Equivalence testing for functional data with an application to comparing pulmonary function devices

Equivalence testing for scalar data has been well addressed in the literature, however, the same cannot be said for functional data. The resultant complexity from maintaining the functional structure of the data, rather than using a scalar transformation to reduce dimensionality, renders the existing literature on equivalence testing inadequate for the desired inference. We propose a framework for equivalence testing for functional data within both the frequentist and Bayesian paradigms. This framework combines extensions of scalar methodologies with new methodology for functional data. Our frequentist hypothesis test extends the Two One-Sided Testing (TOST) procedure for equivalence testing to the functional regime. We conduct this TOST procedure through the use of the nonparametric bootstrap. Our Bayesian methodology employs a functional analysis of variance model, and uses a flexible class of Gaussian Processes for both modeling our data and as prior distributions. Through our analysis, we introduce a model for heteroscedastic variances within a Gaussian Process by modeling variance curves via Log-Gaussian Process priors. We stress the importance of choosing prior distributions that are commensurate with the prior state of knowledge and evidence regarding practical equivalence. We illustrate these testing methods through data from an ongoing method comparison study between two devices for pulmonary function testing. In so doing, we provide not only concrete motivation for equivalence testing for functional data, but also a blueprint for researchers who hope to conduct similar inference.