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Equivalence testing for scalar data has been well addressed in theliterature, however, the same cannot be said for functional data. The resultantcomplexity from maintaining the functional structure of the data, rather thanusing a scalar transformation to reduce dimensionality, renders the existingliterature on equivalence testing inadequate for the desired inference. Wepropose a framework for equivalence testing for functional data within both thefrequentist and Bayesian paradigms. This framework combines extensions ofscalar methodologies with new methodology for functional data. Our frequentisthypothesis test extends the Two One-Sided Testing (TOST) procedure forequivalence testing to the functional regime. We conduct this TOST procedurethrough the use of the nonparametric bootstrap. Our Bayesian methodologyemploys a functional analysis of variance model, and uses a flexible class ofGaussian Processes for both modeling our data and as prior distributions.Through our analysis, we introduce a model for heteroscedastic variances withina Gaussian Process by modeling variance curves via Log-Gaussian Process priors.We stress the importance of choosing prior distributions that are commensuratewith the prior state of knowledge and evidence regarding practical equivalence.We illustrate these testing methods through data from an ongoing methodcomparison study between two devices for pulmonary function testing. In sodoing, we provide not only concrete motivation for equivalence testing forfunctional data, but also a blueprint for researchers who hope to conductsimilar inference.

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