  When conducting a paired $2\times2$ crossover design, each subject is paired
with another subject with similar characteristics. The pair is then randomized
to the same sequence of two treatments. That is, the two subjects receive the
first experimental treatment, and then they cross over and receive the other
experimental treatment(s). The paired $2\times2$ crossover design that was used
in the Beta Adrenergic Response by GEnotype (BARGE) Study conducted by the
National Heart, Lung and Blood Institute's Asthma Clinical Research Network
(ACRN) has been described elsewhere. When the data arising from such a design
are balanced and complete -- or if at least any missingness that occurs is at
random -- general linear mixed-effects model methods can be used to analyze the
data. In this paper, we present a method based on a pattern-mixture model for
analyzing the data arising from a paired $2\times2$ crossover design when some
of the data are missing in a non-ignorable fashion. Because of its inherent
scientific interest, we focus our particular attention on the estimation of the
treatment-by-type of subject interaction term. Finally, we illustrate the
pattern-mixture model methods described in this paper on the data arising from
the BARGE study.
