Multilevel Cross-Dependent Binary Longitudinal Data

Abstract

We provide insights into new methodology for the analysis of multilevel binary data observed longitudinally, when the repeated longitudinal measurements are correlated. The proposed model is logistic functional regression conditioned on three latent processes describing the within- and between-variability, and describing the cross-dependence of the repeated longitudinal measurements. We estimate the model components without employing mixed-effects modeling but assuming an approximation to the logistic link function. The primary objectives of this paper are to highlight the challenges in the estimation of the model components, to compare two approximations to the logistic regression function, linear and exponential, and to discuss their advantages and limitations. The linear approximation is computationally efficient whereas the exponential approximation applies for rare events functional data. Our methods are inspired by and applied to a scientific experiment on spectral backscatter from long range infrared light detection and ranging (LIDAR) data. The models are general and relevant to many new binary functional data sets, with or without dependence between repeated functional measurements.

Significance tests for functional data with complex dependence structure

“…So there is a lot of interest in knowing whether the two types of clouds are different. The data have been previously described in Carroll et al (2012) and discussed recently in Serban et al (2013) and Xun et al (2013).…”

“…Because of physical properties, the backscatter efficiency can be viewed as a function of the wavelength for each burst (see Serban et al (2013)). Define the response Y ijl as the backscatter efficiency for CO 2 laser wavelength t ijl for the j th burst sampled at s ij within cloud i .…”

Exponential Family Functional Data Analysis via a Low-Rank Model

“…() proposed a Latent Gaussian Process (LGP) model to analyze irregularly spaced non‐Gaussian observations, where the relations between longitudinal measurements are characterized by an underlying Gaussian process. Later, the method was generalized to rare events (Serban et al., ) and multilevel data (Goldsmith et al., ). More recently, Gertheiss et al.…”

A note on modeling sparse exponential-family functional response curves

“…Analyzing repeated exponential-family outcomes using functional data approaches is currently an area of intensive research [12, 13, 14, 15, 16, 17, 18]. Hall et al [12] extended model (1) to handle non-Gaussian functional data.…”

“…$$Efalse[Y_{i}false(tfalse)\mid {\xi }_{i}false]=h\left\{\mu false(tfalse)+\sum _{k=1}^M{\xi }_{ik}{\psi }_k(t)\right\},$$ with a known response function h (·); more recently, this model framework has been extended to account for rare events [14] and repeated functional observations on each subject [17]. As in model (1), it is assumed that, conditional on the subject-specific scores ξ ik , the responses Y i ( t )’s are independent over t .…”