## Smoothing Spline Models for the Analysis of Nested and Crossed Samples of Curves (1998)

Venue: | Journal of the American Statistical Association |

Citations: | 80 - 1 self |

### BibTeX

@ARTICLE{Brumback98smoothingspline,

author = {Babette A. Brumback and John A. Rice},

title = {Smoothing Spline Models for the Analysis of Nested and Crossed Samples of Curves},

journal = {Journal of the American Statistical Association},

year = {1998},

volume = {93},

pages = {961--976}

}

### Years of Citing Articles

### OpenURL

### Abstract

We introduce a class of models for an additive decomposition of groups of curves strati ed by crossed and nested factors, generalizing smoothing splines to such samples by associating them with a corresponding mixed e ects model. The models are also useful for imputation of missing data and exploratory analysis of variance. We prove that the best linear unbiased predictors (BLUP) from the extended mixed e ects model correspond to solutions of a generalized penalized regression where smoothing parameters are directly related to variance components, and we show that these solutions are natural cubic splines. The model parameters are estimated using a highly e cient implementation of the EM algorithm for restricted maximum likelihood (REML) estimation based on a preliminary eigenvector decomposition. Variability of computed estimates can be assessed with asymptotic techniques or with a novel hierarchical bootstrap resampling scheme for nested mixed e ects models. Our methods are applied to menstrual cycle data from studies of reproductive function that measure daily urinary progesterone; the sample of progesterone curves is strati ed by cycles nested within subjects nested within conceptive and non-conceptive groups.

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Citation Context ... Imbalance can also occur within the observational units, as it is not uncommon for cycles to be measured with missing data. In this paper, we extend the smoothing spline model for individual curves (=-=Silverman 1985-=-; Wahba 1978) to generalized smoothing spline models for samples of curves strati ed by nested and crossed factors. In so doing, we create for the menstrual data a smoothing spline estimation of group... |

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Citation Context ...ines will rst be given, with particular emphasis on the correspondence between smoothing splines and mixed e ects models, which arises from the 3sderivation of a smoothing spline as a Bayes estimate (=-=Kimeldorf and Wahba 1970-=-; Wahba 1978). While the setting of these papers was in continuous time, Silverman (1985) developed the discrete time analog and Speed (1991) pointed out the connection with BLUP and REML in a mixed e... |

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Citation Context ...st contemporary research in the area of samples of curves assumes, either tacitly or explicity, that the individual curves are pairwise uncorrelated (Anderson and Jones 1995; Ramsay and Dalzell 1991; =-=Rice and Silverman 1991-=-). An exception can be found in Barry (1996), where ni individuals indexed by j and assigned to treatment i contribute curves that are modeled as the sum of a random mean for treatment i plus a random... |

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Citation Context ...ith mean 0 and variance 2 = gING 2 = wINW 2 = cIN (where is the direct sum operation for matrices), the solutions to (21) are those to arg min , 2logp(yju)p(u) and hence BLUP (Lindley and Smith 1972; =-=Robinson 1991-=-) for the model in (11). Therefore the BLUP tted curves for (11) summarized by ^y = X ^ + Z ^u are equivalent to the tted curves given by the penalized regression in (14). 2 Proof of Theorem 2 The pro... |

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Citation Context ...s, analysis of variance, and samples of curves. Most contemporary research in the area of samples of curves assumes, either tacitly or explicity, that the individual curves are pairwise uncorrelated (=-=Anderson and Jones 1995-=-; Ramsay and Dalzell 1991; Rice and Silverman 1991). An exception can be found in Barry (1996), where ni individuals indexed by j and assigned to treatment i contribute curves that are modeled as the ... |

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