FMEM: Functional Mixed Effects Models for Longitudinal Functional Responses

Abstract

The aim of this paper is to conduct a systematic and theoretical analysis of estimation and inference for a class of functional mixed effects models (FMEM). Such FMEMs consist of fixed effects that characterize the association between longitudinal functional responses and covariates of interest and random effects that capture the spatial-temporal correlations of longitudinal functional responses. We propose local linear estimates of refined fixed effect functions and establish their weak convergence along with a simultaneous confidence band for each fixed-effect function. We propose a global test for the linear hypotheses of varying coefficient functions and derive the associated asymptotic distribution under the null hypothesis and the asymptotic power under the alternative hypothesis are derived. We also establish the convergence rates of the estimated spatial-temporal covariance operators and their associated eigenvalues and eigenfunctions. We conduct extensive simulations and apply our method to a white-matter fiber data set from a national database for autism research to examine the finite-sample performance of the proposed estimation and inference procedures.

Modeling multivariate age‐related imaging variables with dependencies

“…Second, this article has not mentioned the repeated measured longitudinal design that is commonly available in medical imaging analyses. To date, few works 24,25 have been conducted along this line. We hope our approach could be used as a tool for deeper investigations to shed light on age‐induced brain changes.…”

A novel longitudinal rank-sum test for multiple primary endpoints in clinical trials: Applications to neurodegenerative disorders

“…The first, the individual test approach, assesses longitudinal outcomes either independently or jointly to determine treatment effects for each outcome, followed by multiplicity adjustment. Analytical techniques include parametric methods such as univariate or multivariate GEE and linear mixed models (LMMs), as well as nonparametric methods such as univariate or multivariate functional mixed models (FMMs) (Zhu and others , 2019) and various rank-based tests (Akritas and Brunner, 1997; Konietschke and others , 2010; Brunner and others , 2017; Zhuang and others , 2018; Umlauft and others , 2019; Rubarth and others , 2022 a , b ). Multiplicity adjustment may employ co-primary endpoint strategy, requiring significance across all endpoints for treatment efficacy assertion, adopted in studies such as Semagacestat (Doody and others , 2013), EXPEDITION (Doody and others , 2014), LMTM (Gauthier and others , 2016), and ADMET 2 (Mintzer and others , 2021), or multiple endpoint strategy, requiring at least one significant outcome after multiple comparison corrections (e.g., Bonferroni, Holm, or Hochberg procedures, and fixed-sequence methods) per FDA guidelines (GUIDANCE, 2022).…”

Fixed‐effects inference and tests of correlation for longitudinal functional data

“…The average activity measured at multiple visits gives rise to standard longitudinal data, whereas minute-by-minute activity profiles observed at multiple visits give rise to longitudinal functional data. Another example is diffusion tensor imaging (DTI), [6][7][8][9][10] where imaging data along brain tracts were collected for patients of multiple sclerosis and healthy controls at a number of visits. The physical activity measurements and the DTI data are usually collected at sparse longitudinal visits with densely observed functions, the data structure the article focuses on.…”

“…Longitudinal functional data often exhibit complex within-and between-visit correlations. To accommodate such correlations, various functional mixed effects models 6,8,9,[11][12][13][14][15][16] have been proposed. These approaches, after data transformation or projections, allow inference on the fixed effects parameters using the linear mixed effects (LMEs) inferential machinery.…”

“…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.…”