条件付き曝露オッズ比による条件付き発症リスク比の近似

共変量の値が $z$ のときのリスク比は、定義式より、 $$\begin{array}{rl}{\mathrm{RR}}_{\mathrm{z}}& =\frac{P\left(D\right|E,z)}{P\left(D\right|\overline{E},z)}\\ & =\frac{P(D\cap E|z)}{P\left(E\right|z)}÷\frac{P(D\cap \overline{E}|z)}{P\left(\overline{E}\right|z)}\\ & =\frac{P(D\cap E|z)\cdot P\left(\overline{E}\right|z)}{P(D\cap \overline{E}|z)\cdot P\left(E\right|z)}\\ & =\frac{P\left(E\right|D,z)\cdot [P\left(\overline{E}\right|D,z)\cdot P\left(D\right|z)+P\left(\overline{E}\right|\overline{D},z)\cdot P\left(\overline{D}\right|z)]}{P\left(\overline{E}\right|D,z)\cdot [P\left(E\right|D,z)\cdot P\left(D\right|z)+P\left(E\right|\overline{D},z)\cdot P\left(\overline{D}\right|z)]}\\ & =\frac{P\left(E\right|D,z)\cdot [P\left(\overline{E}\right|D,z)\cdot P\left(D\right|z)+P\left(\overline{E}\right|\overline{D},z)\{1-P\left(D\right|z)\}]}{P\left(\overline{E}\right|D,z)\cdot [P\left(E\right|D,z)\cdot P\left(D\right|z)+P\left(E\right|\overline{D},z)\{1-P\left(D\right|z)\}]}\\ & =\frac{P\left(E\right|D,z)\cdot P\left(\overline{E}\right|D,z)\cdot P\left(D\right|z)+P\left(E\right|D,z)\cdot P\left(\overline{E}\right|\overline{D},z)\{1-P\left(D\right|z)\}}{P\left(\overline{E}\right|D,z)\cdot P\left(E\right|D,z)\cdot P\left(D\right|z)+P\left(\overline{E}\right|D,z)\cdot P\left(E\right|\overline{D},z)\{1-P\left(D\right|z)\}}\end{array}$$ 共変量の値が $z$ のときの全体の発症確率を $P\left(D\right|z)={\delta }_z$ とおくと、 $$\begin{array}{rl}{\mathrm{RR}}_{\mathrm{z}}& =\frac{{\varphi }_{DE}\cdot {\varphi }_{D\overline{E}}\cdot {\delta }_z+{\varphi }_{DE}\cdot {\varphi }_{\overline{D}\overline{E}}(1-{\delta }_z)}{{\varphi }_{DE}\cdot {\varphi }_{D\overline{E}}\cdot {\delta }_z+{\varphi }_{\overline{D}E}\cdot {\varphi }_{D\overline{E}}(1-{\delta }_z)}\\ & =\frac{{\varphi }_{DE}\cdot {\varphi }_{D\overline{E}}\cdot {\delta }_z+{\varphi }_{12}(1-{\delta }_z)}{{\varphi }_{DE}\cdot {\varphi }_{D\overline{E}}\cdot {\delta }_z+{\varphi }_{21}(1-{\delta }_z)}\end{array}$$

希少疾患 ${\delta }_z\to 0$ の仮定のもとで、 $$\begin{array}{rl}\underset{{\delta }_z\to 0}{lim}{\mathrm{RR}}_{\mathrm{z}}& =\frac{{\varphi }_{DE}\cdot {\varphi }_{D\overline{E}}\cdot 0+{\varphi }_{12}(1-0)}{{\varphi }_{DE}\cdot {\varphi }_{D\overline{E}}\cdot 0+{\varphi }_{21}(1-0)}\\ & =\frac{{\varphi }_{12}}{{\varphi }_{21}}\\ & ={\mathrm{OR}}_{\mathrm{CRetro}}\end{array}$$ $◼$