ANNALSoF HUMANBror-ocy, 1978, vor. 5, No. 5, 421-434

Analysisof the adolescentgrowth spurt usingsmoothing spline functions

R. H. LARGO, TH. GASSER, and A. PRADER Departmentof Paediatrics,University of Zurich and W. STUETZLE and P. J. HUBER Departmentof Mathematical Statistics, SwrssFederal Institute of Tech- nology,Zurich

lReceivedI August1977; revised 30 January1978]

Summary. growth velocity curvesbetween 4.5 and 17.75 yearswere estimated, using smoothing spline functions, for 112boys and I l0 girls from the Zurich Longitudinal Study (1955-1976). Parameterscharacterizing the growth process,such as peak height velocity and age at peak height velocity, were calculateddirectly from the estimatedcurves. The variability of parameters describing the adolescentgrowth spurt is large, both betweenand within sexes.Peak height, defined as increaseof height velocityduring the growth spurt, and age at peak height velocity both characterizethe sex difference in growth in a highly significant manner. Peak height of at least 4 cmfyear is found in70\ of the boys, but in only 11\ of the girls. The age at peak height velocity averages72'2 yearsin girls and 13.9 yearsin boys and has a wide rangeof 5.7 yearsand 3. 8 yearsrespectively. The sexdifference in adult height of 12.6 cm is composedof the following 4 factors: * I '6 cm causedby more prepubertalgrowth in boys, * 6'4 cm by the boys' delayin spurt, * 6.0 cm by the more extensivespurt in boys and - I .4 cm by more post-spurtgrowth in girls. Correlationsbetween parameters indicate that the adult height dependsneither on the duration of growth, nor on the duration and height of the peak. Minimal pre-spurt height velocity and peak height velocity, but not peak height, are age- and height-dependent. Partial correlationsgiven adult height revealtwo compensating mechanismsbetween growth in the prepubertal and in the pubertal period. Small prepubertal height and low height velocity with respect to adult height are followed by a late adolescentspurt and vice versa. Small height at the onset of the spurt with respect to adult height is followed by a longer lasting, but not higher spurt and vice versa.

1. Introduction The analysisof longitudinal growth data usually consistsof two steps: first, parameters have to be defined which summarize the important properties of the growth process; second,a method has to be found which assignsto each individual 422 R. H. Largo et al. seriesof measurementsthe correspondingset of parametervalues. The distribution of these parametersover the sample can then be analysedby classicalstatistical procedures. Hitherto, the commonly used method to perform the first step was to choosea parametricfamily of curvesand to ascertainthe parametervalues for eachindividual by the method of leastsquares. However, prepubertal growth is completelydifferent from pubertal growth, and the pubertal component shows features with a high variability over the population. This makes it difficult to find a parametricfamily of curveswhich meetsthe following two requirements:(a) it should fit all individual seriesof measurementswithout systematicerrors; (6) its parametersshould summarize thoseproperties of the curve in which one is interested. In view of requirement(a), most authors (Deming, 1957; Marubini, Reseleand Barghini, 1971: Marubini, Tanner and Whitehouse, 1972; Tanner, Whitehouse, Marubini and Resele,1976) restrict themselves to a descriptionof the growth spurt. Bock, Wainer, Petersen,Thissen, Murray and Roche (1973)use the superpositionof two logistic functions to describeboth the prepubertal and the pubertal period. However,their family of curvesdoes not meet requirement(a). An averagesquared residualof 2'0 cm2 in boys and 0'3 cm2 in girls indicatesa clear lack of fit. As far as requirement(b) is concerned,the parametersof the logistic function accounting for prepubertal growth vary greatly between sexes.This indicates that they do not summarizethe propertiesof growth in earlychildhood very well. Prepubertalgrowth is known to be almost the same for both sexes. In our study,the methodologicalapproach is different.We estimatethe individual curves of growth velocity non-parametrically,using smoothing spline functions. Parameterssuch as peak height velocity and age at peak height velocity are then calculateddirectly from the estimatedindividual curves.With the help of thesepara- meters, the variability over the population and sex differencesin the growth spurt aredescribed. In a further analysisofprepubertal and pubertalgrowth, the parameters are presentedas linear functions of four independent factors and the relationships betweenthe parametersare evaluated. 2, Subjects and methods Subjects In I 955a prospectivelongitudinal study of growth and development of 412healthy Swiss children was initiated at the Kinderspital Zurich and is still in progress.The work is being done in coordination with four other longitudinal growth studiesin (Brussels,London, Paris and Stockholm),and the Centre International de I'Enfance(CIE) in Paris. In the presentwork, data of ll2 boys and 110 girls are included. In none of these children were more than three measurementsor two consecutivemeasurements missing between 4 and 18 years.No child was suffering from a diseasewhich might have hamperedgrowth. Six boys and two girls of very tall stature were treated with high doses of sex hormones in order to reduce predicted adult height and were therefore excludedfrom the study. Measurements Height was measuredaccording to the direction of the CIE (Falkner, 1960), using gentleupward pressureunder the mastoidprocess. After the age of 8 yearsall measurementswere done by the sametrained anthropometrist(Miss M. Willisegger). Before the age of 9 years in girls and 10 years in boys the children were measured yearly at birthday + 14 days. Afterwards they were measured6-monthly within the Adolescentspurt analysetlby smoothingspline functions 423 same limits until the annual incrementin height was lessthan 0'5cm/year' There- after, the measurementswere done again annually and were discontinued when the incrementhad becomeless than 0'5 cm in 2 years'

Editing Gross errors were looked for by searchingfor negativeincrements in height and by visualinspection of the raw and smoothedvelocity curves on the computerdisplay. The few which could not be correctedwith the help of the original sheetswere treated as missingdata. Missing observationswere completedusing an iterativeprocedure which assuredthat the filled-in pseudo-observationshad zero residualswhen the seriesof height measurementswas smoothed.Results of cross-validation(Wahba and Wold. 1975)and visualcomparison of measuredvelocities with simulatedvelocities, where the varianceof the "random" errors was known, suggestthat the "random" fluctuations in height measurementshave a of about 4 mm' Al1 computationswere performed on the CDC 6400/6500of the Computer Centre of the SwissFederal Institute of Technology(ETH), Zurich. For the graphicaloutput a Textronix 4014display and a Bensonplotter were used. Methods (a) Snoothingsplinefunctions.For detailed description of the statisticalmethods and the algorithmsused, see Stuetzle (1977). Since the methodsare quite important with respectto the presentfindings, a qualitative explanation follows' In any statisticalanalysis of longitudinaldata, one tries to determinethose traits in the individual curves which are consistent over the population, and to disregard the superposedfluctuations which stem from the histories of single individuals and arenot relatedto the growth processper se.This leadsto the following decomposition: f ,(t,): f ,* (t,) 1-e,(t,), wheref : ssriesof measurementsat times /i,,4* : unknown growth pattern of indivi- dual j, and e;:random term, includingmeasurement error' We would like to estimate/l* as accurately as possible.There exist two different approaches,the regressionand the smoothing method, both with specificadvantages and disadvantages. The regressionmethod postulates one functional form for the growth process over a certainepoch for all individuals;e.g. the logisticfunction (Marubini et al.,197l) *(t): - f al{t+exp[- D(t c)]] or the Gompertz function (Deming, 1957),both of which are used for fitting height measurements: *(t): .f aexp {-exp [- b(r- c)]] For each individual, parametersa, b, and c are estimatedby least squares.The above functions are no more than a clever guess,and a serious bias may result due to the differences between the estimated curve and the real growth process' A residual analysison individuals,e.g. by a run test, is not likely to reveala distinct bias over the population; this is due to the low discrimination power for small sample size. The inlroduction of regressionfunctions with more parametersmay lower the bias, but at the price of higher variability of the estimatedparameters' The smoothing approach does not assumea unique form of the growth process for all individuals, and we neednot postulatea functional form. A graphical procedure, as usedin Tanner et at. (1966),may be biasedand is only reproducible in skilled hands. 424 R. H. Largo et al.

We estimate"fi* by cubic smoothing spline functions (Reinsch, 1967).Such a function (s) allows for an arbitrary squareddeviation C from the measuredvalues: f, [s(r,)-/(r,)],

8 I '6 o

'5

Aec (Yco)

Figure l. Raw height velocities (divided differences of measured ) and estimated height velocity curve of a who had one of the most prominent growth spurts. Adolescentspurt analysedby smoothingspline functions 425

as we remain with descriptiveprocesses; for real modelling, parametric methods become necessaryand fruitful, as will be demonstratedin a forthcoming paper (Stuetzle,Gasser, Largo, Praderand Huber, in the press). Hitherto, growth curveshave been analysed using series of measurementsof height attained.In this study,the smoothingspline functions are appliedto seriesof values of heightvelocity. Velocity curves may presentmore difficultieswith regardto estima- tion and interpretationthan distancecurves because of large stochasticvariations (which are inverselyrelated to the time intervals betweenthe measurements).But thesedifhculties are more than outweighedby the detailedpicture of the dynamicsof the growth processwhich velocitycurves offer. Unequal varianceand the correlation structure,introduced by forming velocities,can both be accountedfor in splinecompu- tations. In the caseof growth velocity curveswe expectthe largestbias at the age of peak height velocity (APHV) (Figures l, 2). The population averageof the bias at this point can be estimatedby the following procedure:

(a) For eachindividual the residuals(differences between measurements and estimated curve)are determined. (b) T'heindividual curvesand the residualsare shiftedon the time scaleuntil all the APHV coincide. (c) The residualsare smoothedby a moving average. The rnaximalmean residualin our samplewas about 5 mm/yr. The sameidea of alignmentis usedby Tanneret al. (1966)to constructtheir individual type standards.

E E

I

Agc (yot)

Figure 2. 50th percentileof height velocity in boys (peak height centred).At the bottom mean smoothedbias curve (seemethods, section (c)).

2r 426 R. H. Largo et al.

(b;) Definition of the parameters (Figure 3 and Table 1) Prepubertal height velocity (PV) is defined as the mean annual increment between age 4 and 6j years.This age period was chosenbecause during this period there were no major changes in height velocity, especiallythose due to the onset of pubertal growth. Minimal pre-spurtheight velocity (MHV) signifiesthe onset of the growth spurt; ageat minimal pre-spurtheight velocity (AMHV) denotesthe ageat which the spurt begins.Peak height velocity(PHV) is definedas the maximum height velocity during the growth spurt and ageat peak height velocity(APHV) as the ageat which PHV occurs.Peak height (PH), or increasein heightvelocity during the growth spurt, is defined as the differencebetween peak height velocity (PHV) and minimal pre-spurt height velocity (MHV). Minimal pre-spurtheight velocity return (MHVR) and age at minimal pre-spurtheight velocity return (AMHVR) are height velocity and age when after the spurt, MHV is again attained. AMHVR defines-somewhat arbit- rarily-the end of the spurt and is a measurefor the duration of growth. Peak basis (PB) is definedas AMHVR-AMHV and is a measurefor the duration of the spurt. Peakarea (PAR) is definedas PH x PB and is a measurefor the intensityof the spurt. H4 is height at age4. HMHV, HPHV and HMHVR are definedas height at the onset, at the peak and at the end of the growth spurt. HA (adult height) is defined in 57 boys and 78 girls as the height measuredafter a 2-yearperiod with an increment in height of lessthan 0. 5 cm. In the other 55 boys and 32 girls who have not yet reached that stageat 20 years of age HA representsheight at the age of 20 years.

Heighl Velocity

-+t H4

Figure 3. Definition of the parameters.

Results General description of the adolescentspurt (Table I and Figure 4) From 4 to 9] yearsthere is only a slight differencein height velocity betweenboys and girls. The adolescentspurt starts in girls at a mean age of 9'6 years,ranging frorn 6.6 to 12.9 years(AMHV). In boys the spurt begins1'4 yearslater than in girls, at a mean age of ll'0 years,with the earliestonset at 7'8 yearsand the latestat 13'5 years. As the peak curve has an asymmetrical shape,the peak height velocity (PHV) is attainedonly at the beginningof the third part of the peak,that is, at 12'2 yearsin Adolescentspurt analysedby smoothingspline functions 427

Parameters Mean Range

AMHV Age at minimal prespurtheight velocity (yr) m ll.0 1.2 7'8- 13.5 f 9.6 1.1 6.6- 12.9 APHV Age at peak heightvelocity (yr) m 13.9 0'8 I2.0- 15.8 f 12.2 1.0 9.3- 15.0 AMHVR Age at minimal prespurtheight velocityreturn (yr) m 15.5 0.9 13'5- 17.4 f 13.5 1.1 10.0-16.2 H4 Height at age4 (cm) m 104.3 3.'/ 96.3 114.0 f 103.1 3.7 93.4-111.9 HMHV Height at minimal prespurtheight velocity(cm) m 143.8 1.7 122.7-164.9 f 135.8 7.3 112.2-151.0 HPHV Height at peak heightvelocity (cm) m 16l.9 6.2 149.3-177 . 5 f 150.5 5.7 135.9-161.8 HMHVR Height at minimal prespurtheight velocity return (cm) m l'73.0 5'9 161.4-187 . 6 f 159.0 5.6 144'9-l'70.7 HA Adult height(cm) m 1'77.4 6.2 165.8-193.5 f 164.8 5.7 149.8-176.6 PV Prepubertalheight velocityaI age4-6! (cm/year) m 6.3 0.6 4.9- 8.1 f 6.3 0.6 4.8- 8.1 MHV Minimal prespurtheight velocity (cm/yr) m 4.2 0.6 2.9- 6.0 f 4.8 0.7 3.0- 7.0 PHV Peakheight velocity (cm/yr) m 9.0 l'1 6.7- 12.4 f 7.1 1.0 5.0- 10.1 PH Peakheight (cmiyr) m 4.8 1.3 1.9- 7.7 f 2.4 1.0 o.'7- 5-6 PB Peakbasis (yr) m 4.5 0.9 2.6- 8.2 f 3'9 0.8 l .3- 6.5 PAR Peakarea: PH x PB (cm) m 21.7 6.8 8.2- s3.2 f 9.4 4.5 0.1- 25.2

Table 1. Means, SD and range of the parameters.

ChronologicolAge, yeors 4 5 6 7 8 9 10 lt 12 t3 14 t5 16 17 ttlttl

9_

-'Ft =i6_ o5, >4

'o I?_

I

I z 6 5 4 -r z i pru-ifz -.-o llrl d: .a 5 -4 -3 -2 -t pHV.r .2 .J Yeors from PHV

Figure 4. 50th percentileof height velocity in boys and girls (peak height centred).

2r2 428 R. H. Largo et al. girls and ar 13.9 years in boys (APHV). Peak height (PH), or increasein height velocityduring the growth spurt, in boys is 4'8 cm/year,which is twice as high as in girls. Peak height velocity (PHV), the sum of the minimal pre-spurtheight velocity (MHV) and the peakheight (PH), averages9'0cm/year in boysand 7'1cm/year in girls.The end of the adolescentspurt (AM HVR), deflnedas the agewhen the minimal prepubertalvelocity (MHVR) is again attained,is reached3'8 yearsafter the onset of the spurt in girls and 4'5 yearsthereafter in boys. The adult height(HA) is ll7.4cm in boysand 164.8cm in girls.Adult heightis mildly underestimatedfor two reasons.Firstly, accordingto our definition of adult height (seemethods), growth ceasedonly in two thirds of the girls and in half of the boys at 20 yearsof age.Secondly, the exclusionfrom the study of six boys and two girls of very tall staturedecreases the mean adult height by about 8 mm in boys and 2 mm in girls.

Variations in the adolescentspurt (Table I and Figure 5) The age at peak height velocity (APHV) averages72'2 yearsin girls and 13'9 yearsin boys and has a wide rangeof 5.7 yearsor 3'8 yearsrespectively. The peak basis(PB), which is a measureof the duration of the growth spurt, may be as short as 1.3 yearsin girls or 2.6years in boysand may reach6.5 yearsor 8.2 yearsres- pectively.ln 2\ of the boys and 33/. of the girls, the peak height (PH) is lessthan 2 cmfyear (Figure 5). A peak height of at least4 cm/year,or in terms of peak height velocity(PHV), a doubling of the minimal pre-spurtvelocity (MHV), is found in 70il of the boysand only I I I of the girls.A peakheight higher than 6 cm/yearis observed in 11l of the boys and in none of the girls.

Adolescentspurt antl sex dffirences in adult height (Table 2) At age4 the boys are 1.2 cm taller than the girls. The differencebetween the height of the girls at the onset of their spurt (AMHV) and the height of the boys at the corres- ponding age,namely 1'4 yearsbefore the onsetof the boys' spurt (heightat (AMHV minus 1.4 years))is l.6cm. The sexdifference in height at the age of minimal pre- spurt heightvelocity (AMHV) is 8.0cm;at rhe ageof peakheight velocity (APHV)

?5

20

15

l0

E -25 Girls E2A z 15

10

5

0r2345678 PpnkHPrnhl cm/vPor

Figure 5. Histogramsof peak height in boys and girls. Adolescentspurt analysed by smoothing spline.functions 429

Sex Height l( of HA difference (cm) (cm)

H4 f 103.1 62.6 m 104.3 58.8 HMHV f 135.8 82.4 H (AMHV- 1'4 yr) m t3'7.4 77.5 1'6 HMHV m r43.8 8l.l 8.0 HPHV f t50.5 9t.3 m t6t.9 91.3 ll .4 HMHVR f 159.0 96.5 m 173.0 97.5 l4'0 HA f t64.8 100'0 m r'77'4 100'0 lz.6

Table2. Sexdifference. it is 1l'4cm; at the end of the adolescentspurt, definedas the age when minimal pre-spurtheight velocity is again attained(AMHVR), it is l4'0 cm. After the adoles- cent spurt, the sex differencedrops to 12'6cm at the age when growth has ceased. Thus the sexdifference of l2.6cm in adult height is composedof the following four components: *l'6cm causedby the more extensivegrowth in boys during the prepubertalperiod, *6'4cm by the boys'delay in spurt, *6'0cm by the greater male spurt and - I .4 cm by the more extensivegrowth of the girls in the post-spurt period. To further clarify the sex difference in height growth it is interesting to look at the percentagesof adutt height attained at differentages. At the girls' onset of the adolescentspurt (AMHV),82'4% of the adult height is attainedby girls; in boys, only 77.5/" of the adult height is attained.Because of the delay in spurt, the boys have a lower percentageof adult height than girls at the same chronological pre- pubertal ages.So, at the ageof 4, girls have attained62'6% and boys 58'8/" of the adult height.At the onsetof the spurt (AMHV) in both sexesthe percentageof adult heightattained is82'4/" for girlsand 8l'l\for boys.At the end of the spurt,the boys have reached97'7% and the girls 96'5l". This confirms that the adolescent spurt in boys contributesmore to the adult height than it does in girls, whereasin the post-spurtperiod the contribution to adult height in girls is larger than in boys. The latter finding may be in part due to the fact that two thirds of the girls, but only half of the boys haveattained adult heightaccording to our definition. Using only peak height (PH) and age at peak height velocity (APHV) in discrimi- nant analysis,90/, of the individualscould be classifiedcorrectly by sex.If all derived parameterswere used,this was possiblein97 I of the children.

Relationshipsbetween the diferent parameters (Figure 6, Table 3 and 4) A factor analysis yielded 4 independent factors (only the results for boys are given, the pattern in girls is the same).These factors reflectthe parametersin linear combinationsand explain about 90\ of the variancein the l4-dimensionalfactor space. After appropriate rotation (Quartimax) the factors may be interpreted as follows: factor I is height(essentially identical with adult height); factor 2 is duration of growth (essentiallyidentical with AMHVR); factor 3 is peak height (PH), and factor 4 is peak basis(PB). Figure 6 and Table 3 illustratethat adult height (HA-fl) is uncorrelatedwith the timing (AMHV, APHV, AMHVR -f2),the height(PH-f3), 430 R. H. Largo et al.

. PAR

HMHV. HPH\ PAR . .P I rurvn" - u^ I . HMHVR HA rz rv.-.li?';!Orrrr"-iIlV": 'H4 .PHv APH' . MHV

. PHV

HMXVR f1 pe.,L.ou"uru. -.APHV . fu AMHVR HMHV H4 HPH

. MHV

.pB f3 l4

HPHV |rdil,,}g"xilTr AMHVR' Pv. HA

PAR.

PB' P8. f4 f4

Figure6. Factoranalysis in boys.t.i:Hrt:.rfiJ"?fi:l: adultheight, duration of growth,peak the duration (PB-f4) and the intensity(PAR) of the spurt. Minimalpre-spurt height velocity (MHV) and peak height velocity(PHV) are both negativelycorrelated with age; in remarkablecontrast, peak height (PH) is independentof height and age. A negative partial correlation given adult height was found between the age of minimal prepubertalheight velocity (AMHV) and the height at age 4 (H4), and betweenthe prepubertalheight velocity (PV) and the age at the onset of the spurt (AMHV) (Table4). The sameapplies to the ageat peak height velocity(APHV) and the age at the end of the spurt. (AMHVR). Further there is a negativepartial corre- lation given adult height betweenthe height at the onset of the spurt (HMHV) and the peak basis(PB), a measurefor the duration of the spurt, but thereis no significant correlationbetween HMHV and peak height (PH).

Discussion In this study, the onset of the adolescentgrowth spurt, defined as the age of minimal pre-spurtheight velocity (AMHV), is 11.0 yearsin boys and 9.6 yearsin girls.Thus the onsetof the pubertalgrowth spurt is slightlyearlier than the appearance of the first sexcharacteristics and the clinicallydiscernible growth spurt. In our children,the spurt startsabout I .0 year earlierin boys and 0. 7 year earlier Adolescentspurt analysedby smoothingspline functions +Jl

o$€n+@hr\o@€{\o I **.*"**:'*'*'*'* o O d n* at !f,* n* F-* h* < o; o o o c)* c)* o o o c)* c)* c)* c)* arI llllll

ro\rn\oholclqloalo\ io* 6 ; F aI 6 o o oI o o o o o 6* o oI llllllllll

o\*o\Noi'o\\oa6\o - 6 : c.l O O {* €* d 6+ E: a a o o 6* o o o oI ;I oI oI rtllllll

>: 3* 5* 5 3 3 I = 3 5 F* 3 h* oI oI o o o o o o o oI o oI Al rtlllll

-+€O\rrnalar:O\hOh ) ox Q* n+ -* E;I oi oI oI o oI oI oI oI ;I oI oI oi

\oNrr\ocl-iSricl\c N -* 6* $* o* h* n* h* h* o $* h* n* L oi o; o+ o* c)* o* c)* €)* o* o c)x c)* o* ttllll.a

r+ G N.* <9-^ + F c)* o \8 I .- E =7 r )n6€ o 6-8 eA: : ? J o o oo& lEo t-\ii v .-6 O\a >? * = N 'l' '-' '* c)* o c)x o F3 E o* o-

\o >+++rhO\O*-\O<'6 i- \b* d -: \O* €* r* r* d* O \O* o* *! 'J.lx.*.**.*t*"*'+ *d =oi o* o o* c)* o+ o* o* o o o c)* c)* F llll t

@Oro\OO\clollQq,a o? T t* 91 9I lI YI aI ? | TI tr o c)* o* o* o* c)* o* <)* c)* o o* o* "Io* ltl "I I

5ru \o O \o n i c) @ a 9Q r-| 99 h* r* o+ N* $* o* 2 \d* dx-. -* N o* O ' '* Fl a* * . * . . * * . '* ' * ' * * * t oi o;i e* o o* o* o c)* o* <)* o* c)* o* ztttl

Z €* 5* K K S* 3* 3 $* S* 3* I $* X E al oI oI oI oI oi o oi oI oI o of 5* 432 R. H. Largo et al,

MHV

AMHV m -0.25 -0.37 - 0'41

f -0'26 - 0.28 -0 65

APHV m -0.38 - 0'40 -0.54

f - 0.40 -0.50 -0.78

AMHVR m -0.43 -0.43 -0 64

f -0.44 -0.57 - 0.83

HMHV m - 0.08 -0.85

f -0.19 .-0'6'7

*** ** P<0.001; P<0.01 ; * P<0.05.

Table 4. Conditional correlationsgiven adult height (in boys). in girls than in the Harpenden study (Tanner et al., 1976).The reasonfor this difference may be that the ageof minimal pre-spurtheight velocity (AMHV) cannotbe discerned accuratelyby fitting logisticcurves to the growth spurt, but can only be estimatedby eyeas done by Tanner et al. (1976).However, it cannotbe ruled out that our smoothing procedure may have slightly underestimatedthe age at which the spurt takes off. As a consequenceof the earlier onset of the spurt all our parameterswhich dependon AMHV are somewhatdifferent from those of the Harpendenstudy. The height at the onsetof the spurt(HMHV) is 2.3 cm smallerin boys and2.l cm smallerin girls than in the Harpendenstudy. The adolescentgain in our children, definedas adult heightminus height at minimal prespurtheight velocity is 33.6cm in boysand 29.0 cm in girls.This is 6'0 cm more, and 3.8 cm more respectivelythan in the Harpenden study. In addition to the earlieronset of the spurt this differenceis a reflectionof the greateradult heightof 177.4cm in boysand 164'8cm in girlsin our study. In the analysisof the derivedparameters, we weremainly interestedin the following three questions:(l) How largeis the individual variation in the adolescentspurt? (2) Which aspectsof the growth spurt bestcharacterize sex differences, and how do these relate to the sex differencein adult height? (3) Which parametersbest define the adolescentspurt and what are the relationshipsbetween these parameters ? Besidesthe well-knownwide variation in the ageof the adolescentspurt, our data demonstratethat thereis also a greatvariation in the duration and heightof the peak. The peak basis,as a measureof the duration of the adolescentspurt, may be as short as I '3 yearsin girls and 2.6 yearsin boys or may exceed6 and 8 yearsrespectively. Peak height, or the increasein height velocity during the growth spurt, may be as small as 0'7 cm/yearin girls and I .9 cm/yearin boys or may be as large as 5. 6 and 7'7 cmfyear respectively. In agreementwith Deming (1957)and Tanner et al. (1976),our data show that the pubertal growth spurt is much more impressivein boys than in girls. A peak height of at feast 4 cmfyear, which means,in termsof peakheight velocity (PHV), a doubling of the minimal pre-spurtvelocity, is found in70'l of the boys and in only I I \ of the girls. Adolescentspurt analyseclby smoothingspline functions 433

According to Tanner et al. (1962, 1976)the adult sex differencein height is due to the later onsetrather than to the more ertensivespurt in boys. We founclthat the longer lasting prepubertalgrowth and the greater extent of the growth spurt in malesboth contribute 6 cm to the final sex differencein adult height. Furthermore, during the post-spurtperiod the girls show a largergain in heightthan the boys.Thus, the maximum sex differenceof l4 cm at the end of the spurt is reducedto the final differencein adult heightof l2'6 cm. This analysisof the contribution of the growth spurt to the sexdifference in adult height can be confirmed by looking at the percentageof adult height attained at different ages.During the prepubertalperiod the boys have a lower percenta-seof adult heightthan the girls at the samechronological age. At the ageof peak height velocity,both sexeshave reached 9l'3% of the adult heightand at the end of the spurt the boys have reached91 '5% and the girls 96'5o./uof their respectiveadult heights.Thus, in boys the prepubertalperiod requiredto attain the samepercentage of adult height is longer; there is more growth during the pubertal period and less in the post-spurtperiod than observedin girls. Thesefindings coincide with the fact that in discriminantanalysis peak height (PH) and ageat peakheight velocity (APHV) allow a good discriminationbetween boys and girls. Correlationsindicate that adult height doesnot dependeither on the duration of growthor on the durationand heightof the peak.Thus, a child doesnot becometall becauseof the late onsetof a long lastingand high growth spurt. Furthermore,peak height, or increasein height velocityduring the growth spurt, is independentof age and heightof thechilu. Partial correlationsgiven adult height revealedtwo compensatingmechanisms between the prepubertal and pubertal period. Firstly, a child with long-lasting growth, i.e. with a late growth spurt, tendsto grow slower in the prepubertalperiod with respectto his adult height. Secondly,a child with a low prepubertalpercentage of adult height has a longer-lastingbut not higher spurt than a child with a higher prepubertalpercentage of adult height.

Acknowledgments We are gratefulto Miss Williseggerwho for the last | 3 yearshas been the unfailing secretaryand anthropometristof our study.W. Stuetzlewas supportedin part by the Fritz Hoffmann La Roche Foundation.

References Bock. R. D., Wainer, H., Petersen,A., Thissen, D', Murray, J., and Roche' A. (1973). A para- meterization for individual growth ctrves. Hunan Biology,45' 63'80. Deming, J. (1957).Application of the Gompertz curve to the observed pattern of growth in length of 48 individual boys and girls during the adolescentcycle of growth. Human Biolog4",29, 83 t22. Falkner, F. (1960). Chilcl Developrnent. An Internationol Method of Study. Modern Problenrs in Pediatrics.Basel: Karger. Marubini, E., Resele,L. F., and Barghini, G. (1971). A comparative fitting of the Gompertz and logistic functions to longitudinal height data during adolescencein girls. Hunnn Biologv,43, 237 252. Marubini,E., Resele,L. F.,Tanner, J. M., andWhitehouse, R. H. (1972)The fit of Gompertzand logistic curvesto longitudinal data during adolescenceon height, sitting height and biacromial diimeter in boys and girls of the Harpenden Growth sludy. ITuman Biolog1,,44,5Il- 524. Reinsch, ch. (1967).Smoothing by spline functions. NunterischeMothematik,l0, 177 183. Stuetzle,W. (1977). Estinration and parameterizationof growth curves. Ph.D. thesis,Su'iss Federal Institute of Technology, Zurich. Stuetzle, W., Casser, Th., Largo, R. H., Prader, A., and Huber, P. J. Models for human height growth: overview and suggestions.Annals o.f Human Biologl' (in the press)' 434 R. H. Largo et al.

Tanner, J. M. (1962). Crowth at ,2ndedition. Oxford:Blackwell ScientificPublications. Tanner, J. M., Whitehouse, R. H., and Takaishi, M. (1966). Standards from birth to maturity for height, weight, height velocity and weight velocity: British children 1965. Archives of Disease in Childhood, 41, 454-4'7| ; 613-635. Tanner. J. M", Whitehouse, R. H., Marubini, E., and Resele,L. F. (1976). The adolescentgrowth spurt of boys and girls of the Harpenden study. Annals of Human Biology,3' 109-126. Wahba, G., and Wold, S. (1975). A completely automatic french curve: fitting spline functions by cross-validation Communications in Statistics, 4, l-17.

Address correspondence to: Professor A. Prader, Kinderspital Zurich, Steinweisstrasse75, 8052 Zurich, Switzerland.

Zusammenfassung. Ftir I l2 Knaben und 107 Mddchen der Ztircher Liingsschnittstudie (1955-1976) wurden die Korperhohen-Zuwachskurven zwischen 4,5 und 17,5 Jahren mit Hilfe gliittender Splinefunktionen geschAtzt. Die Parameter, die den Wachstumsvorgang charakterisieren, wie groBter Korperhohenzuwachs und Alter beim groBten Korperhohenzuwachs, wurden direkt aus den Kurven berechnet. Die Variabilitiit der Parameter, die den jugendlichen Wachstumsschub beschreiben, ist groB, sowohl zwischen als auch innerhalb der Geschlechter. GroBter Korperhohenzuwachs, definiert als Korperhohen-Geschwindigkeit wiihrend des Wachstumsschubes, und AIter beim hochsten Korperhdhenzuwachs charakterisieren den Geschlechtsunterschied im Wachstum hochsignifikant. Ein groBter Korperhohenzuwachs von mindestens 4 Zentimetern pro Jahr findet sich bei 70% der Knaben, aber nur bei 11\ der Miidchen. Das Alter beim groBten Korperhohenzuwachs betriigt im Durchschnitt 12,2 Jahre bei Miidchen und 13,9 Jahre bei Knaben und hat die weite Spanne von 5,7 bzw. 3,8 Jahren. Der Ceschlechtsunterschied der erwachsenen Korperhohe von 12,6 Zentimetern setzt sich aus folgenden 4 Faktoren zusammen: *l,6cm durch stiirkeres pr?ipuberales Wachstum bei Knaben, *6,4cm durch die Verziigerung des Wachstumsschubs bei Knaben, *6,0cm durch den sttirkeren Wachstumsschub bei Knaberr und -l,4cm durch stiirkeres Wachstum der Miidchen nach dem Wachstumsschub. Die Korrelationen zwischen den Parametern zeigen, daB die erwachsene Kdrperhohe weder von der Dauer des Wachstums abhiingt, noch von der Dauer und der Hohe des Cipfels. Der geringste Korperhohenzuwachs vor dem Schub und der gro8te Korperhohenzuwachs, nicht aber die Gipfel- hohe, hiingen von Alter und Korperhdhe ab. Die Partialkorrelationen mit der erwachsenen Korperhdhe ergeben zwei Kompensationsmechanismen zwischen dem Wachstum in der priipuberalen und in der puberalen Phase. Geringe priipuberale Korperhdhe und geringer Korperhtjhenzuwachs im Verhtiltnis zur erwachsenen Korperhohe werden gefolgt von einem sptiten jugendlichen Schub und umgekehrt. Geringe Korperhohe beim Einsetzen des Wachstumsschubes, im Verh?iltnis zur erwachsenen Kiirperhohe, wird von einem liinger andauernden, aber nicht stiirkeren Schub gefolgt und umgekehrt.

R6sum6. Des courbes de vitesse de croissance en taille de 4.5 ir 17,75 ans ont 6t6 estim6es6l'aide de fonction de lissage des irregularit6s pour I l2 gargons et I l0 filles de I'Etude Longitudinale de Zurich (1955-1976). Des paramdtres caract6risant le processus de croissance, tels que le pic de vitesse de croissance en taille et I'ige ir ce pic, ont 6te calcules directement ?t partir des courbes. La variabilit6 des paramdtres d€crivant la pouss€e de croissance ir I'adolescence est grande, a la fois entre les sexes et dans chaque sexe. La hauteur du pic, d6finie comme I'augmentation de la vitesse de croissance en taille durant la poussee de croissance, et I'ige au pic de croissance en taille caract€risent tous deux la diffdrence sexuelle de croissance de faEon hautement significative. Une hauteur du pic d'au moins 4 cm par an est trouvee chez70il" des gargons,mais chez seulement1l'l des filles. L'Age au pic de vitesse de la taille est en moyenne de 12,2 ans chez les filles et 13,9 ans chez les gargons avec une large arnplitude de 5,7 ans et 3,8 ans respectivement. La diff€rence sexuelle de taille adulte, de l2,6cm, est compos6e des quatre facteurs suivants: *l,6cm caus6 par plus de croissancepr6-pubertaire chez les garqons, *6,4cm par le retard des garqons en poussee pubertaire, *6,0 cm par la poussdeplus intense chez'les garqons, et - 1,4 cm par davantage de croissance apr6s la pousseechez les filles. Les corr6lationsentre les paramdtresindiquent que la taille adulte ne d6pend ni de la dur€e de la croissance ni de la dur6e et de la hauteur du pic. La vitesse de croissance minimale avant la pouss6e et la vitesse de pointe, mais non la hauteur du pic, dependent de I'Age et de la taille. Les co116lationspartielles A taille adulte constante 16vdlentdeux m6canismescompensatoires entre la croissance ir la periode pr6-pubertaire et 2r la pdriode pubertaire. Une petite taille pre-pubertaire et une basse vitesse de croissance en taille par rapport ir la taille adulte sont suivis par une pouss6e pubertaire tardive et vice-versa. Une taille petite au ddbut de la pouss6e par rapport b la taille adulte est suivie par une pouss6e durant plus longtemps, mais plus intense, et vice-versa.