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submitted to the University of Marburg in Spring 1985 as well Completeness is not attempted, as on published and unpublished work neither in the text nor in the references The following persons have been particularly generous in sharing research or giving advice: Roussas, U Th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 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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,