  Frequently, empirical studies are plagued with missing data. When the data
are missing not at random, the parameter of interest is not identifiable in
general. Without additional assumptions, we can derive bounds of the parameters
of interest, which, unfortunately, are often too wide to be informative.
Therefore, it is of great importance to sharpen these worst-case bounds by
exploiting additional information. Traditional missing data analysis uses only
the information of the binary missing data indicator, that is, a certain data
point is either missing or not. Nevertheless, real data often provide more
information than a binary missing data indicator, and they often record
different types of missingness. In a motivating HIV status survey, missing data
may be due to the units' unwillingness to respond to the survey items or their
hospitalization during the visit, and may also be due to the units' temporarily
absence or relocation. It is apparent that some missing types are more likely
to be missing not at random, but other missing types are more likely to be
missing at random. We show that making full use of the missing types results in
narrower bounds of the parameters of interest. In a real-life example, we
demonstrate substantial improvement of more than 50% reduction in bound widths
for estimating the prevalence of HIV in rural Malawi. As we illustrate using
the HIV study, our strategy is also useful for conducting sensitivity analysis
by gradually increasing or decreasing the set of types that are missing at
random. In addition, we propose an easy-to-implement method to construct
confidence intervals for partially identified parameters with bounds expressed
as the minimums and maximums of finite parameters, which is useful for not only
our problem but also many other problems involving bounds.
