Restricted Likelihood Ratio Tests for Functional Effects in the Functional Linear Model

Swihart, Bruce J.; Goldsmith, Jeff; Crainiceanu, Ciprian M.

doi:10.1080/00401706.2013.863163

Cited by 25 publications

(31 citation statements)

References 67 publications

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“…However as we show above, the association is not significant if the model accounts for the corpus-callosum fractional anisotropy. Interestingly, both findings are in agreement with Swihart et al (2014), who used the fractional anisotropy along the two tracts of the multiple sclerosis subjects measured at all the available hospital visits and a restricted likelihood ratio-based testing approach.…”

Section: Real Data Applicationsupporting

confidence: 86%

“…Usually, this statistic is defined as −2{ L n ( η̃, σ̃ 2 ) − L n ( η̂, σ̂ 2 )} which simplifies to n log( σ̃ 2 / σ̂ 2 ). This test is similar to the ‘restricted’ likelihood ratio test when there is a single functional covariate, discussed in Swihart et al (2014); in this scenario, their proposed restricted likelihood ratio test becomes a likelihood ratio test. Using the same argument as in Wald test, in this case also, we consider the restricted maximum likelihood estimate for σ 2 for both the null and the full model, and define a slightly modified likelihood ratio statistic…”

Section: Methodsmentioning

confidence: 94%

“…To gain more insight, we compared numerically our proposed methods and the minimax adaptive testing method proposed in Hilgert et al (2013); this comparison is included in the Supplementary Material, Section 5.2, due to space limitation of the paper. Furthermore we compared the F test with our discussed asymptotic null distribution with two other alternatives: 1) the F test with the bootstrap-based approximation of its null distribution discussed in González-Manteiga et al (2014) and 2) the likelihood ratio test of Swihart et al (2014) for single functional covariate. The results are described in the Supplementary Material.…”

Section: Simulation Studymentioning

confidence: 99%

“…Additionally, as we observe in our simulation studies, the Wald test statistic is not very reliable for small sample sizes and exhibits significantly inflated Type I error rate even when the functional covariate is observed on very fine grids and without error. Swihart, Goldsmith, and Crainiceanu (2014) addressed a similar testing problem when the setting involves multiple functional covariates; they discussed the restricted likelihood ratio test and investigated its performance numerically, via simulation studies, but did not present its theoretical properties.…”

Section: Introductionmentioning

confidence: 99%

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Classical testing in functional linear models

Kong

Staicu

Maity

2016

Journal of Nonparametric Statistics

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We extend four tests common in classical regression - Wald, score, likelihood ratio and F tests - to functional linear regression, for testing the null hypothesis, that there is no association between a scalar response and a functional covariate. Using functional principal component analysis, we re-express the functional linear model as a standard linear model, where the effect of the functional covariate can be approximated by a finite linear combination of the functional principal component scores. In this setting, we consider application of the four traditional tests. The proposed testing procedures are investigated theoretically for densely observed functional covariates when the number of principal components diverges. Using the theoretical distribution of the tests under the alternative hypothesis, we develop a procedure for sample size calculation in the context of functional linear regression. The four tests are further compared numerically for both densely and sparsely observed noisy functional data in simulation experiments and using two real data applications.

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Section: Real Data Applicationsupporting

confidence: 86%

Section: Methodsmentioning

confidence: 94%

Section: Simulation Studymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Classical testing in functional linear models

Kong

Staicu

Maity

2016

Journal of Nonparametric Statistics

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“…In the literature, tests based on pre-specified spline basis representations have also been proposed, for example a permutation F-test (Ramsay et al, 2009) and restricted likelihood ratio tests (Swihart et al, 2014). The former one requires a relatively high computational cost, and the later ones are shown to be outperformed by FPCA-based tests in terms of power (Swihart et al, 2014). Thus, we will focus on FPCA-based approaches in this paper.…”

Section: Introductionmentioning

confidence: 99%

Hypothesis Testing in Functional Linear Models

Hsu

2017

Biometrics

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Summary Functional data arise frequently in biomedical studies, where it is often of interest to investigate the association between functional predictors and a scalar response variable. While functional linear models (FLM) are widely used to address these questions, hypothesis testing for the functional association in the FLM framework remains challenging. A popular approach to testing the functional effects is through dimension reduction by functional principal component (PC) analysis. However, its power performance depends on the choice of the number of PCs, and is not systematically studied. In this paper, we first investigate the power performance of the Wald-type test with varying thresholds in selecting the number of PCs for the functional covariates, and show that the power is sensitive to the choice of thresholds. To circumvent the issue, we propose a new method of ordering and selecting principal components to construct test statistics. The proposed method takes into account both the association with the response and the variation along each eigenfunction. We establish its theoretical properties and assess the finite sample properties through simulations. Our simulation results show that the proposed test is more robust against the choice of threshold while being as powerful as, and often more powerful than, the existing method. We then apply the proposed method to the cerebral white matter tracts data obtained from a diffusion tensor imaging tractography study.

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Functional Data Analysis: An Introduction and Recent Developments

Gertheiss,

Rügamer,

Liew

et al. 2024

Biometrical J

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Functional data analysis (FDA) is a statistical framework that allows for the analysis of curves, images, or functions on higher dimensional domains. The goals of FDA, such as descriptive analyses, classification, and regression, are generally the same as for statistical analyses of scalar‐valued or multivariate data, but FDA brings additional challenges due to the high‐ and infinite dimensionality of observations and parameters, respectively. This paper provides an introduction to FDA, including a description of the most common statistical analysis techniques, their respective software implementations, and some recent developments in the field. The paper covers fundamental concepts such as descriptives and outliers, smoothing, amplitude and phase variation, and functional principal component analysis. It also discusses functional regression, statistical inference with functional data, functional classification and clustering, and machine learning approaches for functional data analysis. The methods discussed in this paper are widely applicable in fields such as medicine, biophysics, neuroscience, and chemistry and are increasingly relevant due to the widespread use of technologies that allow for the collection of functional data. Sparse functional data methods are also relevant for longitudinal data analysis. All presented methods are demonstrated using available software in R by analyzing a dataset on human motion and motor control. To facilitate the understanding of the methods, their implementation, and hands‐on application, the code for these practical examples is made available through a code and data supplement and on GitHub.

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Restricted Likelihood Ratio Tests for Functional Effects in the Functional Linear Model

Cited by 25 publications

References 67 publications

Classical testing in functional linear models

Classical testing in functional linear models

Hypothesis Testing in Functional Linear Models

Functional Data Analysis: An Introduction and Recent Developments

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