Likelihood Ratio Tests for Dependent Data with Applications to Longitudinal and Functional Data Analysis

type="main" xml:id="sjos12075-abs-0001"> This paper introduces a general framework for testing hypotheses about the structure of the mean function of complex functional processes. Important particular cases of the proposed framework are as follows: (1) testing the null hypothesis that the mean of a functional process is parametric against a general alternative modelled by penalized splines; and (2) testing the null hypothesis that the means of two possibly correlated functional processes are equal or differ by only a simple parametric function. A global pseudo-likelihood ratio test is proposed, and its asymptotic distribution is derived. The size and power properties of the test are confirmed in realistic simulation scenarios. Finite-sample power results indicate that the proposed test is much more powerful than competing alternatives. Methods are applied to testing the equality between the means of normalized δ-power of sleep electroencephalograms of subjects with sleep-disordered breathing and matched controls.

[1]  Ana-Maria Staicu,et al.  Fast methods for spatially correlated multilevel functional data. , 2010, Biostatistics.

[2]  G. S. Watson,et al.  Smooth regression analysis , 1964 .

[3]  E. Nadaraya.  On Estimating Regression , 1964 .

[4]  Jianqing Fan,et al.  Semiparametric Estimation of Covariance Matrixes for Longitudinal Data , 2008, Journal of the American Statistical Association.

[5]  S. Wood.  Generalized Additive Models: An Introduction with R , 2006 .

[6]  Brian S Caffo,et al.  Nonparametric Signal Extraction and Measurement Error in the Analysis of Electroencephalographic Activity During Sleep , 2009, Journal of the American Statistical Association.

[7]  M. Wand,et al.  Semiparametric Regression: Parametric Regression , 2003 .

[8]  Brian S. Caffo,et al.  Multilevel functional principal component analysis , 2009 .

[9]  Helmut Küchenhoff,et al.  Size and power of tests for a zero random effect variance or polynomial regression in additive and linear mixed models , 2008, Comput. Stat. Data Anal..

[10]  J. Samet,et al.  The Sleep Heart Health Study: design, rationale, and methods. , 1997, Sleep.

[11]  H. D. Patterson,et al.  Recovery of inter-block information when block sizes are unequal , 1971 .

[12]  P. Hall,et al.  Properties of principal component methods for functional and longitudinal data analysis , 2006, math/0608022.

[13]  T. Hsing,et al.  Uniform convergence rates for nonparametric regression and principal component analysis in functional/longitudinal data , 2010, 1211.2137.

[14]  Wensheng Guo.  Functional Mixed Effects Models , 2002 .

[15]  D. Billheimer.  Functional Data Analysis, 2nd edition edited by J. O. Ramsay and B. W. Silverman , 2007 .

[16]  Jin-Ting Zhang,et al.  Statistical inferences for functional data , 2007, 0708.2207.

[17]  T. Cai,et al.  A Constrained ℓ1 Minimization Approach to Sparse Precision Matrix Estimation , 2011, 1102.2233.

[18]  Wolfgang Jank,et al.  Functional Data Analysis in Electronic Commerce Research , 2006, math/0609173.

[19]  Jeffrey S. Morris,et al.  Wavelet‐based functional mixed models , 2006, Journal of the Royal Statistical Society. Series B, Statistical methodology.

[20]  Catherine A. Sugar,et al.  Principal component models for sparse functional data , 1999 .

[21]  C. Crainiceanu,et al.  Modeling multilevel sleep transitional data via Poisson log-linear multilevel models , 2009 .

[22]  D. Ruppert.  Selecting the Number of Knots for Penalized Splines , 2002 .

[23]  Kung-Yee Liang,et al.  On the asymptotic behaviour of the pseudolikelihood ratio test statistic with boundary problems. , 1996, Biometrika.

[24]  P. Diggle,et al.  Analysis of Longitudinal Data , 2003 .

[25]  David Ruppert.  Local Polynomial Regression and Its Applications in Environmental Statistics , 1996 .

[26]  Leonard M. Adleman,et al.  Proof of proposition 3 , 1992 .

[27]  Ana-Maria Staicu,et al.  Bootstrap‐based inference on the difference in the means of two correlated functional processes , 2012, Statistics in medicine.

[28]  Jianqing Fan,et al.  Local polynomial modelling and its applications , 1994 .

[29]  Gene H. Golub,et al.  Matrix computations , 1983 .

[30]  D. Harville.  Matrix Algebra From a Statistician's Perspective , 1998 .

[31]  P. Billingsley,et al.  Convergence of Probability Measures , 1970, The Mathematical Gazette.

[32]  Jianqing Fan,et al.  Generalized likelihood ratio statistics and Wilks phenomenon , 2001 .

[33]  John A. Rice,et al.  FUNCTIONAL AND LONGITUDINAL DATA ANALYSIS: PERSPECTIVES ON SMOOTHING , 2004 .

[34]  Jianqing Fan,et al.  Test of Significance When Data Are Curves , 1998 .

[35]  J. Ware,et al.  Random-effects models for longitudinal data. , 1982, Biometrics.

[36]  P. Phillips,et al.  Restricted Likelihood Ratio Tests in Predictive Regression , 2014 .

[37]  Kari Karhunen,et al.  Über lineare Methoden in der Wahrscheinlichkeitsrechnung , 1947 .

[38]  B. Ripley,et al.  Semiparametric Regression: Preface , 2003 .

[39]  H. Müller,et al.  Functional Data Analysis for Sparse Longitudinal Data , 2005 .

[40]  Ricardo Fraiman,et al.  An anova test for functional data , 2004, Comput. Stat. Data Anal..

[41]  D. Ruppert,et al.  Likelihood ratio tests in linear mixed models with one variance component , 2003 .

[42]  B. Silverman,et al.  Functional Data Analysis , 1997 .

[43]  K. Liang,et al.  On the asymptotic behaviour of the pseudolikelihood ratio test statistic with boundary problems , 2010 .

[44]  S. Greven,et al.  Restricted likelihood ratio testing in linear mixed models with general error covariance structure , 2011 .

[45]  D. Bosq.  Linear Processes in Function Spaces: Theory And Applications , 2000 .

[46]  K. B. Kulasekera.  Comparison of Regression Curves Using Quasi-Residuals , 1995 .

[47]  Alan Y. Chiang,et al.  Generalized Additive Models: An Introduction With R , 2007, Technometrics.

[48]  M. Wand,et al.  Exact likelihood ratio tests for penalised splines , 2005 .

[49]  P. Diggle,et al.  Analysis of Longitudinal Data. , 1997 .

[50]  P. Achermann,et al.  Sleep Homeostasis and Models of Sleep Regulation , 1999 .

[51]  K. Liang,et al.  Asymptotic Properties of Maximum Likelihood Estimators and Likelihood Ratio Tests under Nonstandard Conditions , 1987 .

[52]  Anestis Antoniadis,et al.  Estimation and inference in functional mixed-effects models , 2007, Comput. Stat. Data Anal..

[53]  Jane-ling Wang.  Nonparametric Regression Analysis of Longitudinal Data , 2005 .

[54]  M. C. Jones,et al.  Spline Smoothing and Nonparametric Regression. , 1989 .

[55]  Ciprian M Crainiceanu,et al.  Multilevel sparse functional principal component analysis , 2014, Stat.

[56]  P. Hall,et al.  On properties of functional principal components analysis , 2006 .