  Imputation methods for dealing with incomplete data typically assume that the
missingness mechanism is at random (MAR). These methods can also be applied to
missing not at random (MNAR) situations, where the user specifies some
adjustment parameters that describe the degree of departure from MAR. The
effect of different pre-chosen values is then studied on the inferences. This
paper proposes a novel imputation method, the Random Indicator (RI) method,
which, in contrast to the current methodology, estimates these adjustment
parameters from the data. For an incomplete variable $X$, the RI method assumes
that the observed part of $X$ is normal and the probability for $X$ to be
missing follows a logistic function. The idea is to estimate the adjustment
parameters by generating a pseudo response indicator from this logistic
function. Our method iteratively draws imputations for $X$ and the realization
of the response indicator $R$, to which we refer as $\dot{R}$, for $X$. By
cross-classifying $X$ by $R$ and $\dot{R}$, we obtain various properties on the
distribution of the missing data. These properties form the basis for
estimating the degree of departure from MAR. Our numerical simulations show
that the RI method performs very well across a variety of situations. We show
how the method can be used in a real life data set. The RI method is automatic
and opens up new ways to tackle the problem of MNAR data.
