On log-linear modeling for an incomplete two-way contingency table with one variable subject to nonresponse

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.

Keywords:

• Boundary solutions
• incomplete contingency table
• log-linear model
• missing data mechanism

MATHEMATICS SUBJECT CLASSIFICATION:

• 62H17; 62J12

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 ${\beta }_{ij}={\pi }_{ij2}/{\pi }_{ij1}$. From NMAR model, it is immediate that $${\beta }_{ij}=\hspace{0.17em}\text{exp}\hspace{0.17em}\left[{\lambda }_R^2-{\lambda }_R^1+{\lambda }_{YR}^{j2}-{\lambda }_{YR}^{j1}\right],$$ which does not depend on the subscript i and can be written by ${\beta }_{ij}={\beta }_{·j}$. For a $I\times I\times 2$ contingency table, Baker and Laird (1988); Baker, Rosenberger, and Dersimonian (1992) showed that ML estimates of ${\beta }_{·j}$ denoted by ${\widehat{\beta }}_{·j}$ satisfied (11) $$\sum _{j}N{\widehat{\pi }}_{ij1}{\widehat{\beta }}_{·j}=y_{i+2}\hspace{1em}\hspace{1em} \text{for} i=1,\dots ,I$$(11) where ${\widehat{\pi }}_{ij1}=y_{ij1}/N$ which is ML estimates in the interior of the parameter space. Thus, ${\widehat{\beta }}_{·j}$ can be obtained by solving system of equations in Eq. (11)(11) $$\sum _{j}N{\widehat{\pi }}_{ij1}{\widehat{\beta }}_{·j}=y_{i+2}\hspace{1em}\hspace{1em} \text{for} i=1,\dots ,I$$(11) . They also showed ML estimates of ${\pi }_{ij2}$ fall on the boundary solution if any ${\widehat{\beta }}_{·j}$ is nonpositive for all j. For the $I\times I\times 2$ contingency table, Eq. (11)(11) $$\sum _{j}N{\widehat{\pi }}_{ij1}{\widehat{\beta }}_{·j}=y_{i+2}\hspace{1em}\hspace{1em} \text{for} i=1,\dots ,I$$(11) gives $$\widehat{\nu }(i,i\prime )=\frac{y_{i+2}}{y_{i\prime +2}}=\frac{\sum _{j}N{\widehat{\pi }}_{ij1}{\widehat{\beta }}_{·j}}{\sum _{j}N{\widehat{\pi }}_{i\prime j1}{\widehat{\beta }}_{·j}}=\frac{\sum _j{\widehat{\pi }}_{ij1}{\widehat{\beta }}_{·j}}{\sum _j{\widehat{\pi }}_{i\prime j1}{\widehat{\beta }}_{·j}}.$$

This yields (12) $${\widehat{\nu }}_m(i,i\prime )-\widehat{\nu }(i,i\prime )=\frac{\sum _{j\ne m}({\widehat{\pi }}_{im1}{\widehat{\pi }}_{i\prime j1}-{\widehat{\pi }}_{i\prime m1}{\widehat{\pi }}_{ij1}){\widehat{\beta }}_{·j}}{{\widehat{\pi }}_{i\prime m1}\sum _j{\widehat{\pi }}_{i\prime j1}{\widehat{\beta }}_{·j}},$$(12) (13) $$\widehat{\nu }(i,i\prime )-{\widehat{\nu }}_n(i,i\prime )=\frac{\sum _{j\ne n}({\widehat{\pi }}_{ij1}{\widehat{\pi }}_{i\prime n1}-{\widehat{\pi }}_{in1}{\widehat{\pi }}_{i\prime j1}){\widehat{\beta }}_{·j}}{{\widehat{\pi }}_{i\prime n1}\sum _j{\widehat{\pi }}_{i\prime j1}{\widehat{\beta }}_{·j}}$$(13) where the subscripts m and n indicate the categories of Y corresponding to ${\widehat{\nu }}_m(i,i\prime )$ and ${\widehat{\nu }}_n(i,i\prime )$, respectively. The relationship between ${\widehat{\nu }}_m(i,i\prime ),{\widehat{\nu }}_n(i,i\prime )$ and $\widehat{\nu }(i,i\prime )$ given by $${\widehat{\nu }}_m(i,i\prime )=\frac{{\widehat{\pi }}_{im1}}{{\widehat{\pi }}_{i\prime m1}}>\frac{{\widehat{\pi }}_{ij1}}{{\widehat{\pi }}_{i\prime j1}}={\widehat{\nu }}_j(i,i\prime )>\frac{{\widehat{\pi }}_{in1}}{{\widehat{\pi }}_{i\prime n1}}={\widehat{\nu }}_n(i,i\prime ),$$

also yields (14) $${\widehat{\pi }}_{im1}{\widehat{\pi }}_{i\prime j1}>{\widehat{\pi }}_{i\prime m1}{\widehat{\pi }}_{ij1}\hspace{1em}\text{and}\hspace{1em}{\widehat{\pi }}_{ij1}{\widehat{\pi }}_{i\prime n1}>{\widehat{\pi }}_{in1}{\widehat{\pi }}_{i\prime j1}\hspace{1em} \text{for} j\ne m,n.$$(14)

We prove Theorem 3.1 by contrapositive. First, we suppose ${\widehat{\beta }}_{·j}>0$ for all $j=1,\dots ,I$. Plugging these two inequalities in Eq. (14)(14) $${\widehat{\pi }}_{im1}{\widehat{\pi }}_{i\prime j1}>{\widehat{\pi }}_{i\prime m1}{\widehat{\pi }}_{ij1}\hspace{1em}\text{and}\hspace{1em}{\widehat{\pi }}_{ij1}{\widehat{\pi }}_{i\prime n1}>{\widehat{\pi }}_{in1}{\widehat{\pi }}_{i\prime j1}\hspace{1em} \text{for} j\ne m,n.$$(14) into Eqs. (12)(12) $${\widehat{\nu }}_m(i,i\prime )-\widehat{\nu }(i,i\prime )=\frac{\sum _{j\ne m}({\widehat{\pi }}_{im1}{\widehat{\pi }}_{i\prime j1}-{\widehat{\pi }}_{i\prime m1}{\widehat{\pi }}_{ij1}){\widehat{\beta }}_{·j}}{{\widehat{\pi }}_{i\prime m1}\sum _j{\widehat{\pi }}_{i\prime j1}{\widehat{\beta }}_{·j}},$$(12) and (13)(13) $$\widehat{\nu }(i,i\prime )-{\widehat{\nu }}_n(i,i\prime )=\frac{\sum _{j\ne n}({\widehat{\pi }}_{ij1}{\widehat{\pi }}_{i\prime n1}-{\widehat{\pi }}_{in1}{\widehat{\pi }}_{i\prime j1}){\widehat{\beta }}_{·j}}{{\widehat{\pi }}_{i\prime n1}\sum _j{\widehat{\pi }}_{i\prime j1}{\widehat{\beta }}_{·j}}$$(13) together with positiveness of ${\widehat{\beta }}_{·j}$, then ${\widehat{\nu }}_m(i,i\prime )-\widehat{\nu }(i,i\prime )>0$ and $\widehat{\nu }(i,i\prime )-{\widehat{\nu }}_n(i,i\prime )>0$. This completes the proof. □

Appendix B. Parameter values used in the simulation study for $2\times 2\times 2$ and $3\times 3\times 2$ contingency tables

Table A1. Values of ${\pi }_{ij1}$ in a $2\times 2\times 2$ table.

Table A2. Values of ${\pi }_{ij2}$ for the NMAR model in a $2\times 2\times 2$ table.

Table A3. Parameters for three nonresponse models in a $2\times 2\times 2$ table.