標本対数人口寄与危険割合オッズの漸近分布

二項分布の正規近似により、標本比率は漸近的に $$\begin{array}{c}{p}_1\sim \mathrm{N}[{\pi }_1,\frac{{\pi }_1(1-{\pi }_1)}{n_1}]\\ p_0\sim \mathrm{N}[{\pi }_0,\frac{{\pi }_0(1-{\pi }_0)}{n_0}]\end{array}$$ 標本比率ベクトルを $$\begin{array}{r}\mathit{p}=\left(\begin{array}{c}{p}_1\\ p_0\end{array}\right)\end{array}$$ 期待値ベクトルを $$\begin{array}{r}\mathit{\pi }=\left(\begin{array}{c}{\pi }_1\\ {\pi }_0\end{array}\right)\end{array}$$ 分散・共分散行列を $$\begin{array}{r}\mathbf{\Omega }=\left[\begin{array}{cc}\frac{{\pi }_1(1-{\pi }_1)}{n_1}& 0\\ 0& \frac{{\pi }_0(1-{\pi }_0)}{n_0}\end{array}\right]\end{array}$$ として、 $$\begin{array}{c}G\left(\mathit{\pi }\right)=\theta =\frac{{\alpha }_1({\pi }_1-{\pi }_0)}{{\pi }_0}\\ G\left(\mathit{p}\right)=\hat{\theta }=\frac{{\alpha }_1(p_1-p_0)}{p_0}\end{array}$$ 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、 $$\begin{array}{r}\mathit{H}\left(\mathit{\pi }\right)=\left(\begin{array}{c}\frac{G\left(\mathit{\pi }\right)}{\partial {\pi }_1}\\ \frac{G\left(\mathit{\pi }\right)}{\partial {\pi }_2}\end{array}\right)=\left[\begin{array}{c}\frac{1}{{\pi }_1-{\pi }_0}\\ -\frac{{\pi }_1}{{\pi }_0({\pi }_1-{\pi }_0)}\end{array}\right]\end{array}$$ 多変量のデルタ法の期待値と分散の公式より、 $$\begin{array}{r}E\left[G\left(\mathit{p}\right)\right]\cong G\left(\mathit{\pi }\right)=\frac{{\alpha }_1({\pi }_1-{\pi }_0)}{{\pi }_0}\end{array}$$ $$\begin{array}{rl}V\left[G\left(\mathit{p}\right)\right]& =\left[\begin{array}{cc}\frac{1}{{\pi }_1-{\pi }_0}& -\frac{{\pi }_1}{{\pi }_0({\pi }_1-{\pi }_0)}\end{array}\right]\left[\begin{array}{cc}\frac{{\pi }_1(1-{\pi }_1)}{n_1}& 0\\ 0& \frac{{\pi }_0(1-{\pi }_0)}{n_0}\end{array}\right]\left[\begin{array}{c}\frac{1}{{\pi }_1-{\pi }_0}\\ -\frac{{\pi }_1}{{\pi }_0({\pi }_1-{\pi }_0)}\end{array}\right]\\ & =\frac{1}{{({\pi }_1-{\pi }_0)}^2}\cdot \frac{{\pi }_1(1-{\pi }_1)}{n_1}+\frac{{\pi }_1^2}{{\pi }_0^2{({\pi }_1-{\pi }_0)}^2}\cdot \frac{{\pi }_0(1-{\pi }_0)}{n_0}\\ & =\frac{{\pi }_1(1-{\pi }_1)}{n_1{({\pi }_1-{\pi }_0)}^2}+\frac{{\pi }_1^2(1-{\pi }_0)}{n_0{\pi }_0{({\pi }_1-{\pi }_0)}^2}\\ & =\frac{{\pi }_1}{{({\pi }_1-{\pi }_0)}^2}\{\frac{1-{\pi }_1}{n_1}+\frac{{\pi }_1(1-{\pi }_0)}{n_0{\pi }_0}\}\\ & =\frac{{\pi }_1}{{({\pi }_1-{\pi }_0)}^2}\left\{\frac{n_0{\pi }_0(1-{\pi }_1)+n_1{\pi }_1(1-{\pi }_0)}{n_1{n}_0{\pi }_0}\right\}\end{array}$$ 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、 $$\begin{array}{r}\hat{V}\left[\hat{\theta }\right]=\frac{p_1}{{(p_1-p_0)}^2}\left\{\frac{n_0{p}_0(1-p_1)+n_1{p}_0(1-p_0)}{n_1{n}_0{p}_0}\right\}\end{array}$$ $◼$