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Analysis of the Adolescent Growth Spurt Using Smoothing Spline

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Analysis of the Adolescent Growth Spurt Using Smoothing Spline 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. Height 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 Europe (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 standard deviation 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,)],<C and in addition has to be made as smooth as possible: I s" 1t1'dr: min! Mathematicscan prove that s consistspiecewise of cubic polynomials,and that the piecesare pasted together very smoothly (with continuous first and second order derivatives).Two
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