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The null distribution of the test statistic in (4) for the null hypothesis that there is no effect of age on activity. The red dashed line is the 95 percent quantile of the null distribution of the test statistic.
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We propose simple inferential approaches for the fixed effects in complex functional mixed effects models. We estimate the fixed effects under the independence of functional residuals assumption and then bootstrap independent units (e.g. subjects) to estimate the variability of and conduct inference in the form of hypothesis testing on the fixed ef...
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... use the proposed testing statistic, T = { µ A (t, x) − µ 0 (t)} 2 dtdx as detailed in Section 4. The estimate µ A (t, x) is based on the tensor product of 15 cubic basis functions in t and 5 cubic basis functions in x and the estimate µ 0 (t) is based on 15 cubic basis functions. Figure 1 shows the null distribution. The observed test statistic is T = 0.041 and the corresponding p-value is less than 0.001 based on 1000 MC samples. ...
Citations
... Much of the existing methodologies on LFD concentrates on modelling through functional principal component analysis (FPCA) (Park and Staicu, 2015;Scheffler et al., 2020). To the best of our knowledge, the only functional testing procedure developed for LFD tests for invariance of the smooth bivariate mean µ(s, d) along the longitudinal component d (Park et al., 2018;Koner et al., 2021). In this paper, we develop a pseudo Generalized F (pGF) test for the significance of smooth bivariate effect of treatment on the PA in a hierarchically structured longitudinal functional crossover design. ...
Wearable devices for continuous monitoring of electronic health increased attention due to their richness in information. Often, inference is drawn from features that quantify some summary of the data, leading to a loss of information that could be useful when one utilizes the functional nature of the response. When functional trajectories are observed repeated over time, it is termed longitudinal functional data. This work is motivated by the interest to assess the efficacy of a noninflammatory medication, meloxicam, on the daily activity levels of household cats with a pre-existing condition of osteoarthritis under a crossover design. These activity profiles are recorded at a minute level by accelerometer over the entire study period. To this aspect, we propose an orthogonal projection-based test pseudo generalized F test for significance of the functional treatment effect under a functional additive crossover model after adjusting for the carryover effect and other baseline covariates. Under mild conditions, we derive the asymptotic null distribution of the test statistic when the projection function for the underlying Hilbert space is estimated from the data. In finite sample numerical studies, the proposed test maintains the size, is powerful to detect the significance of the smooth effect of meloxicam, and is very efficient compared to bootstrap-based alternatives.
... These approaches, after data transformation or projections, allow inference on the fixed effects parameters using the linear mixed effects (LMEs) inferential machinery. Alternatively, bootstrap of study participants 9,17,18 can be used to construct confidence bands and conduct hypothesis tests for fixed effect functions. In spite of the growing interest in this area of research, modeling longitudinal functional data continues to be daunting. ...
... For each table, we compare results using five methods. The ind_obs method refers to the method implemented in the pffr function; we use the bootstrap method 18 to obtain the confidence bands for the coefficient functions; and the independent, exchangeable and unspecified methods refer to the proposed methods with independent, exchangeable and unspecified covariances, respectively. We first focus on the point estimation, which is partially captured by the MISE in the three tables. ...
We propose an inferential framework for fixed effects in longitudinal functional models and introduce tests for the correlation structures induced by the longitudinal sampling procedure. The framework provides a natural extension of standard longitudinal correlation models for scalar observations to functional observations. Using simulation studies, we compare fixed effects estimation under correctly and incorrectly specified correlation structures and also test the longitudinal correlation structure. Finally, we apply the proposed methods to a longitudinal functional dataset on physical activity. The computer code for the proposed method is available at https://github.com/rli20ST758/FILF.
... The adjustment made on the pointwise and setwise p-values is only one of the possible approaches presented in the literature. Some works recently focused on providing simultaneous confidence bands for functional data: Degras (2017) develop asymptotic confidence bands, Rathnayake and Choudhary (2016) focus on parametric confidence bands, and Crainiceanu et al. (2012) and Park et al. (2017) use bootstrap confidence bands. Confidence sets based on random field theory have also been considered in, for example, Telschow and Schwartzman (2022) and Liebl and Reimherr (2020). ...
Functional data are smooth, often continuous, random curves, which can be seen as an extreme case of multivariate data with infinite dimensionality. Just as component‐wise inference for multivariate data naturally performs feature selection, subset‐wise inference for functional data performs domain selection. In this paper, we present a unified testing framework for domain selection on populations of functional data. In detail, p‐values of hypothesis tests performed on point‐wise evaluations of functional data are suitably adjusted for providing a control of the family‐wise error rate (FWER) over a family of subsets of the domain. We show that several state‐of‐the‐art domain selection methods fit within this framework and differ from each other by the choice of the family over which the control of the FWER is provided. In the existing literature, these families are always defined a priori. In this work, we also propose a novel approach, coined threshold‐wise testing, in which the family of subsets is instead built in a data‐driven fashion. The method seamlessly generalizes to multidimensional domains in contrast to methods based on a‐priori defined families. We provide theoretical results with respect to consistency and control of the FWER for the methods within the unified framework. We illustrate the performance of the methods within the unified framework on simulated and real data examples, and compare their performance with other existing methods. This article is protected by copyright. All rights reserved
... Testing procedures for independent functional data focus on assessing the significance of a smooth functional effect (Shen & Faraway 2004, Zhang 2011 These tests are either constructed by taking point-wise supremum of the classical ANOVA based F-test or by considering a L 2 -norm based Global Pointwise F (GPF) statistic that has a mixture of chi-square null distribution. For correlated functional data, to the best of authors knowledge, testing for the time-varying mean function has been only considered by Park et al. (2018), where the authors proposed an L 2 -norm based distance between the estimated mean response under the null and the alternative hypotheses, and used bootstrap to approximate the test's null distribution. The testing procedure is thus computationally intensive, making it unfeasible to study its size and power properties in large sample sizes. ...
... package(Scheipl et al. 2008), and is much faster compared to the analogous bootstrap based procedures. The bootstrap based approach(Park et al. 2018) takes enormous amount of time (about 150 times slower than PROFIT), by comparison, as it requires fitting the bivariate smoother multiple times for each simulation. ...
In many modern applications, a dependent functional response is observed for each subject over repeated time, leading to longitudinal functional data. In this paper, we propose a novel statistical procedure to test whether the mean function varies over time. Our approach relies on reducing the dimension of the response using data-driven orthogonal projections and it employs a likelihood-based hypothesis testing. We investigate the methodology theoretically and discuss a computationally efficient implementation. The proposed test maintains the type I error rate, and shows excellent power to detect departures from the null hypothesis in finite sample simulation studies. We apply our method to the longitudinal diffusion tensor imaging study of multiple sclerosis (MS) patients to formally assess whether the brain's health tissue, as summarized by fractional anisotropy (FA) profile, degrades over time during the study period.
... The two primary methodologic challenges in fitting FoSR models are estimation of smooth fixed effects and accounting for within-subject correlation. As our interest here is in the estimation of population-level marginal models, we take a a bootstrap procedure for both estimation and inference on fixed effects [31]. We use cyclic cubic regression splines for estimating f 0 , and a tensor product smooth of marginal cyclic cubic splines and cubic splines for estimating f 1 . ...
The ability of individuals to engage in physical activity is a critical component of overall health and quality of life. However, there is a natural decline in physical activity associated with the aging process. Establishing normative trends of physical activity in aging populations is essential to developing public health guidelines and informing clinical perspectives regarding individuals’ levels of physical activity. Beyond overall quantity of physical activity, patterns regarding the timing of activity provide additional insights into latent health status. Wearable accelerometers, paired with statistical methods from functional data analysis, provide the means to estimate diurnal patterns in physical activity. To date, these methods have been only applied to study aging trends in populations based in the United States. Here, we apply curve registration and functional regression to 24 h activity profiles for 88,793 men (N = 39,255) and women (N = 49,538) ages 42–78 from the UK Biobank accelerometer study to understand how physical activity patterns vary across ages and by gender. Our analysis finds that daily patterns in both the volume of physical activity and probability of being active change with age, and that there are marked gender differences in these trends. This work represents the largest-ever population analyzed using tools of this kind, and suggest that aging trends in physical activity are reproducible in different populations across countries.
Modern longitudinal studies collect multiple outcomes as the primary endpoints to understand the complex dynamics of the diseases. Oftentimes, especially in clinical trials, the joint variation among the multidimensional responses plays a significant role in assessing the differential characteristics between two or more groups, rather than drawing inferences based on a single outcome. We develop a projection-based two-sample significance test to identify the population-level difference between the multivariate profiles observed under a sparse longitudinal design. The methodology is built upon widely adopted multivariate functional principal component analysis to reduce the dimension of the infinite-dimensional multi-modal functions while preserving the dynamic correlation between the components. The test applies to a wide class of (non-stationary) covariance structures of the response, and it detects a significant group difference based on a single p-value, thereby overcoming the issue of adjusting for multiple p-values that arise due to comparing the means in each of components separately. Finite-sample numerical studies demonstrate that the test maintains the type-I error, and is powerful to detect significant group differences, compared to the state-of-the-art testing procedures. The test is carried out on two significant longitudinal studies for Alzheimer’s disease and Parkinson’s disease (PD) patients, namely, TOMMORROW study of individuals at high risk of mild cognitive impairment to detect differences in the cognitive test scores between the pioglitazone and the placebo groups, and Azillect study to assess the efficacy of rasagiline as a potential treatment to slow down the progression of PD.
Modern studies from a variety of fields record multiple functional observations according to either multivariate, longitudinal, spatial, or time series designs. We refer to such data as second-generation functional data because their analysis—unlike typical functional data analysis, which assumes independence of the functions—accounts for the complex dependence between the functional observations and requires more advanced methods. In this article, we provide an overview of the techniques for analyzing second-generation functional data with a focus on highlighting the key methodological intricacies that stem from the need for modeling complex dependence, compared with independent functional data. For each of the four types of second-generation functional data presented—multivariate functional data, longitudinal functional data, functional time series and spatially functional data—we discuss how the widely popular functional principal component analysis can be extended to these settings to define, identify main directions of variation, and describe dependence among the functions. In addition to modeling, we also discuss prediction, statistical inference, and application to clustering. We close by discussing future directions in this area.
Modern longitudinal studies collect multiple outcomes as the primary endpoints to understand the complex dynamics of the diseases. Oftentimes, especially in clinical trials, the joint variations among the multidimensional responses play a significant role in assessing the differential characteristics between two or more groups, rather than drawing inferences based on a single outcome. Enclosing the longitudinal design under the umbrella of sparsely observed functional data, we develop a projection-based two-sample significance test to identify the difference between the typical multivariate profiles. The methodology is built upon widely adopted multivariate functional principal component analysis to reduce the dimension of the infinite-dimensional multi-modal functions while preserving the dynamic correlation between the components. The test is applicable to a wide class of (non-stationary) covariance structures of the response, and it detects a significant group difference based on a single p-value, thereby overcoming the issue of adjusting for multiple p-values that arises due to comparing the means in each of components separately. Finite-sample numerical studies demonstrate that the test maintains the type-I error, and is powerful to detect significant group differences, compared to the state-of-the-art testing procedures. The test is carried out on the longitudinally designed TOMMORROW study of individuals at high risk of mild cognitive impairment due to Alzheimer's disease to detect differences in the cognitive test scores between the pioglitazone and the placebo groups.
We develop a generalized partially additive model to build a single semiparametric risk scoring system for physical activity across multiple populations. A score comprised of distinct and objective physical activity measures is a new concept that offers challenges due to the nonlinear relationship between physical behaviors and various health outcomes. We overcome these challenges by modeling each score component as a smooth term, an extension of generalized partially linear single-index models. We use penalized splines and propose two inferential methods, one using profile likelihood and a nonparametric bootstrap, the other using a full Bayesian model, to solve additional computational problems. Both methods exhibit similar and accurate performance in simulations. These models are applied to the National Health and Nutrition Examination Survey and quantify nonlinear and interpretable shapes of score components for all-cause mortality.
