Long𝒞ri𝒮𝒫: a test for bump hunting in longitudinal data

We propose an extension of the Harezlak and Heckman (J. Comput. Graph. Statist. 2001; 10(4): 713–729) test for detecting local extrema to the longitudinal data setting. We use penalized spline regression techniques (Statist. Sci. 1996; 11:89–102) to provide a computationally efficient method of testing for relatively large data sets. We estimate the p‐values of our test, Long𝒞ri𝒮𝒫, with a smoothed bootstrap. Our simulation studies indicate that the test is generally conservative and has power exceeding 70 per cent at the α=0.1 nominal level in most considered settings. Finally, we apply our testing procedure to the longitudinal measurements of body mass index of former prisoners of war in Vietnam and conclude that the mean population curve exhibits non‐monotone behaviour. Copyright © 2006 John Wiley & Sons, Ltd.

[1]  D. R. Cox,et al.  NOTES ON THE ANALYSIS OF MIXED FREQUENCY1 DISTRIBUTIONS , 1966 .

[2]  J. Hartigan,et al.  Percentage Points of a Test for Clusters , 1969 .

[3]  S. Wolfe,et al.  On the Unimodality of $L$ Functions , 1971 .

[4]  M. Priestley,et al.  Non‐Parametric Function Fitting , 1972 .

[5]  I. Good,et al.  Density Estimation and Bump-Hunting by the Penalized Likelihood Method Exemplified by Scattering and Meteorite Data , 1980 .

[6]  B. Silverman,et al.  Using Kernel Density Estimates to Investigate Multimodality , 1981 .

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

[8]  Bernard Walter Silverman.  Convergence of a Class of Empirical Distribution Functions of Dependent Random Variables , 1983 .

[9]  H. Müller,et al.  Nonparametric Regression Analysis of Growth Curves , 1984 .

[10]  Andrew R. Solow,et al.  Bootstrapping correlated data , 1985 .

[11]  J. Hartigan,et al.  The Dip Test of Unimodality , 1985 .

[12]  H. Müller.  Nonparametric regression analysis of longitudinal data , 1988 .

[13]  G. Sawitzki,et al.  Excess Mass Estimates and Tests for Multimodality , 1991 .

[14]  N. Heckman.  Bump hunting in regression analysis , 1992 .

[15]  J. Hartigan,et al.  The runt test for multimodality , 1992 .

[16]  D. W. Scott,et al.  The Mode Tree: A Tool for Visualization of Nonparametric Density Features , 1993 .

[17]  J. Ware,et al.  Prediction and creation of smooth curves for temporally correlated longitudinal data , 1995 .

[18]  Paul H. C. Eilers,et al.  Flexible smoothing with B-splines and penalties , 1996 .

[19]  Michael C. Minnotte,et al.  Nonparametric testing of the existence of modes , 1997 .

[20]  M. C. Jones,et al.  Testing Monotonicity of Regression , 1998 .

[21]  J. Rice,et al.  Smoothing spline models for the analysis of nested and crossed samples of curves , 1998 .

[22]  J. Marron,et al.  SiZer for Exploration of Structures in Curves , 1999 .

[23]  David Ruppert,et al.  Variable Selection and Function Estimation in Additive Nonparametric Regression Using a Data-Based Prior: Comment , 1999 .

[24]  Irène Gijbels,et al.  Tests for monotonicity of a regression mean with guaranteed level , 2000 .

[25]  P. Hall,et al.  Linear functions , 2018, Algebra and Geometry.

[26]  N. Heckman,et al.  CriSP: A Tool for Bump Hunting , 2001 .

[27]  Stephanie J. Fonda,et al.  Patterns of body weight in middle-aged and older Americans, by gender and race, 1993–2000 , 2003, Sozial- und Präventivmedizin/Social and Preventive Medicine.

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

[29]  Peter Hall,et al.  Bump hunting with non-Gaussian kernels , 2004 .

[30]  J. Durnin,et al.  A longitudinal study of changes in body composition and basal metabolism in physically active elderly men , 2004, European Journal of Applied Physiology and Occupational Physiology.

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