標本リスク比・標本オッズ比の漸近分散

二項分布の正規近似により、標本比率は漸近的に $$\begin{array}{c}{p}_1\sim N[{\pi }_1,\frac{{\pi }_1(1-{\pi }_1)}{n_1}]\\ p_0\sim N[{\pi }_0,\frac{{\pi }_0(1-{\pi }_0)}{n_0}]\end{array}$$ 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、 $$\begin{array}{c}G\left(\mathit{\pi }\right)=\mathrm{RR}=\frac{{\pi }_1}{{\pi }_0}\\ G\left(\mathit{p}\right)=\widehat{\mathrm{RR}}=\frac{p_1}{p_0}\end{array}$$ 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、 $$\begin{array}{r}\mathit{H}\left(\mathit{\pi }\right)=\left(\begin{array}{c}\frac{G\left(\mathit{\pi }\right)}{\partial {\pi }_1}\\ \frac{G\left(\mathit{\pi }\right)}{\partial {\pi }_0}\end{array}\right)=\left[\begin{array}{c}\frac{1}{{\pi }_0}\\ -\frac{{\pi }_1}{{\pi }_0^2}\end{array}\right]\end{array}$$ 多変量のデルタ法の期待値と分散の公式より、 $$\begin{array}{r}E\left[G\left(\mathit{p}\right)\right]\cong G\left(\mathit{\pi }\right)=\frac{{\pi }_1}{{\pi }_0}\end{array}$$ $$\begin{array}{rl}V\left[G\left(\mathit{p}\right)\right]& =\left[\begin{array}{cc}\frac{1}{{\pi }_0}& -\frac{{\pi }_1}{{{\pi }_0}^2}\end{array}\right]\left[\begin{array}{cc}\frac{{\pi }_1(1-{\pi }_1)}{n_1}& 0\\ 0& \frac{{\pi }_0(1-{\pi }_0)}{n_0}\end{array}\right]\left[\begin{array}{c}\frac{1}{{\pi }_0}\\ -\frac{{\pi }_1}{{{\pi }_0}^2}\end{array}\right]\\ & =\left[\begin{array}{cc}\frac{1}{{\pi }_0}& -\frac{{\pi }_1}{{{\pi }_0}^2}\end{array}\right]\left[\begin{array}{c}\frac{{\pi }_1(1-{\pi }_1)}{n_1{\pi }_0}\\ -\frac{{\pi }_1(1-{\pi }_0)}{n_0{\pi }_0}\end{array}\right]\\ & =\frac{1}{{\pi }_0}\frac{{\pi }_1(1-{\pi }_1)}{n_1{\pi }_0}+\frac{{\pi }_1}{{{\pi }_0}^2}\frac{{\pi }_1(1-{\pi }_0)}{n_0{\pi }_0}\\ & =\frac{{\pi }_1(1-{\pi }_1)}{n_1{\pi }_0^2}+\frac{{\pi }_1^2(1-{\pi }_0)}{n_2{\pi }_0^3}\end{array}$$ いっぽう、 $$\begin{array}{rl}{\mathrm{RR}}^{2}V(\log \widehat{\mathrm{RR}})& =\frac{{\pi }_1^2}{{\pi }_0^2}(\frac{1-{\pi }_1}{n_1{\pi }_1}+\frac{1-{\pi }_0}{n_0{\pi }_0})\\ & =\frac{{\pi }_1(1-{\pi }_1)}{n_1{\pi }_0^2}+\frac{{\pi }_1^2(1-{\pi }_0)}{n_0{\pi }_0^3}\end{array}$$ したがって、漸近的に $$\begin{array}{r}V\left(\widehat{\mathrm{RR}}\right)\cong {\mathrm{RR}}^{2}V(\log \widehat{\mathrm{RR}})\end{array}$$ $◼$

二項分布の正規近似により、標本比率は漸近的に $$\begin{array}{c}{p}_1\sim N[{\pi }_1,\frac{{\pi }_1(1-{\pi }_1)}{n_1}]\\ p_0\sim N[{\pi }_0,\frac{{\pi }_0(1-{\pi }_0)}{n_0}]\end{array}$$ 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、 $$\begin{array}{c}G\left(\mathit{\pi }\right)=\mathrm{OR}=\frac{{\pi }_1}{1-{\pi }_1}\cdot \frac{1-{\pi }_0}{{\pi }_0}\\ G\left(\mathit{p}\right)=\widehat{\mathrm{OR}}=\frac{p_1}{1-p_1}\cdot \frac{1-p_0}{p_0}\end{array}$$ 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、 $$\begin{array}{r}\mathit{H}\left(\mathit{\pi }\right)=\left(\begin{array}{c}\frac{G\left(\mathit{\pi }\right)}{\partial {\pi }_1}\\ \frac{G\left(\mathit{\pi }\right)}{\partial {\pi }_0}\end{array}\right)=\left[\begin{array}{c}\frac{1-{\pi }_0}{{\pi }_0{(1-{\pi }_1)}^2}\\ -\frac{{\pi }_1}{{\pi }_0^2(1-{\pi }_1)}\end{array}\right]\end{array}$$ 多変量のデルタ法の期待値と分散の公式より、 $$\begin{array}{r}E\left[G\left(\mathit{p}\right)\right]\cong G\left(\mathit{\pi }\right)=\frac{{\pi }_1}{1-{\pi }_1}\cdot \frac{1-{\pi }_0}{{\pi }_0}\end{array}$$ $$\begin{array}{rl}V\left[G\left(\mathit{p}\right)\right]& =\left[\begin{array}{cc}\frac{1-{\pi }_0}{{\pi }_0{(1-{\pi }_1)}^2}& -\frac{{\pi }_1}{{\pi }_0^2(1-{\pi }_1)}\end{array}\right]\left[\begin{array}{cc}\frac{{\pi }_1(1-{\pi }_1)}{n_1}& 0\\ 0& \frac{{\pi }_0(1-{\pi }_0)}{n_0}\end{array}\right]\left[\begin{array}{c}\frac{1-{\pi }_0}{{\pi }_0{(1-{\pi }_1)}^2}\\ -\frac{{\pi }_1}{{\pi }_0^2(1-{\pi }_1)}\end{array}\right]\\ & =\left[\begin{array}{cc}\frac{1-{\pi }_0}{{\pi }_0{(1-{\pi }_1)}^2}& -\frac{{\pi }_1}{{\pi }_0^2(1-{\pi }_1)}\end{array}\right]\left[\begin{array}{c}\frac{{\pi }_1(1-{\pi }_0)}{n_1{\pi }_0(1-{\pi }_1)}\\ -\frac{{\pi }_1(1-{\pi }_0)}{n_0{\pi }_0(1-{\pi }_1)}\end{array}\right]\\ & =\frac{1-{\pi }_0}{{\pi }_0{(1-{\pi }_1)}^2}\frac{{\pi }_1(1-{\pi }_0)}{n_1{\pi }_0(1-{\pi }_1)}+\frac{{\pi }_1}{{\pi }_0^2(1-{\pi }_1)}\frac{{\pi }_1(1-{\pi }_0)}{n_0{\pi }_0(1-{\pi }_1)}\\ & =\frac{{\pi }_1{(1-{\pi }_0)}^2}{n_1{\pi }_0^2{(1-{\pi }_1)}^3}+\frac{{\pi }_1^2(1-{\pi }_0)}{n_0{\pi }_0^3{(1-{\pi }_1)}^2}\end{array}$$ いっぽう、 $$\begin{array}{rl}{\mathrm{OR}}^{2}V(\log \widehat{\mathrm{OR}})& =\frac{{\pi }_1^2}{{(1-{\pi }_1)}^2}\cdot \frac{{(1-{\pi }_0)}^2}{{\pi }_0^2}[\frac{1}{n_1{\pi }_1(1-{\pi }_1)}+\frac{1}{n_0{\pi }_0(1-{\pi }_0)}]\\ & =\frac{{\pi }_1{(1-{\pi }_0)}^2}{n_1{\pi }_0^2{(1-{\pi }_1)}^3}+\frac{{\pi }_1^2(1-{\pi }_0)}{n_0{\pi }_0^3{(1-{\pi }_1)}^2}\end{array}$$ したがって、漸近的に $$\begin{array}{r}V\left(\widehat{\mathrm{OR}}\right)\cong {\mathrm{OR}}^{2}V(\log \widehat{\mathrm{OR}})\end{array}$$ $◼$