in the first phase， is highly p・・itiv・1y・・rr・1・t・d with the ag・at m・narch・㌔・ AMS ヱ980 szthゴeot oZassifioαtions．』Primaエγ．92AO 7； Secondary 92AO9 Key wo？cls． Age at menarche ， a1皿ost−one parameter 〔ALONPA） hypothesis， 1i皿earized growth 口rve 〔1・GC）， proper age． 1． Introduction． 01d data， but of good quality， on white American ［1］were available， so that we tried to reconfim our findings［2，3，4］，whi（畑ere derived fr㎝routine work at a Japanese j unior high schoo1． L・t hirW， f・・i一ヱ・2・・…い・th・ h・ight Of th・i−th f・m・1・at h・・・h・・n・1・gical・g・t ・nd ・i ・bit b・her p・・per ag・・bi b・ing・p・・itive c・n・t・nt・ Let bS denote the皿ean with respect to the index”i”by a”dottt i皿plaCe of ． i． Assume a f㎝ale with the mean height h．（t］…md call her the meξm female．、 Likewise， we define the mean male． We assume that the individualized velocity（防Z／cZsi is free of the individua− 1ity， i．e．

〔1）．咋／吟＝de・／dt・

By integration， we get ゴ、．

89

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90 _{L珊皿IZゆGRO㎜㎝理▲m▲GE AT’㎜凪C旺}

（2）㌧「tノ＝％チ》・「tノ〔th・fi・・t f迦㎞ent・1・・1・ti・n），吻「e％＝0血娠first phお…r in．・th・ Childh・・d・、飢d thr・・rrelati・…effi− cient〔CC）・bet・een・i and bi i・ th・ sec・nd phase・・r iP ti・e adalescence・i・ highly negative， according to the allnost−one parameter （in short， ALCNPA） hypo− thesis． ALCNPA means one para皿eter， or two or more highly correlated parameters． Assum山lg that r＝一ヱ， we obtain a τelation among three standard deviations 〔SD）°f h・・αごβ・・輌ed by％・s・．and sb ；espect’vrly：．〔3） sh「tノ＝±［・α一sbh・「tノ］（［fhe second fundarn・nt・1 re1・ti・n〕・ Look at Fig． 1． 1・th・fo11㎝i・9・・n・1y・i・， w・n・glect’the・t・・n・i七i・・al ph寂b・tween l t・・ phases， whi《th is denoted by isolated points「in Fig． 1．． Two indexes ”θ，t’and ・・atr血Table ］， stand fbr the dhilCho6d and the qdoles− cence respectively．． ’ 2．Data analysis with its results． σbserving the first figure， whi（血represents the second fundarnental relation 〔3），we have． decided to use the fbllowing intervals： th・fi・・t ph・・e 3rOgY「s（・・glecti・9 2Y’s ar・d 8．5Y”s血th・・rigi皿・1 rec。rd） the、 seCond phase ヱ4．OrO．5ノヱ8． rβ「s．．脆d・fin・th・n・㎝・1ity・f th・g・・wth by th・． CC Ri b・加een傷r幻・nd・h・・（t）・ Wh・n聖≧0・990r say・we call th・i−th fem・1・0・990−n・m・1・ Then， we obtain the fbllowing results．

￡；、。㌫芸；t．P㌘；，a鑑、1’。蕊1「1。1：：謬鵬。霊、R編。、e

is ca．50％， whidh shows the validity of our hypo七hesis 〔1〕．But in the second phase， the percentage of O L 990−nomal female is㎝1y 50％， probably， because the 「esi曲1 s㎝゜f・q・qre〔SS）・say・SR・b・ing th・・・…th・SS・f㌧・say玩「i・ small ill the relation

〔4）・．㌔一玩r・一・㌶ノ，．・．

wh・n the regressi・n・・effi・i・nt毎・f㌧・砿is sm・11er・ TWo normal and two abnormal cases are shown in Fig． 2 ∼・5． Fig・4suggest・u・th・t thi・i・・iti・11y・w・11−d・v・1・p・d gi・1〔毎・刀h・d suffered from some disease， which caUsed a sudden increase fb11㎝ed by a retar− dati・n・f g・・敵咋・ヱノ血th・fi・・t phase・W・・h・v・ seen・・迦i1・・P・tt・rn ir・ the reference ［3， p．12］，曲ere the subj ect had been prenatally irradiated and died of malignant cerebral tumor after 23 years．

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瓢． H▲SUY▲貰▲・

91

Table la

301

302

303

304

305

306

307

308

309

310

311

312

313

314

315

316

317

318

319

320

321

322

323

324

567

2つ乙つ乙

333

328

329

330

b a

tO．344

★ユ．032 de n．ヱ03 de n．76ヱ ★1．352 ★0．7ヱ4 de n．771 3．482 0．565 0．566 de n．374 ★0．242 0．734 0．562 ★2．520 ★−1．409 de n．724

t2．778

ヱ．638 0．673 2．040 ★ヱ．794 ★ヱ．9ヱ9 ★0．452

kO．871

★0．453 ★0．923 0．449 ★2．288 0．542 α θm 10ヱ．5 −5．6 ヱ44．9 40．9 −54．4 45．9 39．2 −4ヱ5．9 69．0 88．6 ヱ04．9 ヱ24．1 51．7 69．9 −260．4 191．4 54．4 −507．7 一ヱ02．5 58．4 −164．3 一ヱ42．8 128．5 95．4 11．7 89．9 ヱ6．2 96．7 −220．5 73．4

R2

α 0．99ヱ 0．962 0．476 0．928 0．984 0．972 0．909 0．998 0．966 0．959 0．857． 0．594 0．⑨83 0．777 0．987 0．312 0．956 0．989 0．99ヱ 0．975 0．993 0．953 0．893 0．960 0．977 0．969 0．98ヱ 0．856， 0．997 0．962

b

e O．926 ヱ．005 ヱ．059 ヱ．0ひ8 0．595 0．942 0．9．85 0．905 0．955 a e om_8．068 一ヱ．241 −1．723 0．495 45．241 2．752 2．478 5．803 4．252 ヱ・ヱ52 −3●957 ★ ヱ．ヱ45−17．002 0．974 ヱ．099 0．973 0．965 3．823 一5．382 −2．456 3．380 ヱ．083一ヱ0・832 0・954 4．256 0・837 10．441 ★ 0．999 ヱ．265 ★ ヱ．205−22．607 ヱ●ヱ11一ヱ0．215 0．806 1．031 1iヱ02 0．976 0．992 ★0．999 ヱ5．655 −4・061 −4．389 一7．128 0．627 2．014 ヱ・133一ヱ3．064 0．925 4．0ヱ8 ヱ●05ヱー7・863

R2

0 0．999 0．999 0．991 0．945 0．797 0．999 0．997 0．999 0．998 0．999 0．999 1．000 0．998 0．998 0．999 0．998 1．000 ・ヱ．000 0．999 0．986 0．997 0．998 0．999 0．999 0．99ヱ 0．999 0．997 ヱ．000 0．999 ・0．994 t ヱ1．5Y？s 23．2．ヱ0．6 15．2 ヱ3．8 ヱ3．0 2 ヱ5．8 ヱ3．0 10．2 ヱ2．8 ヱ2．8 1ヱ．8 ヱ3．0 14．5 ヱヱ．7 ヱ3．4 14．4 ヱ4．0 ヱ1．6 ヱ4．2 13．9 11．8 1ヱ．6 13．2 11．2・ヱ2．5 13．0 14．8 1ヱ．5 ★ InterpOlated value（s） is （are） included in the original record ［1］． inteηpolated value may be slightly biased， mmch to these estimates in Table 1． ”le

tb

ヱ4

17

14．5 ユ6．5 ヱ7．0 ヱ6．5 ヱ6 2

16

15．5 ヱ6．5 ユ4．5 ヱ6．5 ヱ6．5 17．5

18

16．5

18

ヱ7 ヱ6．5 ヱ7 ヱ7 13．5 ユ5 16．5 14．5 ヱ6．5 ヱ2

18

ヱ5．5 but this bias does not contribute

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92

331

332

333

334

335

351

352

353

354

567

rOrO﹁⊃

333 OO90

5く︼＜U

3マ﹂3

−占フ臼3 ’0’OZO

ワ﹂33

45’010106

333

367

368

369

370

371

372

373

374

375

● LINE」皿工ZED GRO1ml C皿▼E ▲HD ▲GE▲丁皿1▲駐C圓匡 b a ヱ．255

tO．851

de n．579 0．967 0．418 α a om −39．7 30．2 66．ヱ 5．8 87．5 ヱ．805一ユヱ9・3 ヱ．348 −66・4 0◆850 24．1 de n．848 25●7 ヱ◆982一ヱ73◆5 de 1・374−60．5 ★ヱ．17ヱー29．6 ★O〔2ヱ9 ヱ27・ヱヱ・890−14ヱ・2 ★0・852 22・0 0．758 ・ 0148ヱ

t1．007

42．7 90．ヱー1．9 2●682一二Zヱ5．9

0．693 60．5

0．567 76．4 0．5ヱ7 0．55ヱ 0．796 0．889 0．677 83．0 80．3 30．4 6．5 59．7 ★3・689−438．5 tO．592 68．8 ★0．729 42●8 ★0・499 79．0

R2

a O．948 0．781 0．987 0．986 0．989 0．945 0．997 0．982 0．992 0．998 0．982 0．989 0．595 0．997 0．994 0．972 0．947 0．982 0．999 0．936 0．989 0．990 0．755 0．973 0．975 0．971 0．996・ 0．959 0．957 0．957 Table lb

b α R2

θ o o 1●078 −8．7ヱ4 c・M O．993 ★1．088 −5．879 0．9θ9 0・9’80 0．025 0・999 ヱ・007 0．695 0．θ62 −0・6ヱ4 1・077 −6・920 0．878 6・4ヱ4 0・θ92 3●54ヱ 0．882 ヱ4．8ヱ8 0・849 6・902 0・946 5．465 0・942 7●515 ★二Z．120一ヱヱ・547 ヱ●002 2・508 0◆936 6・545 1・084 −7．857 0．934 ヱ2・827 0◆960 2・808 0．θ68 −1・228 ヱ・17ヱー13・555 ヱ・024 −0．329 ヱ．063 ヱ．ヱヱ5 0．969 0．872 0．960 0．98ヱ 0．990 0．954 ヱ．033 一4．ヱヱ4 −7．ヱ60 1．646 5．132 20．042 2●274 2．833 3．826 −6．857 0．999 0．990 0．994 0．997 0．998 0．999 0．998 0．999 0．998 0．999 1．000 0．999 ヱ．000 0．999 0．998 0．999 ヱ．000 0．998 0．998 0．999 0．998 0．997 0．999 0．996 0．998 0．996 0．998 t

m

ヱ2．8 ヱ2．6 12．0 ヱ2．0 ヱ2．4 15．0 12．8 12．4 12．0 ヱ4．6 15．2 ヱ2．4 1ヱ．2 13．6 ユヱ．8・ 12．8 ヱ2．4 13．2 ユ4．2 13．0 12．1 ヱヱ．6 ヱ0．6 2 12．8 ヱヱ．8 ヱ5．3 12．8 12．4 ヱ2．4

゜ρ766

カヱ7一7’

16

ヱ6．5 ヱ7．5 ヱ7．5

17

18

ヱ7．5 ヱ6 ヱ8 13．5 ヱ8 ヱ7 15．5

17

ヱ6．5 ヱ6．5 ヱ6 15．5 ユ5．5 14．5 ヱ5．5 26．5 15．5 ヱ8．2 16．5 ユ7 ヱ5．5

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H．．H▲SUY▲H▲

93

Table lc

107n◎

777333

379

380

381

382

383

384

385

b α ★0．898 0．6ヱ3 ＊ヱ．589 ＊ヱ．031 1．453 ★ヱ．284 ★0．629 de n．386 0．663 0．550 α a om ヱ2．2 5ヱ．ヱー87．2 一2．4 −68．7 −47．7 61．3 98．6 57．9 77．3 R 2 α 0．921 0．817 0．991 0．942 0．996 0．939 0．976 0．845 0．937 0．960 b o ＊0．97ヱ 0．9ヱ7 1．0ユ0 0．97ユ 0．942 0．987 0．961 1．071 ヱ．08ヱ α o θm −0．58ユ 2．275 0．06ヱ 4．253 8．272 2．606 4．406 −5．089 −3．15ヱ de 早E084 −5・0ヱ5 R 2 0 0．998 0．994 0．999 0．999 ヱ．000 0．999 0．999 0．978 0．999 0．999 t ヱ2．3 12．4 13．4 13．2 ヱ3．3 2 ヱ2．8 ヱ2．4 ユ3．6 ヱ2．2

tb

ヱ4．5 15．5 17．5 ヱ6．5 ヱ7．5 ヱ7 ヱ6．5 15 ヱ5 15．5 〔ii）W・ get・？「hi’ai）＝−0・999ヱfbr O・990−n・mal females’砲i・h sh・ws the vali・五ty of the ALONPA hypothesis． This high correlation i1呼）1ies the existence of a quasi− fixed point rD， 02 close to every linearized g士owth curve in the second phase，砲ere we have O＝ヱ65．82±2．33CM and D＝165．93±ヱ．57㎝， so that we may set C＝ 1）． in practice． Dually speaking・point （bi’ αt） 1ies close to the quasi−fixed line

（5〕 y＝o−Dx．

Look at Fig． 6．（iii）In th・・ec・nd・ph・・e・th・CC b・tween bi ・nd tnd・the・g・at｝・e・m・n・・ch・・ is ・qu・1 t・0・93 f・r O・990−n°㎝・1 f・ma1…The regressi・n・f㌔。n bi・eem・ t・b・ n・n一迦・ar（Fig・7〕・and th・1mea・regressi・n est迦・t・・f㌦fb・㌧＝Oisヱ1・33 CYTsノ，which is・・unplausible・We propose e珂）irically the regression ofヱ＝ Zo≦7ヱOtmi ㎝㏄＝Z・91“i・WhiCh ・ay・〔6）ヱ＝ 1否ヱ070 チ 0・ヱ342x ． ±0．003ヱ ±0．0ユ23 The CC between x andヱis O．92． tm is shown by an arrow in Fig． 2∼5． We note that the CC between Z）i andカη己 in the first phase is small negative， i．e． −0．50 for n＝ 50 0．999−normal girls． See Fig． 8．〔iv） The CC between two bi†s in two phases is also almost nil．

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94・

L工NE▲RIZED GROUTH CURΨE 」Lm ▲GE ▲丁貰ENARCHE （・）L・t琉b・・he age a・・hi・h・her・b・n・・g・ reaCh…6・25y”s・th・n・th・CC b・tweenカZ孤d㌦i・equ・1 t・0・67 fO・0・990−n・・m・1 f・m・1e・・（vi）Th・qC b・tween㌦皿d力尻、 is equal t・0・60 fb・0・990−−fem・1es・〔vii） The CC P between male and female mean person I s heights is equal to O。9998 fbr the period 3 r力gy−？s・an’d O．9978 fbr the periodヱ4rヱ）1Sβ「s． Hence， the regression of the female h． on the male h． is almost linear not only in the first phase but also in the second phase． Note that the female is mo］〔’e． precocious than the male， which is shown b）T b k ヱ in the second phase． See Fig． 9 and 〔iii）・ 3．Conclusion．、． Ainericai｝ white women’s data gf good quality endorse the validity of tWo hyp・theses p卿・sed・by th・・uth・r・A・d・the・9・ at．m・nar・h・㌔i・1・rger・・n・verag・・ when b， the ratio of the proper age to the chronological age， i皿 the second phase is larger， whidh is a new result． Adlmowledgment． I aln grateful to Dr．工． Kawabe災who kindly sent me the original report ［1］， after reading my preprint on the linearized growth curve preparedやr a special lecture at IWA［［E miversity． Note： Strictly speaking， we should define the proper age of the i−th individua1 』by％＝βiち曲…βiエ・ap・・itive c・n・t・nt・孤d th・m・・n h・ight血th・p°pulati°n

by％「カノ・th・n・u・h）rP・th・・i・i・giv・n by ・

〔N・1）づ／d・z−d・h／dt；

By lntegratエon， we get ・（N・2）毎「t）；％チβ九「tノ・ h．（t’）’being the sample皿ean of N individuals， we get ．〉〉．・．．1‘．・r1 （N・3）．h・rカ‥… B・ Vh「tノ・ Eliminating ph「tノ・we．（）btain（2〕』in the text・wher・we set

〔N・4） bi ＝βノβ・・and・ゲ・i−bia・

Hence， the following identities hold ［3，4］：〔N．5） α．＝ 0， and b．＝ヱ．

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・『狽堰D頁▲SUY▲麗▲「』

95

7cu1 5 3 100 150cm ・i…［h6 sec・・d F輌飽・a・・Re…i㎝ 1SO㎝ 100

／

100 150enl Fig， 2 Normal SUI）j ect

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96

LINE▲RrZlm GROWT：日【CURva ▲仰 ▲GE 1▲T HE貰▲RCHE 150ロn 100 ．阜み

／

100 150aa pig． 3 Nonnaユ SU）ject 1SO㎝ 100 100 1socm Fig． 4 Abnormaユ S1』）ject

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1SO㎝ 100 H．．瞳▲SUlrム麗▲

97

㎝ 100 150 Fig・5や…cm・1 S・bject 200cm 0 一200 ’400 0 ． Fig．6 Quasi−fixed・Line 5

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1 AT HENARCHE ▲GE

㎜

（URva

GROWTH

LINEARIZED

990 narche Age Isy「5 O ゜・武゜● ● ・阜 ● ．⇔芦・ O 10 5 0 the Second Phase ，°m tノ rb． 7 Fig．

d

垢㎏ O 15y「s O O ● ●●●⑨

●・嶋

♂ ．“．

O

●os？ ● ●● ● ● ● ．多 ● ● O 10 1．0 o．5 in the First Phase カ∂ 8．．．Cb， Fig．

98

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貰． H▲surY▲聴▲

99

150㎝ 100 Fe醜ale mean height 100 150㎝ y「s 18 ユ4

／

Male mean height Fig．9 LGC fbr Mean Female ［1］［2工［3］［4］ REFERENCES Tuddenham， R．D． and Snyder， M．M．（1954〕：APzysioaZ Growth o了CaZiforvziαBo1イs ctncl（批Zs脅oηBirth to Eighteen Yect？s．Uhiversity of Califomia Press． Masuyama， M．（1979）：Inference on age at erUption of tooth and height growth，ヱ’RU Mathematies，15−1， 69−76．It is included in Hzmαan BioehendeαZ Vntα1）ZZ− ities and theirρuα．gi−oonstanoy． The ReseaiCch Institute of Life Insurance Welfare， T◎kyo． Masuyama， M．（1985）：Me AZmost−one Par，a7ne ter Hypothesis AρP Zied加AacoZogY and Phcti・mαooたinetics． The Research Institute of Life Insuτance Welfare ， Tokyo． Masuyama， M・〔1989）：ヱ｝tHypo古力εsθ de Plr・esque dn Pa1・cmlatre pour Z，ヱ＞ndeviduaZite’ dg Za Croissanee de Za Tai ZZe et sa Portie 磁 Z，Ef子icacite’． Tokyo．