  Missing data are frequently encountered in various disciplines and can be
divided into three categories: missing completely at random (MCAR), missing at
random (MAR) and missing not at random (MNAR). Valid statistical approaches to
missing data depend crucially on correct identification of the underlying
missingness mechanism. Although the problem of testing whether this mechanism
is MCAR or MAR has been extensively studied, there has been very little
research on testing MAR versus MNAR.A critical challenge that is faced when
dealing with this problem is the issue of model identification under MNAR. In
this paper, under a logistic model for the missing probability, we develop two
score tests for the problem of whether the missingness mechanism is MAR or MNAR
under a parametric model and a semiparametric location model on the regression
function. The score tests require only parameter estimation under the null MAR
assumption, which completely circumvents the identification issue. Our
simulations and analysis of human immunodeficiency virus data show that the
score tests have well-controlled type I errors and desirable powers.
