Mediation Analysis in Categorical Variables under Non-Ignorable Missing Data Mechanisms

Abstract

A mediating variable is a variable that is intermediate in the causal path relating an independent variable to a dependent variable in statistical analysis. The mediation analysis of using a categorical predictor, mediator, and outcome variables has been investigated in the literature. It is extremely common to have missing data even after having a well-controlled study. It is also well known that missingness, especially the non-ignorable missing, in a dataset has often been proven to produce biased results. This paper uses the extended Baker, Rosenberger, and Dersimonian (BRD) model to estimate the mediation effect under non-ignorable missing mechanisms. This paper also proposes four identifiable models to estimate the mediation effect for missingness in one categorical variable with two fully observed categorical variables. We reported the relative bias and Mean Square Error to compare the performance of the proposed BRD models against the Complete Case and Multiple Imputation methods in estimating the mediated effect ($\widehat{a}\widehat{b}$) under the non-ignorable missing mechanism. The application of these models in estimating the mediated effect was demonstrated using the Multiple Risk Factor Intervention Trial datasets.

Appendices

Appendix

Model (${\boldsymbol{\alpha }}_{\dots }$)

$\begin{gathered} L = \left\{ {\prod\limits_{i,j,k} {\frac{{e^{{ - \mu_{{_{ijk111} }} }} (\mu_{{_{ijk111} }} )^{{n_{{_{ijk111} }} }} }}{{^{{n_{{_{ijk111} }} }} }}} \times \prod\limits_{j,k} {\frac{{e^{{ - \mu_{{_{ + jk211} }} }} (\mu_{{_{ + jk211} }} )^{{n_{ + jk211} }} }}{{n_{ + jk211} }}} } \right\} - \mu_{ + + + 111} , \hfill \\ \hfill \\ \end{gathered}$

given ${\mu }_{ijk111}= \widehat{{m}_{ijk}}$ and ${\mu }_{+jk211}= \widehat{{m}_{+jk}}\widehat{{\alpha }_{\dots }}$, the above equation can be written as

$$L = \left\{ {\prod\limits_{i,j,k} {\frac{{e^{{ - m_{{_{ijk} }} }} (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }} )^{{n_{{_{ijk111} }} }} }}{{^{{n_{{_{ijk111} }} }} }}} \times \prod\limits_{j,k} {\frac{{e^{{ - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ + jk} }} \alpha_{{_{...} }} }} (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ + jk} }} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{...} }} )^{{n_{ + jk211} }} }}{{n_{ + jk211} }}} } \right\} - \mu_{ + + + 111}$$

The log-likelihood function can be derived as

$$L = \sum\limits_{i} {\sum\limits_{j} {\sum\limits_{k} {n_{{_{ijk111} }} \log (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }} )} } } + \sum\limits_{j} {\sum\limits_{k} {n_{ + jk211} \log (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ + jk} }} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{...} }} )} } - \sum\limits_{i} {\sum\limits_{j} {\sum\limits_{k} {\left\{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }} (1 + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{...} }} )} \right\}} } } + \Delta$$

By differentiating with respect to $\widehat{{\alpha_{ \ldots } }},$ we have:

$$\therefore \frac{dL}{{d\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{...} }} }} = \frac{{n_{ + jk211} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ + jk} }} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{...} }} }} \bullet \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ + jk} }} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ + + + } }}$$

$$\therefore 0 = \frac{{n_{ + jk211} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{...} }} }} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{ + + + }$$

$$\therefore \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{...} }} = \frac{{n_{ + jk211} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{ + + + } }}.$$

Given each cell count denoted by {n_ijkabc}, where i, j, and k represent the categories for variables I, J, and K, respectively. Hence, we have

$$\begin{gathered} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }} = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{ + + + } = n_{ + + + 111} \hfill \\ n_{ + jk211} = n_{ + + + 211} \hfill \\ \end{gathered}$$

$$\therefore \boxed{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{...} }} = \frac{{n_{ + + + 211} }}{{n_{ + + + 111} }}}$$

By differentiating with respect to $\widehat{{m_{ + jk} }},$ we have:

$$\frac{dL}{{d\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{ijk} }} = \frac{{n_{ijk111} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{ijk...} }} + \frac{{n_{ + jk211} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{ + jk} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{...} }} \bullet \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{...} - (1 + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{...} )$$

$$0 = \frac{{n_{ijk111} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{ijk...} }} + \frac{{n_{ + jk211} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{ + jk} }} - (1 + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{...} )$$

$$\therefore (1 + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{...} ) = \frac{{n_{ijk111} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{ijk...} }} + \frac{{n_{ + jk211} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{ + jk} }}$$

$$\therefore \left( {1 + \frac{{n_{ + + + 211} }}{{n_{ + + + 111} }}} \right) = \frac{{n_{ijk111} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{ijk...} }} + \frac{{n_{ + jk211} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{ + jk} }}$$

$$\left( {\frac{{n_{ + + + 111} + n_{ + + + 211} }}{{n_{ + + + 111} }}} \right) = \frac{{n_{ijk111} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{ijk...} }} + \frac{{n_{ + jk211} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{ + jk} }}$$

$$(\frac{{n_{ + + + + 11} }}{{n_{ + + + 111} }})\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{ijk...} = n_{ijk111} + \frac{{n_{ + jk211} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{ + jk} }}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{ijk}$$

$$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{ijk...} = \left( {n_{ijk111} + \frac{{n_{ + jk211} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{ + jk} }}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{ijk} } \right)\left( {\frac{{n_{ + + + 111} }}{{n_{ + + + + 11} }}} \right)$$

$$\hat{{m}_{+jk}}=\left({n}_{+jk111} + \frac{{n}_{+jk211}}{\hat{{m}_{+jk}}}\hat{{m}_{+jk}}\right)\left(\frac{{n}_{+++111}}{{n}_{++++11}}\right)$$

$$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{ + jk} = \left( {n_{ + jk111} + n_{ + jk211} } \right)\left( {\frac{{n_{ + + + 111} }}{{n_{ + + + + 11} }}} \right)$$

$$\boxed{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ + jk} }} = \frac{{n_{{_{ + jk + 11} }} n_{ + + + 111} }}{{n_{ + + + + 11} }}}$$

This can be simplified further as

$$\frac{{n_{{_{ + jk + 11} }} n_{ + + + 111} }}{{n_{ + + + + 11} }} = \left( {n_{{_{ + jk111} }} + \frac{{n_{{_{ + + + + 11} }} n_{{_{ + jk211} }} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }} }}{{n_{{_{ + jk + 11} }} n_{{_{ + + + 111} }} }}} \right)\left( {\frac{{n_{ + + + 111} }}{{n_{ + + + + 11} }}} \right)$$

$$\frac{{n_{{_{ + jk + 11} }} n_{ + + + 111} }}{{n_{ + + + + 11} }} = \left( {\frac{{n_{{_{ + jk111} }} n_{{_{ + jk + 11} }} n_{{_{ + + + 111} }} + n_{{_{ + + + + 11} }} n_{{_{ + jk211} }} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }} }}{{n_{{_{ + jk + 11} }} n_{{_{ + + + 111} }} }}} \right)\left( {\frac{{n_{ + + + 111} }}{{n_{ + + + + 11} }}} \right)$$

$$\frac{{n_{{_{ + jk + 11} }} n_{ + + + 111} }}{{n_{ + + + + 11} }} = \left( {\frac{{n_{{_{ + + + 111} }} n_{{_{ + jk111} }} n_{{_{ + jk + 11} }} n_{{_{ + + + 111} }} + n_{{_{ + + + 111} }} n_{{_{ + + + + 11} }} n_{{_{ + jk211} }} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }} }}{{n_{{_{ + + + + 11} }} n_{{_{ + jk + 11} }} n_{{_{ + + + 111} }} }}} \right)$$

$n_{{_{ + jk + 11} }} n_{{_{ + + + 111} }} n_{{_{ + + + + 11} }} n_{{_{ + jk + 11} }} n_{{_{ + + + 111} }} = n_{{_{ + + + + 11} }} n_{{_{ + + + 111} }} n_{{_{ + jk111} }} n_{{_{ + jk + 11} }} n_{{_{ + + + 111} }} + n_{{_{ + + + + 11} }} n_{{_{ + + + 111} }} n_{{_{ + + + + 11} }} n_{{_{ + jk211} }} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }}$

$$\frac{{n_{{_{ + jk + 11} }} n_{{_{ + + + 111} }} n_{{_{ + + + + 11} }} n_{{_{ + jk + 11} }} n_{{_{ + + + 111} }} - n_{{_{ + + + + 11} }} n_{{_{ + + + 111} }} n_{{_{ + jk111} }} n_{{_{ + jk + 11} }} n_{{_{ + + + 111} }} }}{{n_{{_{ + + + + 11} }} n_{{_{ + + + 111} }} n_{{_{ + + + + 11} }} n_{{_{ + jk211} }} }} = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }}$$

$$\begin{gathered} \frac{{n_{{_{ + jk + 11} }} n_{{_{ + + + 111} }} n_{{_{ + + + + 11} }} n_{{_{ + jk + 11} }} n_{{_{ + + + 111} }} - n_{{_{ + + + + 11} }} n_{{_{ + + + 111} }} n_{{_{ + jk111} }} n_{{_{ + jk + 11} }} n_{{_{ + + + 111} }} }}{{n_{{_{ + + + + 11} }} n_{{_{ + + + 111} }} n_{{_{ + + + + 11} }} n_{{_{ + jk211} }} }} = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }} \hfill \\ \hfill \\ \end{gathered}$$

Therefore, we have

$$\boxed{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }} = \frac{{n_{{_{ + jk + 11} }} n_{{_{ + + + 111} }} (n_{{_{ + jk + 11} }} - n_{{_{ + jk111} }} )}}{{n_{{_{ + + + + 11} }} n_{{_{ + jk211} }} }}}$$

Model (${\boldsymbol{\alpha }}_{{\varvec{i}}..}$)

$$\begin{gathered} L = \left\{ {\prod\limits_{i,j,k} {\frac{{e^{{ - \mu_{{_{ijk111} }} }} (\mu_{{_{ijk111} }} )^{{n_{{_{ijk111} }} }} }}{{^{{n_{{_{ijk111} }} }} }}} \times \prod\limits_{j,k} {\frac{{e^{{ - \mu_{{_{ + jk211} }} }} (\mu_{{_{ + jk211} }} )^{{n_{ + jk211} }} }}{{n_{ + jk211} }}} } \right\} - \mu_{ + + + 111} \hfill \\ \hfill \\ \end{gathered}$$

$$L = \left\{ {\prod\limits_{i,j,k} {\frac{{e^{{ - m_{{_{ijk} }} }} (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }} )^{{n_{{_{ijk111} }} }} }}{{^{{n_{{_{ijk111} }} }} }}} \times \prod\limits_{j,k} {\frac{{e^{{ - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ + jk} }} \alpha_{{_{...} }} }} (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{i..} }} )^{{n_{ + jk211} }} }}{{n_{ + jk211} }}} } \right\} - \mu_{ + + + 111}$$

$$L = \sum\limits_{i} {\sum\limits_{j} {\sum\limits_{k} {n_{{_{ijk111} }} \log (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }} )} } } + \sum\limits_{j} {\sum\limits_{k} {n_{ + jk211} \log \left( {\sum\limits_{i} {(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{i..} }} )} } \right)} } - \sum\limits_{i} {\sum\limits_{j} {\sum\limits_{k} {\left\{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }} (1 + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{i..} }} )} \right\}} } } + \Delta$$

$$\therefore \frac{dL}{{d\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{i..} }} }} = \sum\limits_{j} {\sum\limits_{k} {\left( {\frac{{n_{ + jk211} }}{{\sum\limits_{i} {(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{i..} }} )} }}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }} } \right)} } - \sum\limits_{j} {\sum\limits_{k} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }} } }$$

$\sum\limits_{j} {\sum\limits_{k} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }} } } = \sum\limits_{j} {\sum\limits_{k} {\left( {\frac{{n_{ + jk211} }}{{\sum\limits_{i} {(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{i..} }} )} }}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }} } \right)} }$,

therefore, we can deduce that given $\boxed{\sum\limits_{i} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{ijk} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{i..} } = n_{ + jk211} }$ $\forall \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{i..} }}$, then

$\frac{dL}{{d\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{ijk} }} = \frac{{n_{ijk111} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{ijk...} }} + \frac{{n_{ + jk211} }}{{\sum\limits_{i} {(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{ijk} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{i..} )} }} \bullet \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{i..} - (1 + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{i..} )$

$$(1 + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{i..} ) = \frac{{n_{ijk111} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{ijk...} }} + \frac{{n_{ + jk211} }}{{\sum\limits_{i} {(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{ijk} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{i..} )} }} \bullet \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{i..}$$

$$(1 + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{i..} )\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{ijk...} = n_{ijk111} + \frac{{n_{ + jk211} }}{{\sum\limits_{i} {(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{ijk} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{i..} )} }}(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{i..} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{ijk...} ).$$

By assuming $\sum\limits_{i} {(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{ijk} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{i..} )} = n_{ + jk211}$, we have:

$$(1 + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{i..} )\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{ijk...} = n_{ijk111} + \frac{{n_{ + jk211} }}{{n_{ + jk211} }}(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{i..} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{ijk...} )$$

$$(1 + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{i..} )\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{ijk...} = n_{{_{ijk111} }} + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{i..} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{ijk...}$$

$$\boxed{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m} ijk_{ijk...} = n_{ijk111} }$$

Model (${\boldsymbol{\alpha }}_{.{\varvec{j}}.}$)

$$L = \sum\limits_{i} {\sum\limits_{j} {\sum\limits_{k} {n_{{_{ijk111} }} \log (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }} )} } } + \sum\limits_{j} {\sum\limits_{k} {n_{ + jk211} \log (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ + jk} }} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{.j.} }} )} } - \sum\limits_{j} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ + j + } }} (1 + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{.j.} }} )} + \Delta$$

$$\therefore \frac{dL}{{d\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{.j.} }} }} = \sum\limits_{k} {\left( {\frac{{n_{ + jk211} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ + jk} }} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{.j.} }} }}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ + jk} }} } \right)} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ + j + } }}$$

$$\therefore 0 = \frac{{n_{ + j + 211} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{.j.} }} }} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ + j + } }}$$

$$\therefore \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ + j + } }} = \frac{{n_{ + j + 211} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{.j.} }} }}$$

$$\boxed{\therefore \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{.j.} }} = \frac{{n_{ + j + 211} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ + j + } }} }}}$$

$\therefore \frac{dL}{{d\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }} }} = \frac{{n_{{_{ijk111} }} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }} }} + \frac{{n_{ + jk211} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ + jk} }} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{.j.} }} }}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{.j.} }} - (1 + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{.j.} }} )$, since $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ + jk} }} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{.j.} }} = n_{ + jk211}$, then:

$$\therefore (1 + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{.j.} }} ) = \frac{{n_{{_{ijk111} }} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }} }} + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{.j.} }}$$

$$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }} (1 + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{.j.} }} ) = n_{{_{ijk111} }} + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{.j.} }} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }}$$

$$\boxed{\therefore \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }} = n_{{_{ijk111} }} }$$

Model (${\boldsymbol{\alpha }}_{..{\varvec{k}}}$)

$$L = \sum\limits_{i} {\sum\limits_{j} {\sum\limits_{k} {n_{{_{ijk111} }} \log (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }} )} } } + \sum\limits_{j} {\sum\limits_{k} {n_{ + jk211} \log (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ + jk} }} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{..k} }} )} } - \sum\limits_{k} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ + + k} }} (1 + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{..k} }} )} + \Delta$$

$$\therefore \frac{dL}{{d\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{..k} }} }} = \sum\limits_{j} {\left( {\frac{{n_{ + jk211} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ + jk} }} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{..k} }} }}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ + jk} }} } \right)} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ + + k} }}$$

$$\therefore 0 = \frac{{n_{ + + k211} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{..k} }} }} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ + + k} }}$$

$$\therefore \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ + + k} }} = \frac{{n_{ + + k211} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{..k} }} }}$$

$$\boxed{\therefore \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{..k} }} = \frac{{n_{ + + k211} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ + + k} }} }}}$$

$\therefore \frac{dL}{{d\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }} }} = \frac{{n_{{_{ijk111} }} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }} }} + \frac{{n_{ + jk211} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ + jk} }} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{..k} }} }}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{..k} }} - (1 + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{..k} }} )$, since $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ + jk} }} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{..k} }} = n_{ + jk211}$, then:

$$\therefore (1 + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{..k} }} ) = \frac{{n_{{_{ijk111} }} }}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }} }} + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{..k} }}$$

$$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }} (1 + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{..k} }} ) = n_{{_{ijk111} }} + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{_{..k} }} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }}$$

$$\boxed{\therefore \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{m}_{{_{ijk} }} = n_{{_{ijk111} }} }$$