Variational regression methods are increasingly popular because they enable efficient estimation in complex statistical models. To characterize their theoretical properties, researchers have established frequentist properties for models fitted using mean-field variational Bayes (MFVB). However, hypothesis testing in this framework remains understudied. We propose a global test for functional effects in scalar-on-function regression. The test combines consistency results for MFVB estimators with an existing test for smoothed semiparametric curves. To implement the procedure, we also develop a coordinate ascent variational inference (CAVI) algorithm for MFVB scalar-on-function regression, rather than relying on existing black-box procedures. Finally, we conduct a simulation study comparing our estimation procedure and global test with fully frequentist and fully Bayesian alternatives in terms of bias, root mean squared error, size, and power.

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