Functional multiple indicators, multiple causes measurement error models

Objective measures of oxygen consumption and carbon dioxide production by mammals are used to predict their energy expenditure. Since energy expenditure is not directly observable, it can be viewed as a latent construct with multiple physical indirect measures such as respiratory quotient, volumetric oxygen consumption, and volumetric carbon dioxide production. Metabolic rate is defined as the rate at which metabolism occurs in the body. Metabolic rate is also not directly observable. However, heat is produced as a result of metabolic processes within the body. Therefore, metabolic rate can be approximated by heat production plus some errors. While energy expenditure and metabolic rates are correlated, they are not equivalent. Energy expenditure results from physical function, while metabolism can occur within the body without the occurrence of physical activities. In this manuscript, we present a novel approach for studying the relationship between metabolic rate and indicators of energy expenditure. We do so by extending our previous work on MIMIC ME models to allow responses that are sparsely observed functional data, defining the sparse functional multiple indicators, multiple cause measurement error (FMIMIC ME) models. The mean curves in our proposed methodology are modeled using basis splines. A novel approach for estimating the variance of the classical measurement error based on functional principal components is presented. The model parameters are estimated using the EM algorithm and a discussion of the model's identifiability is provided. We show that the defined model is not a trivial extension of longitudinal or functional data methods, due to the presence of the latent construct. Results from its application to data collected on Zucker diabetic fatty rats are provided. Simulation results investigating the properties of our approach are also presented.