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The use of digital devices to collect data in mobile health (mHealth) studiesintroduces a novel application of time series methods, with the constraint ofpotential data missing at random (MAR) or missing not at random (MNAR). In timeseries analysis, testing for stationarity is an important preliminary step toinform appropriate later analyses. The augmented Dickey-Fuller (ADF) test wasdeveloped to test the null hypothesis of unit root non-stationarity, under nomissing 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 contextwith more complex missing data mechanisms. Multiple imputation with chainedequations, Kalman smoothing imputation, and linear interpolation have also beenproposed for time series data, however such methods impose constraints on theautocorrelation structure, and thus impact unit root testing. We proposemaximum likelihood estimation and multiple imputation using state space modelapproaches to adapt the ADF test to a context with missing data. We furtherdevelop sensitivity analysis techniques to examine the impact of MNAR data. Weevaluate the performance of existing and proposed methods across differentmissing mechanisms in extensive simulations and in their application to amulti-year smartphone study of bipolar patients.

Testing unit root non-stationarity in the presence of missing data in univariate time series of mobile health studies