  Practical problems with missing data are common, and statistical methods have
been developed concerning the validity and/or efficiency of statistical
procedures. On a central focus, there have been longstanding interests on the
mechanism governing data missingness, and correctly deciding the appropriate
mechanism is crucially relevant for conducting proper practical investigations.
The conventional notions include the three common potential classes -- missing
completely at random, missing at random, and missing not at random. In this
paper, we present a new hypothesis testing approach for deciding between
missing at random and missing not at random. Since the potential alternatives
of missing at random are broad, we focus our investigation on a general class
of models with instrumental variables for data missing not at random. Our
setting is broadly applicable, thanks to that the model concerning the missing
data is nonparametric, requiring no explicit model specification for the data
missingness. The foundational idea is to develop appropriate discrepancy
measures between estimators whose properties significantly differ only when
missing at random does not hold. We show that our new hypothesis testing
approach achieves an objective data oriented choice between missing at random
or not. We demonstrate the feasibility, validity, and efficacy of the new test
by theoretical analysis, simulation studies, and a real data analysis.
