Diagnostics for functional regression via residual processes. (English) Zbl 1162.62394

Summary: Methods of regression diagnostics for functional regression models are developed which relate a functional response to predictor variables that can be multivariate vectors or random functions. For this purpose, a residual process is defined by subtracting the predicted from the observed response functions. This residual process is expanded into functional principal components (FPC), and the corresponding FPC scores are used as natural proxies for the residuals in functional regression models. For the case of a univariate covariate, a randomization test is proposed based on these scores to examine if the residual process depends on the covariate. If this is the case, it indicates lack of fit of the model. Graphical methods based on the FPC scores of observed and fitted functions can be used to complement more formal tests. The methods are illustrated with data from a recent study of Drosophila fruit flies regarding life-cycle gene expression trajectories as well as functional data from a dose-response experiment for Mediterranean fruit flies (Ceratitis capitata).


62J20 Diagnostics, and linear inference and regression
62H25 Factor analysis and principal components; correspondence analysis
62A09 Graphical methods in statistics
62P10 Applications of statistics to biology and medical sciences; meta analysis


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Full Text: DOI


[1] Anscombe, F.J.; Tukey, J.W., The examination and analysis of residuals, Technometrics, 5, 141-160, (1963) · Zbl 0118.13903
[2] Arbeitman, M.N.; Furlong, E.E.M.; Imam, F.; Johnson, E.; Null, B.H.; Baker, B.S.; Krasnow, M.A.; Scott, M.P.; Davis, R.W.; White, K.P., Gene expression during the life cycle of drosophila melanogaster, Science, 297, 72-83, (2002)
[3] Cardot, H.; Sarda, P., Estimation in generalized linear models for functional data via penalized likelihood, J. multivariate anal., 92, 24-41, (2005) · Zbl 1065.62127
[4] Cardot, H.; Ferraty, F.; Mas, A.; Sarda, P., Testing hypotheses in the functional linear model, Scand. J. statist., 30, 241-255, (2003) · Zbl 1034.62037
[5] Carey, J.R.; Liedo, P.; Harshman, L.; Zhang, Y.; Müller, H.G.; Partridge, L.; Wang, J.L., Life history response of Mediterranean fruit flies to dietary restriction, Aging cell, 1, 140-148, (2002)
[6] Chiou, J.-M.; Müller, H.G.; Wang, J.L., Functional quasi-likelihood regression models with smooth random effects, J. roy. statist. soc. B, 65, 405-423, (2003) · Zbl 1065.62065
[7] Chiou, J.-M.; Müller, H.G.; Wang, J.L.; Carey, J.R., A functional multiplicative effects model for longitudinal data, with application to reproductive histories of female medflies, Statist. sinica, 13, 1119-1133, (2003) · Zbl 1034.62097
[8] Chiou, J.-M.; Müller, H.G.; Wang, J.L., Functional response models, Statist. sinica, 14, 675-693, (2004) · Zbl 1073.62098
[9] Cook, R.D., Detection of influential observations in linear regression, Technometrics, 19, 15-18, (1977) · Zbl 0371.62096
[10] Cook, R.D.; Weisberg, S., Residuals and influence in regression, (1982), Chapman & Hall London · Zbl 0564.62054
[11] Cuevas, A.; Febrero, M.; Fraiman, R., Linear functional regression: the case of fixed design and functional response, Canad. J. statist., 30, 285-300, (2002) · Zbl 1012.62039
[12] Draper, N.R.; Smith, H., Applied regression analysis, (1998), Wiley New York · Zbl 0158.17101
[13] Escabias, M.; Aguilera, A.M.; Valderrama, M.J., Principal component estimation of functional logistic regression: discussion of two different approaches, J. nonpara. statist., 16, 365-384, (2004) · Zbl 1065.62114
[14] Faraway, J.J., Regression analysis for a functional response, Technometrics, 39, 254-261, (1997) · Zbl 0891.62027
[15] Ferraty, F.; Vieu, P., Nonparametric models for functional data, with application in regression, time-series prediction and curve discrimination, J. nonpara. statist., 16, 111-125, (2004) · Zbl 1049.62039
[16] Green, J.R., Testing departures from a regression without using replication, Technometrics, 13, 609-615, (1971) · Zbl 0225.62022
[17] Grenander, U., 1950. Stochastic processes and statistical inference. Arkiv för Matematik, 195-276. · Zbl 0058.35501
[18] Hall, P.; Hosseini-Nasab, M., On properties of functional principal components analysis, J. roy. statist. soc. B, 68, 109-126, (2006) · Zbl 1141.62048
[19] Hall, P.; Wilson, S.R., Two guidelines for bootstrap hypothesis testing, Biometrics, 47, 757-762, (1991)
[20] Hart, J.D., Nonparametric smoothing and lack-of-fit tests, (1997), Springer New York · Zbl 0886.62043
[21] He, G.; Müller, H.G.; Wang, J.L., Extending correlation and regression from multivariate to functional data, (), 197-210
[22] James, G.M., Generalized linear models with functional predictors, J. roy. statist. soc. B, 64, 411-432, (2002) · Zbl 1090.62070
[23] Liu, X.; Müller, H.G., Modes and clustering for time-warped gene expression profile data, Bioinformatics, 19, 1937-1944, (2003)
[24] Müller, H.G.; Stadtmüller, U., Generalized functional linear models, Ann. statist., 33, 774-805, (2005) · Zbl 1068.62048
[25] Müller, H.G., Functional modelling and classification of longitudinal data, Scand. J. statist., 32, 223-240, (2005) · Zbl 1089.62072
[26] Ramsay, J.O.; Dalzell, C.J., Some tools for functional data analysis, J. roy. statist. soc. B, 53, 539-572, (1991) · Zbl 0800.62314
[27] Ramsay, J.O.; Silverman, B.W., Functional data analysis, (2005), Springer New York · Zbl 1079.62006
[28] Rice, J.A.; Silverman, B.W., Estimating the Mean and covariance structure nonparametrically when the data are curves, J. roy. statist. soc. B, 53, 233-243, (1991) · Zbl 0800.62214
[29] Yao, F.; Müller, H.G.; Clifford, A.J.; Dueker, S.R.; Follett, J.; Lin, Y.; Buchholz, B.; Vogel, J.S., Shrinkage estimation for functional principal component scores, with application to the population kinetics of plasma folate, Biometrics, 59, 676-685, (2003) · Zbl 1210.62076
[30] Yao, F.; Müller, H.G.; Wang, J.L., Functional data analysis for sparse longitudinal data, J. amer. statist. assoc., 100, 577-590, (2005) · Zbl 1117.62451
[31] Yao, F.; Müller, H.G.; Wang, J.L., Functional linear regression analysis for longitudinal data, Ann. statist., 33, 2873-2903, (2005) · Zbl 1084.62096
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