The use of digital devices to collect data in mobile health (mHealth) studies introduces a novel application of time series methods, with the constraint of potential data missing at random (MAR) or missing not at random (MNAR). In time series analysis, testing for stationarity is an important preliminary step to inform appropriate later analyses. The augmented Dickey-Fuller (ADF) test was developed to test the null hypothesis of unit root non-stationarity, under no missing data. Beyond recommendations under data missing completely at random (MCAR) for complete case analysis or last observation carry forward imputation, researchers have not extended unit root non-stationarity testing to a context with more complex missing data mechanisms. Multiple imputation with chained equations, Kalman smoothing imputation, and linear interpolation have also been proposed for time series data, however such methods impose constraints on the autocorrelation structure, and thus impact unit root testing. We propose maximum likelihood estimation and multiple imputation using state space model approaches to adapt the ADF test to a context with missing data. We further develop sensitivity analysis techniques to examine the impact of MNAR data. We evaluate the performance of existing and proposed methods across different missing mechanisms in extensive simulations and in their application to a multi-year smartphone study of bipolar patients.