Abstract
In this paper we address two issues in the use of the nonresponse log-linear models for the analysis of an incomplete two-way contingency table with one variable subject to nonresponse, the occurrence of nonresponse boundary solutions and the assessment of the missing data mechanism. To this end, we employ a set of response odds from the fully classified counts and nonresponse odds from partially classified counts. We first investigate the role of the set of odds in identifying the occurrence of boundary solutions and assessing the nonresponse log-linear models suitable for the data, and then propose a data analytic guideline for the analysis of an incomplete two-way contingency table. We also examine the theoretical properties of the set of odds when nonresponse boundary solutions occur.
MATHEMATICS SUBJECT CLASSIFICATION:
Disclosure statement
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Funding
This research was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education (NRF-2016R1D1A3B03930392) and National Research Foundation of Korea grant funded by the Korea government (MSIP; Ministry of Science, ICT & Future Planning) (No. 2017R1C1B5077065).
Appendix A. Proof of Theorem 3.1
Proof. Define . From NMAR model, it is immediate that
which does not depend on the subscript i and can be written by
. For a
contingency table, Baker and Laird (Citation1988); Baker, Rosenberger, and Dersimonian (Citation1992) showed that ML estimates of
denoted by
satisfied
(11)
(11)
where
which is ML estimates in the interior of the parameter space. Thus,
can be obtained by solving system of equations in EquationEq. (11)
(11)
(11) . They also showed ML estimates of
fall on the boundary solution if any
is nonpositive for all j. For the
contingency table, EquationEq. (11)
(11)
(11) gives
This yields
(12)
(12)
(13)
(13)
where the subscripts m and n indicate the categories of Y corresponding to
and
, respectively. The relationship between
and
given by
also yields
(14)
(14)
We prove Theorem 3.1 by contrapositive. First, we suppose for all
. Plugging these two inequalities in EquationEq. (14)
(14)
(14) into EquationEqs. (12)
(12)
(12) and Equation(13)
(13)
(13) together with positiveness of
, then
and
. This completes the proof. □
Appendix B. Parameter values used in the simulation study for ![](//:0)
and ![](//:0)
contingency tables
Table A1. Values of in a
table.
Table A2. Values of for the NMAR model in a
table.
Table A3. Parameters for three nonresponse models in a table.