標本条件付き対数オッズ比の漸近分布

分割表に四項分布モデルを仮定するとき、多項分布の周辺分布であることから、 $$\begin{array}{r}f\sim \mathrm{B}(N,{\pi }_{12})\\ g\sim \mathrm{B}(N,{\pi }_{21})\end{array}$$ 二項分布の正規近似により、標本比率は漸近的に $$\begin{array}{c}{p}_{12}\sim N[{\pi }_{12},\frac{{\pi }_{12}(1-{\pi }_{12})}{N}]\\ p_{21}\sim N[{\pi }_{21},\frac{{\pi }_{21}(1-{\pi }_{21})}{N}]\end{array}$$ 標本比率ベクトルを $$\begin{array}{r}\mathit{p}=\left(\begin{array}{c}{p}_{12}\\ p_{21}\end{array}\right)\end{array}$$ 期待値ベクトルを $$\begin{array}{r}\mathit{\pi }=\left(\begin{array}{c}{\pi }_{12}\\ {\pi }_{21}\end{array}\right)\end{array}$$ 分散・共分散行列を $$\begin{array}{r}\mathbf{\Omega }=\left[\begin{array}{cc}\frac{{\pi }_{12}(1-{\pi }_{12})}{N}& -\frac{{\pi }_{12}{\pi }_{21}}{N}\\ -\frac{{\pi }_{12}{\pi }_{21}}{N}& \frac{{\pi }_{21}(1-{\pi }_{21})}{N}\end{array}\right]\end{array}$$ として、 $$\begin{array}{c}G\left(\mathit{\pi }\right)=\theta =\log \frac{{\pi }_{12}}{{\pi }_{21}}\\ G\left(\mathit{p}\right)=\hat{\theta }=\log \frac{p_{12}}{p_{21}}\end{array}$$ と変数変換する。 多変量のデルタ法を用いて $G\left(\mathit{p}\right)$ を期待値 $E\left(\mathit{p}\right)=\mathit{\pi }$ まわりでテイラー展開すると、偏導関数ベクトルは、 $$\begin{array}{r}\mathit{H}\left(\mathit{\pi }\right)=\left(\begin{array}{c}\frac{G\left(\mathit{\pi }\right)}{\partial {\pi }_1}\\ \frac{G\left(\mathit{\pi }\right)}{\partial {\pi }_2}\end{array}\right)=\left[\begin{array}{c}\frac{1}{{\pi }_{12}}\\ -\frac{1}{{\pi }_{21}}\end{array}\right]\end{array}$$ 多変量のデルタ法の期待値と分散の公式より、 $$\begin{array}{r}E\left[G\left(\mathit{p}\right)\right]\cong G\left(\mathit{\pi }\right)=\log \frac{{\pi }_{12}}{{\pi }_{21}}\end{array}$$ $$\begin{array}{rl}V\left[G\left(\mathit{p}\right)\right]& =\left[\begin{array}{cc}\frac{1}{{\pi }_{12}}& -\frac{1}{{\pi }_{21}}\end{array}\right]\left[\begin{array}{cc}\frac{{\pi }_{12}(1-{\pi }_{12})}{N}& -\frac{{\pi }_{12}{\pi }_{21}}{N}\\ -\frac{{\pi }_{21}{\pi }_{12}}{N}& \frac{{\pi }_{21}(1-{\pi }_{21})}{N}\end{array}\right]\left[\begin{array}{c}\frac{1}{{\pi }_{12}}\\ -\frac{1}{{\pi }_{21}}\end{array}\right]\\ & =\left[\begin{array}{cc}\frac{1}{{\pi }_{12}}& -\frac{1}{{\pi }_{21}}\end{array}\right]\left[\begin{array}{c}\frac{1}{N}\\ -\frac{1}{N}\end{array}\right]\\ & =\frac{1}{N{\pi }_{12}}+\frac{1}{N{\pi }_{21}}\end{array}$$ したがって、漸近的に $$\begin{array}{c}\hat{\theta }\stackrel{d}{\to }\mathrm{N}[\log \frac{{\pi }_{12}}{{\pi }_{21}},\frac{1}{N}(\frac{1}{{\pi }_{12}}+\frac{1}{{\pi }_{21}})]\end{array}$$ 母比率 $\pi$ を一致推定量である標本比率 $p_i$ で置き換えると、漸近分散の一致推定量は、 $$\begin{array}{r}{\hat{\sigma }}_1^2=\frac{1}{N{p}_{12}}+\frac{1}{N{p}_{21}}=\frac{1}{f}+\frac{1}{g}\end{array}$$ $◼$

不一致な応答の総数 $M=f+g$ が固定されているとの仮定のもとで、応答不一致のペア数 $f,g$ は、 $$\begin{array}{c}f\sim \mathrm{B}(M,\frac{{\pi }_{12}}{{\pi }_{12}+{\pi }_{21}})\\ g\sim \mathrm{B}(M,\frac{{\pi }_{21}}{{\pi }_{12}+{\pi }_{21}})\end{array}$$ 二項分布の正規近似により、漸近的に $$\begin{array}{c}f\sim \mathrm{N}[\frac{{\pi }_{12}}{{\pi }_{12}+{\pi }_{21}}M,\frac{{\pi }_{12}{\pi }_{21}}{{({\pi }_{12}+{\pi }_{21})}^2}M]\\ g\sim \mathrm{N}[\frac{{\pi }_{21}}{{\pi }_{12}+{\pi }_{21}}M,\frac{{\pi }_{12}{\pi }_{21}}{{({\pi }_{12}+{\pi }_{21})}^2}M]\end{array}$$ 線形変換の性質より、標本比率は漸近的に $$\begin{array}{c}{p}_{12}\sim \mathrm{N}[\frac{{\pi }_{12}}{{\pi }_{12}+{\pi }_{21}},\frac{{\pi }_{12}{\pi }_{21}}{M{({\pi }_{12}+{\pi }_{21})}^2}]\\ p_{21}\sim N\mathrm{N}[\frac{{\pi }_{21}}{{\pi }_{12}+{\pi }_{21}},\frac{{\pi }_{12}{\pi }_{21}}{M{({\pi }_{12}+{\pi }_{21})}^2}]\end{array}$$ ここで、和の分散の公式より、 $$\begin{array}{r}V(p_{12}+p_{21})=V\left(p_{12}\right)+V\left(p_{21}\right)+2\mathrm{Cov}(p_{12},p_{21})\end{array}$$ 不一致な応答の総数が固定されているとの仮定のもとで $$\begin{array}{r}{p}_{12}+p_{21}=\frac{f}{M}+\frac{g}{M}=1\end{array}$$ したがって、 $$\begin{array}{c}V\left(1\right)=\frac{{\pi }_{12}{\pi }_{21}}{M{({\pi }_{12}+{\pi }_{21})}^2}+\frac{{\pi }_{12}{\pi }_{21}}{M{({\pi }_{12}+{\pi }_{21})}^2}+2\mathrm{Cov}(p_{12},p_{21})\\ 0=\frac{2{\pi }_{12}{\pi }_{21}}{M{({\pi }_{12}+{\pi }_{21})}^2}+2\mathrm{Cov}(p_{12},p_{21})\\ \mathrm{Cov}(p_{12},p_{21})=-\frac{{\pi }_{12}{\pi }_{21}}{M{({\pi }_{12}+{\pi }_{21})}^2}\end{array}$$ ここで、標本比率ベクトルを $$\begin{array}{r}\mathit{p}=\left(\begin{array}{c}{p}_{12}\\ p_{21}\end{array}\right)\end{array}$$ 期待値ベクトルを $$\begin{array}{r}\mathit{\pi }=\left(\begin{array}{c}{\pi }_{12}\\ {\pi }_{21}\end{array}\right)\end{array}$$ 分散・共分散行列を $$\begin{array}{r}\mathbf{\Omega }=\left[\begin{array}{cc}\frac{{\pi }_{12}{\pi }_{21}}{M{({\pi }_{12}+{\pi }_{21})}^2}& -\frac{{\pi }_{12}{\pi }_{21}}{M{({\pi }_{12}+{\pi }_{21})}^2}\\ -\frac{{\pi }_{12}{\pi }_{21}}{M{({\pi }_{12}+{\pi }_{21})}^2}& \frac{{\pi }_{12}{\pi }_{21}}{M{({\pi }_{12}+{\pi }_{21})}^2}\end{array}\right]\end{array}$$ として、 $$\begin{array}{c}G\left(\mathit{\pi }\right)=\theta =\log \frac{{\pi }_{12}}{{\pi }_{21}}\\ G\left(\mathit{p}\right)=\hat{\theta }=\log \frac{p_{12}}{p_{21}}\end{array}$$ と変数変換する。 多変量のデルタ法を用いて $G\left(\mathit{p}\right)$ を期待値 $E\left(\mathit{p}\right)=\mathit{\pi }$ まわりでテイラー展開すると、偏導関数ベクトルは、 $$\begin{array}{r}\mathit{H}\left(\mathit{\pi }\right)=\left(\begin{array}{c}\frac{G\left(\mathit{\pi }\right)}{\partial {\pi }_1}\\ \frac{G\left(\mathit{\pi }\right)}{\partial {\pi }_2}\end{array}\right)=\left[\begin{array}{c}\frac{1}{{\pi }_{12}}\\ -\frac{1}{{\pi }_{21}}\end{array}\right]\end{array}$$ 多変量のデルタ法の期待値と分散の公式より、 $$\begin{array}{r}E\left[G\left(\mathit{p}\right)\right]\cong G\left(\mathit{\pi }\right)=\log \frac{{\pi }_{12}}{{\pi }_{21}}\end{array}$$ $$\begin{array}{rl}V\left[G\left(\mathit{p}\right)\right]& =\left[\begin{array}{cc}\frac{1}{{\pi }_{12}}& -\frac{1}{{\pi }_{21}}\end{array}\right]\left[\begin{array}{cc}\frac{{\pi }_{12}{\pi }_{21}}{M{({\pi }_{12}+{\pi }_{21})}^2}& -\frac{{\pi }_{12}{\pi }_{21}}{M{({\pi }_{12}+{\pi }_{21})}^2}\\ -\frac{{\pi }_{12}{\pi }_{21}}{M{({\pi }_{12}+{\pi }_{21})}^2}& \frac{{\pi }_{12}{\pi }_{21}}{M{({\pi }_{12}+{\pi }_{21})}^2}\end{array}\right]\left[\begin{array}{c}\frac{1}{{\pi }_{12}}\\ -\frac{1}{{\pi }_{21}}\end{array}\right]\\ & =\left[\begin{array}{cc}\frac{1}{{\pi }_{12}}& -\frac{1}{{\pi }_{21}}\end{array}\right]\left[\begin{array}{c}\frac{1}{M({\pi }_{12}+{\pi }_{21})}\\ -\frac{1}{M({\pi }_{12}+{\pi }_{21})}\end{array}\right]\\ & =\frac{1}{M{\pi }_{12}({\pi }_{12}+{\pi }_{21})}+\frac{1}{M{\pi }_{21}({\pi }_{12}+{\pi }_{21})}\\ & =\frac{1}{M({\pi }_{12}+{\pi }_{21})}(\frac{1}{{\pi }_{12}}+\frac{1}{{\pi }_{21}})\end{array}$$ したがって、漸近的に $$\begin{array}{c}\hat{\theta }\stackrel{d}{\to }\mathrm{N}[\log \frac{{\pi }_{12}}{{\pi }_{21}},\frac{1}{M({\pi }_{12}+{\pi }_{21})}(\frac{1}{{\pi }_{12}}+\frac{1}{{\pi }_{21}})]\end{array}$$母比率 ${\pi }_i$ を一致推定量である標本比率 $p_i$ で置き換えると、漸近分散の一致推定量は、 $$\begin{array}{rl}\hat{V}\left[\hat{\theta }\right]& =\frac{1}{M(p_{12}+p_{21})}(\frac{1}{p_{12}}+\frac{1}{p_{21}})\\ & =\frac{1}{M{p}_{12}}+\frac{1}{M{p}_{21}}\\ & =\frac{1}{f}+\frac{1}{g}\end{array}$$ $◼$