Testing for linearity and independence in scalar-on-function regression with responses missing at random by generalized distance covariance

Abstract

Scalar-on-function regression models are used to predict a scalar response based on a functional covariate. A procedure for jointly testing the linearity of the relationship between the scalar response and the functional covariate and the independence between the functional covariate and the error term when some responses are Missing at Random (MAR) is proposed. The method uses generalized distance covariance (GDC) to detect dependency between the functional covariate and residuals from a functional linear model fit. The null distribution of the test statistic is calibrated using residual bootstrap. Accurate slope estimation in the functional linear model is crucial for handling MAR responses. Three estimation methods based on Functional Principal Components (FPCs) are evaluated: (i) the simplified method, which omits pairs with missing responses; (ii) the imputed method, which fills in missing responses to increase data usage; and (iii) the inverse probability weighted method, which adjusts for the probability of missing responses. Cross-validation is used to select the optimal number of FPCs for each method.