  Continuous glucose monitors (CGMs) are increasingly used to measure blood
glucose levels and provide information about the treatment and management of
diabetes. Our motivating study contains CGM data during sleep for 174 study
participants with type II diabetes mellitus measured at a 5-minute frequency
for an average of 10 nights. We aim to quantify the effects of diabetes
medications and sleep apnea severity on glucose levels. Statistically, this is
an inference question about the association between scalar covariates and
functional responses. However, many characteristics of the data make analyses
difficult, including (1) non-stationary within-day patterns; (2) substantial
between-day heterogeneity, non-Gaussianity, and outliers; 3) large
dimensionality due to the number of study participants, sleep periods, and time
points. We evaluate and compare two methods: fast univariate inference (FUI)
and functional additive mixed models (FAMM). We introduce a new approach for
calculating p-values for testing a global null effect of covariates using FUI,
and provide practical guidelines for speeding up FAMM computations, making it
feasible for our data. While FUI and FAMM are philosophically different, they
lead to similar point estimators in our study. In contrast to FAMM, FUI is
fast, accounts for within-day correlations, and enables the construction of
joint confidence intervals. Our analyses reveal that: (1) biguanide medication
and sleep apnea severity significantly affect glucose trajectories during
sleep, and (2) the estimated effects are time-invariant.
