Analysis of Functional Data1Mary J. LindstromMarch, 2002University of Wisconsin–MadisonDepartment of Biostatistics & Medical InformaticsTechnical Report #1671This research was supported in part by National Institutes of Health grants Nos. CA75097 and DC00820April 21, 2005 iiContents1 Introduction 32 Linear Models 52.1 Subject Specific model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Subject-specific, random-effects model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Population-average (marginal) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.4 Varying design matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.5 Model summaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.6 Restricted maximum likelihood estimation (REML) . . . . . . . . . . . . . . . . . . . . . . . 312.7 Standard errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.8 Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.9 Nested curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.10 Generalized estimating equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 Nonlinear Models 673.1 Nonlinear least squares review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.2 Nonlinear Mixed Effects Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724 Semiparametric Models 814.1 Self-modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815 Other Approaches 1075.1 Functional principal components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107Appendices 115A FTP directory 115B Notation 117C Mixed Effects Software Summary 119D Software for Self-modeling 127E Recommended Books 1291CONTENTS CONTENTSApril 21, 2005 2Chapter 1IntroductionFunctional data arise when the ideal observation for each experimental unit is a curve or function rather thana single number. Dose-response curves of all kinds fall into this class of data. We cannot usually observethe true curve but instead obtain observations at a number of points along the curve for each individual.A data set is then made up of these data curves, one or more per individual. Typical goals in analyzingfunctional data include estimating the typical curve for the population from which the sample of individualswas drawn, describing the between- and within-curve variability structure, estimating individual curves, andtesting for differences between groups of curves.Example: ramus heightA very simple example of functional data is the ramus height data described in Grizzle and Allen (1969).Ramus (jaw bone) height of 20 boys were measured at 8, 8.5, 9 and 9.5 years of age. Figure 1.1 shows theramus height at the back of the jaw labeled “RH”. We will start with a subset of 3 boys (plotted in Figure1.2) to illustrate some models and methods.> ramus.reducedGrouped Data: ramus.height ~ age | individualindividual age ramus.height2 8.0 46.42 8.5 47.32 9.0 47.72 9.5 48.48 8.0 49.88 8.5 50.08 9.0 50.38 9.5 52.710 8.0 45.010 8.5 47.010 9.0 47.310 9.5 48.33IntroductionFigure 1.1: Jaw diagramAge (years)Ramus height (mm)8.0 8.5 9.0 9.53031323310 2 8Figure 1.2: Ramus height of 3 boys measured at 8, 8.5, 9 and 9.5 years of age.April 21, 2005 4Chapter 2Linear Models2.1 Subject Specific modelFor simplicity we will start out by assuming that each individual has observations at the same set of timepoints t1, . . . tnas is true in the ramus height example. Here t1= 8, t2= 8.5, t3= 9, and t4= 9.5 for eachboy. It seems reasonable to assume a (straight line) linear model for the ithboy’s responsesyi,j= βi,1+ βi,2tj+ ei,jj = 1, 2, 3, 4 i = 1, 2, 3E[ei,j] = 0where yi,jis the jthobservation on the ithindividual and tjis the jthtime point (the same for all individualsto start). E.g. y1,3= 47.7 (mm) and t3,2= 8.5 (years). The models for all of individual i’s observations are:yi,1= βi,1+ βi,2t1+ ei,1yi,2= βi,1+ βi,2t2+ ei,2yi,3= βi,1+ βi,2t2+ ei,2yi,4= βi,1+ βi,2t4+ ei,4Which can be rewritten in matrix form as:yi,1yi,2yi,3yi,4= βi,11111+ βi,2t1t2t3t4+ei,1ei,2ei,3ei,452.1. SUBJECT SPECIFIC MODEL Linear Modelsor equivalentlyyi=1 t11 t21 t31 t4βi,1βi,2+ ei= Xiβi+ eiWhere Xiis 4 × 2 design matrix that is the same for all i. We will keep the subscript for generality. Nowthat we have notation for the complete error vector for individual i we can write down a more completedistributional assumptionei∼ N (04, σ2Λ(φi))Where 04is a 4 × 1 vector of zeros and Λ(φi) is a 4 by 4 correlation matrix.The possibility that Λ(φi) is not the identity makes this a general linear model. Note that this is notthe same as a generalized linear model which does not require normally distributed errors. Some possiblestructures for Λ(φ) are:• General correlation matrix[Λ(φ)]h,k= [Λ(φ)]k,h=φh,kfor h 6= k1 for h = k• IndependenceΛ(φ) = I4×4• Equal correlation (compound symmetric)Λ(φ) =1 φ φ φφ 1 φ φφ φ 1 φφ φ φ 1• Toeplitz (for equally spaced time points)Λ(φ) =1 φ1φ2φ3φ11 φ1φ2φ2φ11 φ1φ3φ2φ11• AR(1) correlation for equally spaced observationsΛ(φ) =1 φ φ2φ3φ 1 φ φ2φ2φ 1 φφ3φ2φ 1or, in general[Λ(φ)]h,k= φ|h−k|April 21, 2005 62.1. SUBJECT SPECIFIC MODEL Linear Modelswhere [Λ(φ)]h,kis the h, k entry of Λ(φ).• …