Nonparametric regression analysis of longitudinal data

Lecture Notes in Statistics Edited by J Berger, S Fienberg, J Gani, K Krickeberg, and B Singer 46 Hans-Georg Muller Nonparametric Regression Analysis of Longitudinal Data Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Author Hans-Georg Muller Institute of Medical Statistics, University of Erlangen-Nurnberg 8520 Erlangen, Federal Republic of Germany and Division of Statistics, University of California Davis, CA 95616, USA AMS Subject Classification (1980): 62GXX ISBN-13: 978-0-387-96844-5 e-ISBN-13: 978-1-4612-3926-0 DOl: 10.1007/978-1-4612-3926-0 This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks Duplication of this publication or parts thereof is only permitted under the provisions of the German Cqpyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid Violations fall under the prosecution act of the German Copyright Law © Springer-Verlag Benin Heidelberg 1988 2847/3140.543210 Preface This monograph reviews some of the work that has been done for longitudinal data in the rapidly expanding field of nonparametric regression The aim is to give the reader an impression of the basic mathematical tools that have been applied, and applications also to Applications provide to the intuition analysis of about the longitudinal methods and studies are emphasized to encourage the non-specialist and applied statistician to try these methods out To facilitate this, FORTRAN programs are provided which carry out some of the procedures described in the text The emphasis of most research work so far has been on the theoretical aspects of nonparametric regression It is my hope that these techniques will gain a firm place in the repertoire of applied convincing applications statisticians and who realize the need to use these the large potential for techniques concurrently with parametric regression This text evolved during a set of lectures given by the author at the Division of Statistics at the University of California, Davis in Fall 1986 and is based on the author's Habilitationsschrift submitted to the University of Marburg in Spring 1985 as well Completeness is not attempted, as on published and unpublished work neither in the text nor in the references The following persons have been particularly generous in sharing research or giving advice: Roussas, U Th Gasser, Stadtmuller, W P Ihm, Y P Stute and R Mack, V Mammi tzsch, to them as well as to numerous other colleagues with whom I discussions I also express my sincere thanks to Colleen had fruitful Criste excellent typing, and to Wilhelm Kleider and Thomas Schmitt for computing assistance Erlangen, December 1987 G G Trautner, and I am very grateful Hans-Georg Muller for ACKNOWLEDGEMENTS The author gratefully acknowledges the permission of the following publishers to reproduce some of the illustrations and tables Almquist and Wiksell International (Scand J Statistics), Institute of Mathematical Statistics, Hayward, California Royal Statistical Society, London F.K Schattauer Verlagsgesellschaft mbH, Stuttgart Taylor & Francis Ltd., London Stockholm Contents Preface Acknowledgements l Introduction Longitudinal data and regression models 2.1 Longitudinal data 2.2 Regression models 2.3 Longitudinal growth curves Nonparametric regression methods 3.1 Kernel estimates 3.2 Weighted local least squares estimates '" 3.3 Smoothing splines 3.4 Orthogonal series estimates 3.5 Discussion 3.6 Heart pacemaker study 15 15 17 19 21 23 24 Kernel and weighted local least squares methods 4.1 Mean Squared Error of kernel estimates for curves and derivatives 4.2 Asymptotic normality 4.3 Boundary effects and Integrated Mean Squared Error 4.4 Muscular activity as a function of force 4.5 Finite sample comparisons 4.6 Equivalence of weighted local regression and kernel estimators 26 26 31 32 36 43 6 38 Optimization of kernel and weighted local regression methods 5.1 Optimal designs 5.2 Ch9ice of kernel functions 5.3 Minimum variance kernels 5.4 Optimal kernels :: 5.5 Finite evaluation of higher order kernels 5.6 Further criteria for kernels 5.7 A hierarchy of smooth optimum kernels 5.8 Smooth optimum boundary kernels 5.9 Choice of the order of kernels for estimating ~~ functions 47 47 49 50 Multivariate kernel estimators 6.1 Definiton and MSE/IMSE 6.2 Boundary effects and dimension problem 6.3 Rectangular designs and product kernels 77 77 84 52 58 63 65 71 73 86 VI Choice of global and local bandwidths 7.1 Overview 7.2 Pilot methods 7.3 Cross-validation and related methods 7.4 Bandwidth choice for derivatives 7.5 Confidence intervals for anthropokinetic data 7.6 Local versus global bandwidth choice Weak convergence of a local bandwidth process 7.7 7.8 Practical local bandwidth choice 91 91 94 98 100 107 110 114 117 Longitudinal parameters 8.1 Comparison of samples of curves 8.2 Definition of longitudinal parameters and consistency 8.3 Limit distributions 122 122 124 126 Nonparametric estimation of the human height growth curve 9.1 Introduction 9.2 Choice of kernels and bandwidths 9.3 Comparison of parametric and nonparametric regression 9.4 Estimation of growth velocity and acceleration 9.5 Longitudinal parameters for growth curves 9.6 Growth spurts 131 131 132 135 141 144 147 10 Further applications 10.1 Monitoring and prognosis based on longitudinal medical data 10.2 Estimation of heteroscedasticity and prediction intervals 10.3 Further developments 151 151 153 155 11 Consistency properties of moving weighted averages 158 11.1 Local weak consistency 158 11.2 Uniform consistency 161 12 FORTRAN routines for kernel smoothing and differentiation 165 12.1 Structure of main routines KESMO and KERN 165 12.2 Listing of programs 169 References 190 If we analyse INTRODUCTION longitudinal data, we are usually interested in the estimation of the underlying curve which produces the observed measurements This curve describes the time course of some measured quantity like the behavior of blood pressure after exercise or the height growth of children If, as usual, the single measurements of the quantity made at different time points are noisy, we have to employ a statistical method in order to estimate the curve specify The classical method here is parametric regression, a class of regression functions depending parameters, the so- called "parametric model" to the data by method, estimating sometimes, if the parameters, realistic on where we finitely many Such a model is then fitted usually by assumptions on the the least squares distribution of the measurement errors are available, by the method of maximum likelihood (Draper and Smith, 1980) For regression models which are nonlinear in the parameters, an iterative numerical algorithm has to be employed in order to obtain the parameter estimates as solutions of the normal equations can lead to computational difficulties when we deal with This sophisticated nonlinear models The main problem with parametric modelling is the search for a suitable parametric model with not too many parameters which gives a reasonable fit to the data Especially in biomedical applications this can be a very difficult task since often there is only little a priori knowledge of the underlying mechanisms that generate the data Fitting an incorrect regression model can lead to completely wrong conclusions as is shown in 2.3 analyse the time courses of a sample of individuals, requires Further, if we a parametric analysis the additional assumption that every individual follows the same parametric model No applied statistician can confine himse1f/herse1f to the task of constructing optimal tests or estimates within a statistical model supplied by the subject-matter scientist The statistician has to play an active role also "appropriate" in the selection of an model, which requires true collaborative efforts Only by such interdisciplinary efforts can the situation of an "interdisciplinary vacuum" (Gasser et a1, 1984b) be avoided, where applied statistician and subject-matter scientist have their own realms and certain models are used mainly because they have been used earlier without critically judging their relevance For the kind of joint efforts required, Ze1en (1983) coined the expression "Biostatistica1 Science" for the biomedical field The methods described in this monograph hopefully serve to bridge the "interdisciplinary vacuum" General basic and practical aspects of longitudinal studies are discussed in the monograph by Goldstein (1979) As parametric modelling encounters fundamental difficulties, attractive alternative are nonparametric curve estimation procedures an Kernel smoothing or kernel estimation is a specific nonparametric curve estimation procedure function In contrast to parametric modelling, to be estimated are differentiability requirements function is not required much weaker, the namely assumptions only on the smoothness and Any further knowledge about the shape of the These methods are therefore especially suited for exploratory data analysis; they let the "data speak for themselves", since only very mild assumptions are needed Sometimes we can come up with a parametric proposal after first carrying out a nonparametric analysis In a second step we could then fit the parametric model to the data However, in many cases the behavior of real life curves is very complicated and may not be possibly fitted by a parametric model, or only by a model with a large number of parameters which cannot be computationally identified, especially if only few data are available In such cases, also the final analysis will have to be carried out with a nonparametric curve estimation procedure In this procedures, monograph we the dicuss several nonparametric emphasis being on kernel estimates promising methods of nonparametric regression, as curve one of estimation the most due to its simplicity, computational advantages and its good statistical properties its We discuss the application of this method to longitudinal growth data and other longitudinal biomedical data Questions of practical relevance like choice of kernels and bandwidths (smoothing parameters) or the estimation of derivatives are addressed The basic approach is the estimation of each individual curve separately Samples of curves can then be compared by means of "10pgitudina1 parameters" Some of the topics discussed bear a more theoretical emphasis, but there is always an applied problem in the background which motivates theory Kernel estimates were introduced by Rosenblatt (1956) in the context of nonparametric density estimation, and for the fixed design regression model occurring in longitudinal studies by Priestley and Chao (1972) A short overview on the literature on nonparametric regression is given by Co11omb (1981) with an update (Co11omb, 1985a) estimation hazard including density rate estimation, estimation besides Prakasa Rao (1983) ideas of spectral nonparametric density regression estimation and is reviewed by Some chapters of Ibragimov and Hasminskii (1981) deal with nonparametric regression, Basic The broad field of curve curve focusing on optimal rates estimation with good intuition of convergence are provided by Rosenblatt (1971), an article which gives an excellent introduction into the field Various aspects of curve estimation can be found in the proceedings of a workshop edited by Gasser and Rosenblatt (1979) A lot of insights, especially towards applications, is contained in the book by Silverman (1986) on density estimation The relation between longitudinal data, the fixed design regression model considered in this monograph and other regression models is discussed in Chapter 2, where in 2.3 these issues are illustrated by means of the human height growth curve which serves as an example to compare the different approaches of parametric and nonparametric curve fitting relevant nonparametric weighted local least regression techniques, squares estimates, namely The practically kernel estimates, smoothing splines and orthogonal series estimates are reviewed in Chapter where a further example of an application to a heart pacemaker study is given in 3.6 In Chapter 4, kernel and weighted local least squares estimators are studied more closely equivalence between these two methods is discussed in 4.6 The The kernel approach to the estimation of derivatives is· described and some statistical properties are derived (Mean Squared Error, rates of convergence and local limit distribution) Of special practical interest is a discussion of boundary effects and boundary modification and a discussion of finite sample results, where e.g smoothing splines and kernel estimates are compared w.r to Integrated Mean Squared Error The kernel estimate depends on two quantities which have to be provided by the user: the kernel function and the bandwidth (smoothing parameter) In Chapter optimization of kernel and weighted local least squares methods w.r to various aspects of the choice of kernels is discussed The order of the kernel determines the rate of convergence of d,e estimate and this is also reflected in finite sample studies Specific problems considered are the leads to various variational problems choice of the orders in case that a Optimlzing the shape of the kernel function is to be estimated (5.9), and ~~ the choice of optimal designs for longitudinal studies (5.1) In Chapter the kernel method is extended to the case of a multivariate predictor variable, including the estimation of partial derivatives A computationally fast algorithm is discussed for the case of a rectangular design Chapter contains an overview over available methods for bandwidth choice Of special interest is the difficult problem of bandwidth choice for derivatives, further the question whether one should choose global or local bandwidths The latter was shown to have better properties in a fully data- adaptive procedure by establishing the weak convergence of a process in the local bandwidths (Muller and Stadtmuller, 1987a) stochastic A practical procedure of local bandwidth choice is discussed in 7.8 Nonparametric estimates for peaks and zeros and the joint asymptotic distribution of estimated location and size of peaks are discussed in Chapter The estimation of peaks ("longitudinal parameters") is important for the analysis and comparison of samples of curves These longitudinal parameters usually have a scientific interpretation (compare Largo et al, 1978) and can be used instead of the parameters of a parametric model to summarize samples of curves An application to follows in Chapter the this study by Gasser et al described data of the Zurich longitudinal growth study The analysis of the growth of 45 boys and 45 girls of (1984a,b, 1985a,b) with the kernel method is The superiority of nonparametric over parametric curve estimation can be demonstrated in this example The pubertal growth spurt and a second "midgrowth" spurt can be quantified; the estimation of derivatives is crucial to assess the dynamics of human growth Further techniques for the analysis of longitudinal medical data pertaining to the problems of prognosis and patient monitoring are summarized 187 I F(MOO(l C, 2) EO 0) A(I·l, KA)=( O+A(I·l, KA»/FLOAT(lC·l) I F(MOO(l C, 2) NE 0) A(I·l, KA)=( O·A( 1·1, KA) )/FLOAT( I C·l) I F(MOO(lC, 2) NE AND NB GT 0) A(I·l,KA)=·A(I·l, KA) 20 CONTINUE IC=KK+KA I F(MOO( IC, 2) EO.O) A(KK, KA)=( O+A(KK,KA) )/FLOAT( IC·l) I F(MOO( IC, 2) NE 0) A(KK, KA)=( 1.0· A(KK, KA) )/FLOAT( I C·l) I F(MOO(lC,2) NE O.AND.NB.GT 0) A(KK,KA)=·A(KK,KA) KM=KA·l DO 30 l=l,KM I F(MOO(l+l, 2) EO 0) A( 1, I )=( O+A( 1,1) )/FLOAT( I) I F(MOO( 1+1,2) NE.O) A( 1,1 )=( 1.0·A(l,1 »fFLOAT(I) I F(MOO(l+l, 2) NE O.AND NB.GT • 0) A( 1,1 )=·A( 1,1) 30 CONTINUE DO 40 1=2,KK DO 50 J=l,KM LC=I+J·l LCC=I+J·KA IF(LC.LE.KA) A(I,J)=A(l,LC) IF(LC.GT KA) A(I,J)=A(LCC,KA) 50 40 CONTINUE CONTINUE IF(NKE.EO.l) GOTO 100 DO 60 l=l,KK DO 70 J=l,KA IN=KA·I+l INN=IN·(NKE·l) A(I N, J )=A(lNN, J) 70 60 CONTINUE CONTINUE DO 80 l=l,KA A( 1,1 )=DBLE( 0) I F(NB GT AND • MOO (I , 2) EO 0) A( 1, I )=·A( 1,1) 80 IF CONTINUE (NKE.EO.2) GOTO 100 DO 90 l=l,KA A(2,1 )=DBLE( FLOAT( 1·1» IF (NB.GT.0.AND.MOO(l,2).EO.0) A(l,I)=·A(l,l) 90 C C CONTINUE CONSTRUCTION OF RH SIDE C 100 NUU=NUE+NKE IF(NUE.EO.O) F=l IF(NUE.EO.l) F=·l IF(NUE.EO.2) F=2 IF (NUE.EO.3) F=·6 DO 110 l=l,KA R(I )=DBLE(O.) 110 CONTINUE R(NUU)=DBLE(F) CALL KERSOL(KA,R,SL,NF) DO 120 l=l,KA Cl (I )=SL(I )/FLOAT(I) CONTINUE 120 RETURN END C C NO.15 188 c····················································· C C SUBROUTI NE KERSOL (ND R SL NF) C C SUBROUTINE FOR SOLUTION OF LINEAR SYSTEM C A*SL=R C C OF DIMENSION ND MOOIFIED AFTER PROGRAM 'LiGLEI' C BY RUTI SHAUSER, ZUER I CH C C PARAMETERS C·········· C SCRATCH NL(20) MEMORY ARRAY FOR PERMUTATION OF COLUMNS C OUTPUT NF=1 IF SYSTEM IS DEGENERATE (SHOULD NOT NF OCCUR IF NO LT 7) C C INTEGER NL(20) DOUBLE PRECISION A(20,2D),R(20),SL(20) DOUBLE PRECISION SUM,MAX,AA,MS,RE COMMON A NF=O D05 J=I,ND NL(J) =J C C CONTINUE SEARCH OF MAXIMAL PIVOT ELEMENT C NDD= NO·1 DO 10 JI=I,NDD MAX=DBLE(O.O) KS=JI+1 KO=JI LO=JI DO 20 J2=JI,ND SUM=DBLE(O.O) DO 30 J3=I,ND SUM=SUM+DABS(A(J2,J3» 30 CONTINUE IF (SUM.EQ.O.D) GOTO 20 DO 40 J4=JI,NO AA=A(J2,J4) IF(AA.LT 0.0) AA=·AA MS=MAX*SUM IF(AA.LE.MS) GOTO 40 MAX=AAJSUM KO=J2 LO=J4 40 20 C C CONTINUE CONTINUE IF (MAX.EQ.O.O) GOTO 500 PERMUTATION OF LI HE AND COLUMN C IF(KO.EQ.JI) GOTO 60 RE=R(KO) R(KO)=R(JI ) R(JI )=RE DO 50 J5=I,ND 189 AA=A(Jl,J5) A(Jl, J5 )=A(KO, J5) A(KO,J5)=AA CONTINUE 50 60 IF (LO.EQ.Jl) GOTO 80 ID=NL(LO) NL(LO)=NL(Jl) NL(J1)=ID DO 70 J7=I,ND AA=A(J7,J1) A(J7,Jl )=A(J7,LO) A(J7, LO)=AA 70 C C CONTINUE MODIFICATION OF MATRIX A C IF (AeJl,Jl).EQ.O.O) GOTO 500 80 DO 90 J9=KS, ND SL(J9)=·A(J9,Jl)/A(Jl,Jl) R( J9)=R(J9)+SL( J9)*R( Jl) DO 100 Jl0=KS,ND A(J9, J 10)=A( J9, Jl0)+SL(J9)*A( Jl, J 10) 100 CONTI NUE 90 CONTINUE 10 CONTINUE C C COMPUTATION OF SOLUTION SL C DO 210 J=I,ND SUM=O.O Jl=ND·J+l KJ=Jl+l IF(J1.LT.ND) GOTO 220 IF (A(Jl,Jl).EQ.O.O) GOTO 500 SL(JI )=R(JI )/A(Jl,Jl) GOTO 210 220 DO 230 J2=KJ,ND 230 CONTI NUE SUM=SUM+A(Jl, J2)*SL(J2) IF (A(Jl,Jl).EQ.O.O) GOTO 500 SL(JI )=(R(Jl) ·SUM)/A(Jl, Jl) 210 CONTINUE DO 300 J=I,ND ID=NL(J) R(lD)=SL(J) 300 CONTI NUE DO 310 J=I,ND SL(J)=R(J) 310 CONTINUE RETURN 500 NF=1 RETURN END REFERENCES Abramson, I (1982a) Arbitrariness of the pilot estimator in adaptive kernel methods J Mult Anal 12, 562-567 Abramson, I (1982b) On bandwidth variation in kernel estimates-a square root law Ann Statist 10, 1217-1223 Anderssen, R.S and Bloomfield, P (1974a) A time series approach to numerical differentiation Technometrics 16, 69-75 Anderssen, R.S and Bloomfield, P (1974b) Numerical differentiation procedures for non-exact data Numer Math 22, 157-182 Backman,G (1934) Das Wachstum der Korperlange des Menschen Kuniglicke Svenska Vetenskapsakademiens Handlingar 14, 1-145 Bartlett, M.S (1963) Statistical estimation of density functions Sankhya A 25, 245-254 Bendetti, J.K (1977) One the nonparametric estimation of regression functions J Roy Statist Soc B 39, 248-253 Beran, R (1981) Nonparametric regression with randomly censored survival data Technical Report, Univ of California, Berkeley Berkey, C.S., Reed, R.B and Valadian, I (1983) Midgrowth spurt in height of Boston children Ann Hum BioI 10, 25-30 Bhattacharya, P.K and Mack, Y.P (1985) A two-stage procedure for nonparametric estimation Statistics and Decisions, Supplement 2, 143-153 Bhattacharya, P.K and Mack, Y.P (1987) Linear functions of nearest neighbor estimators of a univariate regression function Ann Statist 15, 976-994 Bickel, P.J and Wi chura , M.J (1971) Convergence criteria for multiparameter stochastic processes and some applications Ann Math Statist 42, 1656-1670 Billingsley, P (1968) Convergence of Probability Measures Wiley, New York Bock, R.D and Thissen, D (1980) Statistical problems of fitting individual growth curves In: Human Physical Growth and Maturation, Methodologies and Factors, 265-290, Ed F.E Johnston, A.F Roche and C Susanne, Plenum Press, New York Bock, R.D., Wainer, H., Petersen, A., Thissen, D., Murray, J and Roche, A (1973) A parametrization for individual human growth curves Human Biology 45, 63-80 Breiman, L and Friedman, J (1985) Estimating optimal transformations for multiple regression and correlation JASA 80, 580-597 Breiman, L., Friedman, J., Olshen, A and Stone, C.J (1984) CART-classification and regression trees Wadsworth: Belmont, Calif Breiman, L and Meisel, W.S (1976) General estimates of the intrinsic variability of data in nonlinear regression models JASA 71, 301-307 Breiman, L., Meisel, W and Purcell, E (1977) Variable kernel estimates of multivariate densities and their calibration Technometrics 19, 135-144 Cacoullos, R (1966) Estimation of a multivariate density Ann Inst Statist Math 18, 179-189 Carroll, R.J (1982) Adapting for heteroscedasticity in linear models Ann Statist 10, 1224-1233 Castro, P.E., Lawton, W.H and Sylvestre, E.A (1986) Principal modes of variation for processes with continuous sample curves Technomeltrics 28, 329-337 191 Cencov, N.N (1964) Evaluation of an unknown distribution density from observations Soviet Math 3, 1559-1562 Cheng, K.F and Lin, P.E (1981) Nonparametric estimation of a regression function Z Wahr verw Geb 57, 223-233 Clark, R.M (1975) A calibration curve for radiocarbon data Antiquity 49, 251-266 Clark, R.M (1977) Nonparametric estimation of a smooth regression function J Roy Statist Soc B 39, 107-113 Clark, R.M (1980) Calibration, cross-validation and Carbon-14, II J Roy Statist Soc A 143, 177-194 Cleveland, W.S (1979) Robust locally weighted regression and smoothing scatterp1ots JASA 74, 829-836 Co11omb, G (1981) Estimation non-parametrique de 1a regression: revue bib1iographique Int Statist Review 49, 75-93 Co11omb, G (1985a) Nonparametric regression: an up-to-date bibliography Math Operationsforschung und Statistik 16, 305-324 Co11omb, G (1985b) Nonparametric time series analysis and prediction: uniform almost sure convergence of the window and k-NN autoregression estimates Math Operationsforschung und Statistik 16, 297-307 Cook, R.D and Weisberg, S (1982) Residuals and Influence in Regression Chapman and Hall, London Count, E.W (1943) Growth patterns of the human physique: an approach of kinetic anthropometry Human Biology 15, 67-93 Courant, R and Hilbert, D (1953) Methods of Mathematical Physics Interscience New York Cox, D.D (1983) Asymptotics for M-type smoothing splines Ann Statist 11, 530-551 Craven, P and Wahba, G (1979) Smoothing noisy data with spline functions Numerische Mathematik 31, 377-403 Dabrowska, D.M (1987) Nonparametric regression with censored survival time data Preprint Davis, K.B (1975) Mean square error properties of density estimates Ann Statist 3, 1025-1030 Davis, K.B (1977) Mean integrated square error properties of density estimates Ann Statist 5, 530-535 Deheuve1s, P (1977) Estimation non-parametrique de 1a densite par histogrammes generalises Revue de Statistiques App1iquees 25, 5-42 De Peretti, E and Forest, M.G (1976) Unconjugated dehydro-epiandrosterone plasma levels in normal subjects from birth to adolescence in humans: the use of a sensitive radio immuno assay J C1in Endocrino1 Metab 43, 982-991 Deuf1hard, P and Aposto1escu, V (1980) A study of the Gauss-Newton method for the solution of nonlinear least squares problems In: Special Topics of Applied Mathematics, 129-150, Ed Frehse, Pa11aschke, Trottenberg, North Holland, Amsterdam Draper, N.R and Smith, H (1980) Applied Regression Analysis, Wiley, New York Eddy, W.F (1980) Optimum kernel estimators of the mode Ann Statist 8, 870-882 Eddy, W.F (1982) The asymptotic distributions of kernel estimators of the mode Z Wahr verw Geb 59, 279-290 E1 Lozy, M (1978) A critical analysis of the double and triple logistic growth curves Ann Hum Bio1 5, 389-394 Engle, R.F., Granger, C.W.I., Rice, J and Weiss, A (1987) Semiparametric estimates of the relation between weather and electricity sales JASA 81, 310-320 192 Epanechnikov, V.A (1969) Nonparametric estimation of a multivariate probability density Theor Probab App1 14, 153-158 Fa1k, M (1983) Relative efficiency and deficiency of kernel type estimators of smooth distribution functions Statist Neer1andica 37, 73-83 Falkner, F (1960), (ed) Child Development An International Method of Study Karger, Basel Falkner, F and Tanner, J.M (1978), (ed.) Human Growth, Vol 1-3 Plenum, New York Friedman, J and Stutz1e, W (1981) Projection pursuit regression JASA 76, 817-823 Fukunaga, K and Hostetler, L.D (1975) The estimation of the gradient of a density function, with applications in pattern recognition IEEE Trans Inf Theory IT-21, 32-40 Gasser, Th., Jennen-Steinmetz, C and Sroka, L (1986) Residual variance and residual pattern in nonlinear regression Biometrika 73, 625-633 Gasser, Th., Kohler, W., Muller, H.G., Kneip, A., Largo, R., Molinari, L and Prader, A (1984a) Velocity and acceleration of height growth using kernel estimation Ann Hum Bio1 11, 397-411 Gasser, Th., Kohler, W., Muller, H.G., Largo, R and Prader, A (1985a) Human height growth Corre1ationa1 and multivariate structure of velocity and acceleration Ann Hum Bio1 12, 501-515 Gasser, Th and Muller, H.G (1979) Kernel estimation of regression functions Lecture Notes in Mathematics 757, 23-68, Springer-Verlag, Berlin Gasser, Th and Muller, H.G (1984) Estimating regression functions and their derivatives by the kernel method Scand J Statist 11, 171-185 Gasser, Th., Muller, H.G., Kohler, W., Largo, R., Molinari, L and Prader, A (1985b) An analysis of the mid-growth spurt and of the adolescent growth spurt of height based on acceleration Ann Hum Bio1 12, 129-148 Gasser, Th., Muller, H.G., Kohler, W., Molinari, L and Prader, A (1984b) Nonparametric regression analysis of growth curves Ann Statist 12, 210-224 Gasser, Th., Muller, H.G and Mammitzsch, V (1985) Kernels for nonparametric curve estimation J Roy Statist Soc B 47, 238-252 Gasser, Th and Rosenblatt, M (ed.) (1979) Smoothing Techniques for Curve Estimation Lecture Notes in Mathematics 757, Springer-Verlag, Berlin Gelfand, M and Fomin, S.V (1963) Calculus of Variations, Prentice-Hall, Englewood Cliffs, NJ Georgiev, A.A (1984) Speed of convergence in nonparametric kernel estimation of a regression function and its derivatives Ann Inst Statist Math 36, 455-462 Goldstein, H (1979) The Design and Analysis of Longitudinal Studies Academic Press, London Goldstein, H (1986) Efficient statistical modeling of longitudinal data Ann Hum Bio1 13, 129-141 Gosh, M., Grizzle, J.E and Sen, P.K (1973) Nonparametric methods in longitudinal studies JASA 68, 29-36 Granovsky, B.L and Muller, H.G (1987) On the optimality of a class of polynomial kernel functions Preprint Greb1icki, W., Rutkowska, D and Rutkowski, L (1983) An orthogonal series estimate of time-varying regression Ann Inst Statist Math 35, 315-328 Grossmann, W (1985) Diskrimination and K1assifikation von Ver1aufskurven In: Neue Verfahren der nichtparametrischen Statistik, Pflug, G Chr (ed.), Springer-Verlag, Heidelberg-Berlin, 109-129 Hard1e, W (1984) A law of the iterated logarithm for nonparametric regression function estimators Ann Statist 12, 624-635 Hard1e, W., Hall, P and Marron, J.S (1987) How far are automatically chosen regression smoothing parameters from their optimum Preprint 193 HardIe, W and Gasser, Th (1984) Robust non-parametric function fitting J Roy Statist Soc B 46, 42-51 HardIe, W and Marron, J.S (1985a) Asymptotic nonequivalence of some bandwidth selectors in nonparametric regression Biometrika 72, 481-484 HardIe, W and Marron, J.S (1985b) Optimal bandwidth selection in nonparametric regression function estimation Ann Statist 13, 1465-1482 Halasz, G (1976) Statistical interpolation Colloquia mathematica societatis Janos Bolyai 19, Fourier analysis and approximation theory, Budapest 1976, North Holland, 403-410 Hall, P (1982) Comparison of two orthogonal series methods of estimating a density and its derivatives on an interval J Mult Anal 12, 432-449 Hall, P (1983) Orthogonal series distribution function estimation, with applications J Roy Statist Soc B 45, 81-88 Hart, J and Wehrly, T (1986) Kernel regression estimation using repeated measurements data JASA 81, 1080-1089 Hastie, T and Tibshirani, R (1986) Generalized additive models Statistical Science 1, 297-318 Hastie, T and Tibshirani, R (1987) Generalized additive models: some applications JASA 82, 371-386 Hauspie, R.C., Wacholder, A., Baron, G., Cantraine, F., Susanne, C and Graffar, M (1980) A comparative study of the fit of four different functions to longitudinal data of growth in height of Belgian girls Ann Hum BioI 7, 347-358 Hodges, J.L and Lehmann, E.L (1956) The efficiency of some nonparametric competitors of the t test Ann Statist 27, 324-335 Hominal, P and Deheuvels, P (1979) Estimation non-parametrique de la densite compte-tenu d'informations sur Ie support Revue de Statistiques Appliquees 27, 47-59 Huber, P.J (1979) Robust smoothing Robustness in Statistics, Launer und Wilkinson, (ed.) Academic Press, New York Ibragimov, I.A and Hasminskii, R.Z (1981) Statistical Estimation SpringerVerlag, New York Ibragimov, I.A and Hasminskii, R.Z (1982) Estimation of distribution density belonging to a class of entire functions Theor Probab Appl 27, 551-562 Jennen-Steinmetz, Chr and Gasser, Th (1987) A unifying approach to nonparametric regression estimation Preprint Jenss, R.M and Bayley, N (1937) A mathematical method for studying growth in children Hum BioI 9, 556-563 Johnston, G.J (1979) Smooth Nonparametric Regression Analysis Ph.D Dissertation, University of North Carolina at Chapel Hill J~rgensen, M., Nielsen, C.T., Keiding, N and Skakkeback, N.E (1985) Parametrische und nichtparametrische Modelle fur Wachstumsdaten Med Informatik und Statistik 60, Ed G Chr Pflug, Springer-Verlag Berlin, 74-87 Keiding, N and Holst, C (1987) Retrospective estimation of diabetes incidence: Some model experiments Research Report 87/9, Statistical Research Unit, Copenhagen Kiefer, J and Wolfowitz, J (1952) Stochastic estimation of the maximum of a regression function Ann Math Statist 23, 462- 466 Kneip, A and Gasser, Th (1986) Convergence and consistency results for self-modeling nonlinear regression Preprint Krieger, A.M and Pickands, J III (1981) Weak convergence and efficient density estimation at a point Ann Statist 9, 1066-1078 Lai, T.L., Robbins, H and Wei, C.z (1979) Strong consistency of least squares estimates in mUltiple regression II J Mult Anal 9, 343-361 Lamperti, J (1966) Probability W.A Benjamin, New York 194 Largo, R.H., Gasser, Th., Prader, A., Stutzle, W and Huber, P.J (1978) Analysis of the adolescent growth spurt using smoothing spline functions Ann Hum BioI 5, 421-434 Lawton, W.H., Sylvestre, E.A and Maggio, M.S (1972) Self modeling nonlinear regression Technometrics 14, 513-532 Lejeune, M (1984) Optimization in nonparametric regression In Compstat 1984 (Proceedings in Computational Statistics), eds T Havranek, Z Sidak, M Novak Physica-Verlag Wien, 421-426 Lejeune, M (1985) Estimation non-parametrique par noyaux: regression polynomiale mobile Revue de Statistiques Appliquees 33, 43-67 Lenth (1977) Robust splines Commun in Statist A6, 847-854 Macauley, F.R (1931) The smoothing of time series National Bureau of Economic Research, New York Mack, Y.P (1981) Local properties of k-NN regression estimates SIAM J Alg Disc Math 2, 311-323 Mack, Y.P (1983) Rate of strong uniform convergence of k-NN density estimates J Statist Planning and Inference 8, 185-192 Mack, Y.P and Muller, H.G (1987a) Derivative estimation in random-design nonparametric regression Technical Report No 67, Division of Statistics, Univ of California, Davis Mack, Y.P and Muller, H.G (1987b) Convolution type estimators for nonparametric regression Technical Report No 90, Division of Statistics, Univ of California, Davis Mack, Y.P and Muller, H.G (1987c) Adaptive nonparametric estimation of a multivariate regression function J Mult Anal 17, 163-181 Mack, Y.P and Silverman, B.W (1982) Weak and strong uniform consistency of kernel regression estimates Z Wahr verw Geb 61, 405-415 Magnus, W., Oberhettinger, F and Soni, R.P (1966) Formulas and Theorems for the Special Functions of Mathematical Physics Auflage, SpringerVerlag Mallet, A (1986) A maximum likelihood estimation method for random coefficient regression models Biometrika 73, 645-656 Mallows, C (1973) Some comments on Cpo Technometrics 15, 661-663 Mammitzsch, V (1982) Darstellung von Kernfunktionen mittels LegendrePolynomen Technical Report, Univ of Marburg Mammitzsch, V (1983) A note on kernel estimators fulfilling certain moment conditions Proceedings of the 44th Session of the lSI, Madrid, I, 31-34 Mammitzsch, V (1984) On the asymptotically optimal solution within a certain class of kernel type estimators Statistics and Decisions 2, 247-255 Marshall, W.A (1971) Evaluation of growth rates over less than one year Arch Dis Childhood 46, 414-420 Marubini, E., Resele, L.F and Barghini, G (1971) A comparative fitting of the Gompertz and logistic functions to longitudinal height data during adolescence in girls Hum BioI 43, 237-252 Marubini, E., Resele, L.F., Tanner, J.M and Whitehouse, R.H (1972) The fit of Gompertz and logistic curves to longitudinal data during adolescence on height, sitting height and biacromial diameter in boys and girls of the Harpenden growth study Hum BioI 44, 511-524 McDonald, J.A and Owen, A.B (1986) Smoothing with split linear fits Technometrics 28, 195-208 Molinari, L., Largo, R.H and Prader, A (1980) Analysis of the growth spurt at age seven (midgrowth spurt) Helv Paediat Acta 35, 325-334 195 Muller, H.G (1983) Beitrage zur nichtparametrischen Kurvenschatzung Dissertation, Fak fur Naturwissenschaften und Mathematik der Universitat U1m Muller, H.G (1984a) Smooth optimum kernel estimators of regression curves, densities and modes Ann Statist 12, 766-774 Muller, H.G (1984b) Boundary effects in nonparametric curve estimation models Compstat 1984, ed T Havranek et a1., Physica-Ver1ag, 84-89 Muller, H.G (1984c) Optimal designs for nonparametric kernel regression Statist and Probability Letters 2, 285-290 Muller, H.G (1985a) Kernel estimators of zeros and of location and size of extrema of regression functions Scand J Statist 12, 221-232 Muller, H.G (1985b) On the number of sign changes of a real function Periodic a Math Hungarica 16, 209-213 Muller, H.G (1985c) Empirical bandwidth choice for nonparametric kernel regression by means of pilot estimators Statistics and Decisions Supplement 2, 193-206 Muller, H.G (1987a) Weighted local regression and kernel methods for nonparametric curve fitting JASA 82, 231-238 (The reference "Laurent" should be "Lejeune") Muller, H.G (1987b) On the asymptotic mean square error of Ll kernel estimates of ~m functions J Approx Theory 51, 193-201 Muller, H.G (1987c) Weak and universal consistency of weighted moving averages Period Math Hungar 18, 241-250 Muller, H.G and Gasser, Th (1979) Optimal convergence properties of kernel estimates of derivatives of a density function Lecture Notes in Mathematics 757, 144-154 Muller, H.G and Gasser, Th (1986) Nonparametric estimation of spectral densities and of their derivatives: Integrated Mean Squared Error, choice of window width and applications Cahiers du Centre d'Etudes de Recherche Operatione11e 28, 163-173 Muller, H.G and Ihm, P (1985) Kernel estimation techniques for the analysis of clinical curves Methods of Information in Medicine 24, 218-224 Muller, H.G and Schmitt, Th (1986) Kernel and probit estimates in quanta1 bioassay Preprint Muller, H.G and Stadtmu11er, U (1987a) Variable bandwidth kernel estimators of regression functions Ann Statist 15, 182-201 Muller, H.G and Stadtmu11er, U (1987b) Estimation of heteroscedasticity in regression analysis Ann Statist 15, 610-625 Muller, H.G., Stadtmu11er, U and Schmitt, T (1987) Bandwidth choice and confidence intervals for derivatives of noisy data Biometrika 74, 743-750 Nadaraya, E.A (1964) On estimating regression Theor Prob App1 9, 141-142 Nussbaum, M (1982) Nonparametric estimation of a smooth regression function defined on a domain of Rk Technical Report, Akademie der Wissenschaften der DDR Parzen, E (1957) On consistent estimates of the spectrum of a stationary time series Ann Math Statist 28, 329-348 Parzen, E (1962) On estimation of a probability density and mode Ann Math Statist 33, 1065-1076 Petrov, V.V (1975) Sums of independent random variables Springer-Verlag, New York Po1ya, G and Szego, G (1954) Aufgaben und Lehrsatze aus der Analysis I, II Springer, Berlin Potthoff, R.F and Roy, S.N (1964) A generalized multivariate analysis of variance model useful especially for growth curve problems Biometrika 51, 313-326 196 Prader, A (1978) Wachs tum und Entwick1ung In: K1in:!.k der Inneren Sekretion, Ed.: A Labhart, Springer, Berlin, 990-1036 Prader, A (1982) Hormonal regulation of growth and the adolescent growth spurt Talk delivered at the 2nd Int Conference on the Control of the Onset of Puberty, Stresa Prakasa Rao, B.L.S (1983) Nonparametric functional estimation SpringerVerlag, New York Preece, M.A and Baines, M.J (1978) A new family of mathematical models describing the human growth curve Ann Hum Bio1 5, 1-24 Priestley, M.B and Chao, M.T (1972) Nonparametric function fitting J Roy Statist Soc B 34, 384-392 Pruitt, W.E (1966) Summability of independent random variables J Math Mech 15, 769-776 Ramlau-Hansen, H (1983) The choice of a kernel function in the graduation of counting process intensities Scand Actuarial J 165-182 Reinsch, C.H (1967) Smoothing by spline functions Numer Math 10, 177-183 Reinsel, G (1982) Multivariate repeated-measurement or growth curve models with multivariate random-effects covariance structure JASA 77, 190-195 Reiter, E.O., Fuldauer, V.G and Root, A.W (1977) Secretion of the adrenal androgen, dehydroepoandrosterone sufate, during normal infancy, childhood, and adolescence, in sick infants and in children with endocrinological abnormalities J Pediatrics 90, 766-770 Rice, J (1983) Methods for bandwidth choice in nonparametric kernel regression Computer science and statistics: The interface ed J.E Gentle, NorthHolland 1983, 186-190 Rice, J (1984a) Bandwidth choice for nonparametric kernel regression Ann Statist 12, 1215-1230 Rice, J (1984b) Boundary modification for kernel regression Commun Statist Theor.-Meth 13, 893-900 Rice, J (1986a) Bandwidth choice for differentiation J Mu1t Anal 19, 251-264 Rice, J (1986b Convergence rates for partially spline models Statist and Probability Letters 4, 203-208 Rice, J and Rosenblatt, M (1981) Integrated mean squared error of a smoothing spline J Approx Theory 33, 353-369 Rice, J and Rosenblatt, M (1983) Smoothing splines: regression, derivatives and deconvolution Ann Statist 1, 141-156 Robbins, H and Monro, S (1951) A stochastic approximation method Ann Math Statist 22, 400-407 Rosenblatt, M (1956) Remarks on some nonparametric estimates of a density function Ann Math Stat 27, 642-649 Rosenblatt, M (1971) Curve estimates Ann Math Statist 42, 1815-1842 Rutkowski, L (1982) Orthogonal series estimates of a regression function with applications in system identification In: Probability and statistical inference (ed W Grossmann et a1.), North Holland, 343-347 Sacks, J and Y1visaker, D (1970) Designs for regression problems with correlated errors III Ann Math Statist 41, 2057-2074 Schmer1ing, S and Peil, J (1985) Verfahren der loka1en Approximation zur nichparametrischen Schatzung unbekannter stetiger Funktionen aus Mepdaten Gegenbaurs morphologisches Jahrbuch 131, 367-381 Schmer1ing, S and Pei1, J (1986) Improvement of the method of kernel estimation by local polynomial approximation of the empirical distribution function and its application to empirical regression Gegenbaurs morpho1ogisches Jahrbuch 132, 29-35 197 Schoenberg, I.J (1964) Spline functions and the problem of graduation Proc National Acad Sci USA 52, 947-950 Schucany, W.R and Sommers, J.P (1977) Improvement of kernel type density estimators JASA 72, 420-423 Schuster, E and Yakowitz, S (1979) Contributions to the theory of nonparametric regression, with applications to system identification Ann Statist 7, 139-149 Serfling, R.J (1980) Approximation Theorems of Mathematical Statistics Wiley, New York Shapiro, H.S (1969) Smoothing and Approximation of Functions Van Nostrand, New York Shibata, R (1981) An optimal selection of regression variables Biometrika 68, 45-54 Sibson, R (1980) The Dirichlet tessilation as an aid in data analysis Scand J Statist 7, 14-20 Silverman, B.W (1978) Choosing a window width when estimating a density Biometrika 65, 1-11 Silverman, B.W (1984) Spline smoothing: the equivalent variable kernel method Ann Statist 12, 898-916 Silverman, B.W (1985) Some aspects of the spline smoothing approach to nonparametric regression curve fitting J Roy Statist B 47, 1-21 Silverman, B.W (1986) Density Estimation for Statistics and Data Analysis Chapman and Hall, London Singh, R.S (1976) Nonparametric estimation of mixed partial derivatives of a multivariate density J Mult Anal 6, 111-122 Singh, R.S (1979) Mean squared errors of estimates of a density and its derivatives Biometrika 66, 177-180 Singh, R.S (1981) Speed of convergence in nonparametric estimation of a multivariate ~-density and its mixed partial derivatives J Statist Planning Inference 5, 287-298 Sizonenko, P.C (1978) Endocrinology in preadolescents and adolescents Am J Dis Child 132, 704-712 Sizonenko, P.C., Paunier, L and Carmignac, D (1976) Hormonal changes during puberty IV Longitudinal study of adrenal androgen secretions Hormone Res 7, 288-302 Sklar, C.A., Kaplan, S.L and Grumbach, M (1980) Evidence for dissociation between adrenarche and gonadarche: studies in patients with idiopathic precocious puberty, gonadal dysgenesis, isolated gonadotropin deficiency, and constitutionally delayed growth at adolescence J Clin Endocrinol Metab 51, 548-556 Smith, A.F.M and West, M (1983) Monitoring renal transplants: an application of the multi-process Kalman Filter Biometrics 39, 867-878 Stadtmuller, U (1982) Nichtparametrische Schatzung einer Regressionsfunktion in einem Modell mit festem Me~design Habilitationsschrift, Universitat Ulm Stadtmuller, U (1986a) Asymptotic p~operties of nonparametric curve estimates Periodica Math Hung 17, 83-108 Stadtmuller, U (1986b) An inequality between kernel estimators with global and local bandwidths Statistics and Decisions 4, 353-362 Staniswalis, J.G and Cooper, V.D (1986) Kernel estimates of dose response Preprint Stokes, V.P (1984) A method for obtaining 3D kinematics of the pelvis and thorax during locomotion Human Movement Science 3, 77-94 Stone, C.J (1977) Consistent nonparametric regression Ann Statist 5, 505-545 Stone, C.J (1980) Optimal rates of convergence for nonparametric estimators Ann Statist 8, 1348-1360 198 Stone, C.J (1982) Optimal global rates of convergence for nonparametric regression Ann Statist 10, 1040-1053 Stone, C.J (1986) The dimensionality reduction principle for generalized additive models Ann Statist 14, 590-606 Stutzle, W., Gasser, Th., Molinari, L., Largo, R.H., Prader, A and Huber, P.J (1980) Shape-invariant modeling of human growth Ann Hum BioI 7, 507-528 Stutzle, W and Mittal, Y (1979) Some comments on the asymptotic behavior of robust smoothers Lecture Notes in Mathematics 757, 191-195 Stute, W (1982) A law of the iterated logarithm for kernel density estimators Ann Probability 10, 414-422 Stute, W (1986a) Conditional empirical processes Ann Statist 14, 638-647 Stute, W (1986b) On almost sure convergence of conditional empirical distribution functions Ann Probability 14, 891-901 Szego, G (1975) Orthogonal Polynomials American Mathematical Soc., Providence, Rhode Island Tanner, J.M (1981) A History in the Study of Human Growth Cambridge University Press Tanner, J.M.and Cameron, N (1980) Investigation of the mid-growth spurt in height, weight and limb circumference in single-year velocity data from the London 1966-67 growth survey Ann Hum BioI 7, 565-577 Tanner, J.M., Whitehouse, R.H., Marubini, E and Resele, L.F (1976) The adolescent growth spurts of boys and girls of the Harpenden growth study Ann Hum BioI 3, 109-126 Tanner, J.M., Whitehouse, R.H and Takaishi, M (1966a) Standards from birth to maturity for height, weight, height velocity, and weight velocity British Children, 1965-1 Arch Dis Child 41, 454-471 Tanner, J.M., Whitehouse, R.H and Takaishi, M (1966b) Standards from birth to maturity for height, weight, height velocity, and weight velocity British Children, 1965-11 Arch Dis Child 41, 613-635 Trimble, I.M., West, M., Knapp, M., Pownall, R and Smith, A.F.M (1983) Detection of renal allograft rejection by computer British Medical Journal 286, 1695-1699 Tukey, J.W (1977) Exploratory Data Analysis Addison-Wesley, Reading, Mass Utreras, D.F (1980) Sur Ie choix du parametre d'ajustement dans Ie lissage par fonctions spline Numer Math 34, 15-28 Wahba, G (1975) Smoothing noisy data with spline functions Numer Math 24, 383-398 Wahba, G (1979) Convergence rates of "thin plate" smoothing splines when the data are noisy Lecture Notes in Mathematics 757, 233-245 Wahba, G and Wold, S (1975) A completely automatic french curve: fitting spline functions by cross-validation Comm Statist Theor Meth 4, 1-17 Watson, G.S (1964) Smooth regression analysis Sankhya A26, 359-372 Watson, G.S and Leadbetter (1964a) Hazard analysis I Biometrika 51, 175-184 Watson, G.S and Leadbetter (1964b) Hazard analysis II Sankhya A26, 101-116 Whitehouse, R.H., Tanner, J.M and Healey, M.J.R (1974) Diurnal variation in stature and sitting height in 12- to l4-year-old boys Ann Hum Biol 1, 103 Woltring, H.J (1985) On optimal smoothing and derivative estimation from noisy displacement data in biomechanics Human Movement Science 5, 229-245 199 Wong, W.H (1983) On the consistp.ncy of cross-validation in kernel nonparametric regression Ann Statist 11, 1136-1141 Woodroofe, M (1970) On choosing a delta sequence Ann Math Statist 41, 1665-1671 Wu, C.F (1981) Asymptotic theory of nonlinear least squares estimation Ann Statist 9, 501-513 Zacharias, L and Rand, W.M (1983) Adolescent growth in height and its relation to menarche in contemporary American girls Ann Hum Bio1 10, 209-222 Ze1en, M (1983) Biostatistica1 science as a discipline; a look into the future Biometrics 39, 827-837 Lecture Notes in Statistics Vol 1: R A Fisher: An Appreciation Edited by S E Fienberg and D V Hinkley XI, 208 pages, 1980 Vol 2: Mathematical Statistics and Probability Theory Proceedings 1978 Edited by W Klonecki, A Kozek, and J Rosinski XXIV, 373 pages, 1980 Vol 3: B D Spencer, Benefit-Cost Analysis of Data Used to Allocate Funds VIII, 296 pages, 1980 Vol 4: E A van Doorn, Stochastic Monotonicity and Queueing Applications of Birth-Death Processes VI, 118 pages, 1981 Vol 5: T Rolski, Stationary Random Processes Associated with Point Processes VI, 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Acknowledgements l Introduction Longitudinal data and regression models 2.1 Longitudinal data 2.2 Regression models 2.3 Longitudinal growth curves Nonparametric regression methods ... samples of curves An application to follows in Chapter the this study by Gasser et al described data of the Zurich longitudinal growth study The analysis of the growth of 45 boys and 45 girls of (1984a,b,