第５章統計学

\begin{eqnarray} {}_{T_{1\cdot}}\mathrm{C}_{y_{11}}\times{}_{T_{2\cdot}}\mathrm{C}_{y_{21}}/{}_{T}\mathrm{C}_{T_{\cdot1}} &=& \frac{T_{1\cdot}!}{(T_{1\cdot}-y_{11})!y_{11}!}\times\frac{T_{2\cdot} !}{(T_{2\cdot}-y_{21})!y_{21}!}/\left(\frac{T!}{(T-T_{\cdot1})!T_{\cdot1}!}\right) \\ \\ &=& \frac{T_{1\cdot}!T_{2\cdot}!T_{\cdot1}!}{T!y_{11}!y_{21}!}\frac{(T-T_{\cdot1})!}{(T_{1\cdot}-y_{11})!(T_{2\cdot}-y_{21})!} \\ \\ &=& \frac{T_{1\cdot}!T_{2\cdot}!T_{\cdot1}!}{T!y_{11}!y_{21}!}\frac{T_{\cdot2}!}{y_{12}!y_{22}!}&...&T=T_{1\cdot}+T_{2\cdot},T_{1\cdot}=y_{11}+y_{12},T_{2\cdot}=y_{21}+y_{22}\text{を用いた} \\ \\ \end{eqnarray} と導出できる。

\begin{eqnarray} \chi^2 &=& \displaystyle\sum_{i=1}^2\sum_{j=1}^2\frac{(x_{ij}-t_{ij})^2}{t_{ij}} \\ \\ &=& \frac{(x_{11}-t_{11})^2}{t_{11}}+\frac{(x_{12}-t_{12})^2}{t_{12}}+\frac{(x_{21}-t_{21})^2}{t_{21}}+\frac{(x_{22}-t_{22})^2}{t_{22}} \\ \\ &=& (x_{11}-t_{11})^2\frac{T}{T_{1\cdot}T_{\cdot 1}}+(x_{12}-t_{12})^2\frac{T}{T_{1\cdot}T_{\cdot 2}}+(x_{21}-t_{21})^2\frac{T}{T_{2\cdot}T_{\cdot 1}}+(x_{22}-t_{22})^2\frac{T}{T_{2\cdot}T_{\cdot 2}}&...&\text{p.134下より} \\ \\ &=& \frac{T}{T_{1\cdot}T_{\cdot 1}T_{2\cdot}T_{\cdot 2}}\left(\underbrace{(x_{11}-t_{11})^2T_{2\cdot}T_{\cdot 2}+(x_{12}-t_{12})^2T_{2\cdot}T_{\cdot 1}+(x_{21}-t_{21})^2T_{1\cdot}T_{\cdot 2}+(x_{22}-t_{22})^2T_{1\cdot}T_{\cdot 1}}_{(1)}\right) \\ \\ (1) &=& (x_{11}-t_{11})^2T_{2\cdot}T_{\cdot 2}+(x_{12}-t_{12})^2T_{2\cdot}T_{\cdot 1}+(x_{21}-t_{21})^2T_{1\cdot}T_{\cdot 2}+(x_{22}-t_{22})^2T_{1\cdot}T_{\cdot 1} \\ \\ &=& \left(x_{11}-\frac{T_{1\cdot}T_{\cdot1} }{T}\right)^2T_{2\cdot}T_{\cdot 2}+\left(x_{12}-\frac{T_{1\cdot}T_{\cdot 2} }{T}\right)^2T_{2\cdot}T_{\cdot 1}+\left(x_{21}-\frac{T_{2\cdot}T_{\cdot 1} }{T}\right)^2T_{1\cdot}T_{\cdot 2}+\left(x_{22}-\frac{T_{2\cdot}T_{\cdot 2} }{T}\right)^2T_{1\cdot}T_{\cdot 1} \\ \\ &=& \left(x_{11}^2-2x_{11}\frac{T_{1\cdot}T_{\cdot1} }{T}+\frac{T_{1\cdot}^2T_{\cdot1}^2 }{T^2}\right)T_{2\cdot}T_{\cdot 2}+\left(x_{12}^2-2x_{12}\frac{T_{1\cdot}T_{\cdot 2} }{T}+\frac{T_{1\cdot}^2T_{\cdot 2}^2 }{T^2}\right)T_{2\cdot}T_{\cdot 1}+\left(x_{21}^2-2x_{21}\frac{T_{2\cdot}T_{\cdot 1} }{T}+\frac{T_{2\cdot}^2T_{\cdot 1}^2 }{T^2}\right)T_{1\cdot}T_{\cdot 2}+\left(x_{22}^2-2x_{22}\frac{T_{2\cdot}T_{\cdot 2} }{T}+\frac{T_{2\cdot}^2T_{\cdot 2}^2 }{T^2}\right)T_{1\cdot}T_{\cdot 1} \\ \\ &=& \left[x_{11}^2T_{2\cdot}T_{\cdot 2}+x_{12}^2T_{2\cdot}T_{\cdot 1}+x_{21}^2T_{1\cdot}T_{\cdot 2}+x_{22}^2T_{1\cdot}T_{\cdot 1}\right]-2\frac{T_{1\cdot}T_{\cdot1}T_{2\cdot}T_{\cdot2}}{T}\left(x_{11}+x_{12}+x_{21}+x_{22}\right)+\frac{T_{1\cdot}T_{\cdot1}T_{2\cdot}T_{\cdot2}}{T^2}\left(T_{1\cdot}T_{\cdot1}+T_{1\cdot}T_{\cdot2}+T_{2\cdot}T_{\cdot1}+T_{2\cdot}T_{\cdot2}\right) \\ \\ &=& \left[x_{11}x_{11}T_{2\cdot}T_{\cdot 2}+x_{12}x_{12}T_{2\cdot}T_{\cdot 1}+x_{21}x_{21}T_{1\cdot}T_{\cdot 2}+x_{22}x_{22}T_{1\cdot}T_{\cdot 1}\right]-2\frac{T_{1\cdot}T_{\cdot1}T_{2\cdot}T_{\cdot2}}{T}\cdot T+\frac{T_{1\cdot}T_{\cdot1}T_{2\cdot}T_{\cdot2}}{T^2}(T_{\cdot1}+T_{\cdot2})(T_{1\cdot}+T_{2\cdot})&...&x_{11}+x_{12}+x_{21}+x_{22}=T\text{となるため} \\ \\ &=& \left[(T_{1\cdot}-x_{12})(T_{\cdot 1}-x_{21})T_{2\cdot}T_{\cdot 2}+x_{12}x_{12}T_{2\cdot}T_{\cdot 1}+x_{21}x_{21}T_{1\cdot}T_{\cdot 2}+x_{22}x_{22}T_{1\cdot}T_{\cdot 1}\right]-2T_{1\cdot}T_{\cdot1}T_{2\cdot}T_{\cdot2}+\frac{T_{1\cdot}T_{\cdot1}T_{2\cdot}T_{\cdot2}}{T^2}T\cdot T&...&T_{1\cdot}+T_{2\cdot}=T,T_{\cdot1}+T_{\cdot2}=T\text{より} \\ \\ &=& \left[(x_{12}x_{21}-x_{12}T_{\cdot 1}-x_{21}T_{1\cdot})T_{2\cdot}T_{\cdot 2}+T_{1\cdot}T_{\cdot 1}T_{2\cdot}T_{\cdot 2}+x_{12}x_{12}T_{2\cdot}T_{\cdot 1}+x_{21}x_{21}T_{1\cdot}T_{\cdot 2}+x_{22}x_{22}T_{1\cdot}T_{\cdot 1}\right]-2T_{1\cdot}T_{\cdot1}T_{2\cdot}T_{\cdot2}+T_{1\cdot}T_{\cdot1}T_{2\cdot}T_{\cdot2}\\ \\ &=& \left[(x_{12}x_{21}-x_{12}T_{\cdot 1}-x_{21}T_{1\cdot})T_{2\cdot}T_{\cdot 2}+x_{12}x_{12}T_{2\cdot}T_{\cdot 1}+x_{21}x_{21}T_{1\cdot}T_{\cdot 2}+x_{22}x_{22}T_{1\cdot}T_{\cdot 1}\right]+T_{1\cdot}T_{\cdot1}T_{2\cdot}T_{\cdot2}-T_{1\cdot}T_{\cdot1}T_{2\cdot}T_{\cdot2}\\ \\ &=& \left[x_{12}x_{21}T_{2\cdot}T_{\cdot 2}+(-x_{12}T_{\cdot 2}+x_{12}x_{12})T_{2\cdot}T_{\cdot 1}+(-x_{21}T_{2\cdot}+x_{21}x_{21})T_{1\cdot}T_{\cdot 2}+x_{22}x_{22}T_{1\cdot}T_{\cdot 1}\right]\\ \\ &=& \left[x_{12}x_{21}(x_{21}+x_{22})(x_{12}+x_{22})+(-x_{12}(x_{1 2}+x_{2 2})+x_{12}x_{12})(x_{21}+x_{22})(x_{11}+x_{21})+(-x_{21}(x_{21}+x_{22})+x_{21}x_{21})(x_{11}+x_{12})(x_{12}+x_{22})+x_{22}x_{22}(x_{11}+x_{12})(x_{11}+x_{21})\right]&...&x_{1j}+x_{2j}=T_{\cdot j},x_{j1}+x_{j2}=T_{j\cdot}\text{より}\\ \\ &=& \left[x_{12}x_{21}(x_{21}+x_{22})(x_{12}+x_{22})-x_{12}x_{2 2}(x_{21}+x_{22})(x_{11}+x_{21})-x_{21}x_{22}(x_{11}+x_{12})(x_{12}+x_{22})+x_{22}x_{22}(x_{11}+x_{12})(x_{11}+x_{21})\right]\\ \\ &=& \left[x_{12}(x_{21}+x_{22})(x_{21}(x_{12}+x_{22})-x_{2 2}(x_{11}+x_{21}))+x_{22}(x_{11}+x_{12})(x_{22}(x_{11}+x_{21})-x_{21}(x_{12}+x_{22}))\right]\\ \\ &=& \left[x_{12}(x_{21}+x_{22})(x_{21}x_{12}-x_{2 2}x_{11})+x_{22}(x_{11}+x_{12})(x_{22}x_{11}-x_{21}x_{12})\right]\\ \\ &=& \left[-x_{12}(x_{21}+x_{22})(x_{2 2}x_{11}-x_{21}x_{12})+x_{22}(x_{11}+x_{12})(x_{22}x_{11}-x_{21}x_{12})\right]\\ \\ &=& (x_{2 2}x_{11}-x_{21}x_{12})\left[-x_{12}(x_{21}+x_{22})+x_{22}(x_{11}+x_{12})\right]\\ \\ &=& (x_{2 2}x_{11}-x_{21}x_{12})\left[-x_{12}x_{21}+x_{22}x_{11}\right]\\ \\ &=& (x_{2 2}x_{11}-x_{21}x_{12})^2 \\ \\ \therefore \chi^2&=&\frac{T}{T_{1\cdot}T_{\cdot 1}T_{2\cdot}T_{\cdot 2}}(x_{2 2}x_{11}-x_{21}x_{12})^2 \end{eqnarray} と導出できる。

\begin{eqnarray} \frac{(x_{12}-t_{12})^2}{t_{12}}+\frac{(x_{21}-t_{21})^2}{t_{21}} &=& \frac{(x_{12}-\frac{x_{12}+x_{21}}{2})^2}{\frac{x_{12}+x_{21}}{2}}+\frac{(x_{21}-\frac{x_{12}+x_{21}}{2})^2}{\frac{x_{12}+x_{21}}{2}} \\ \\ &=& \frac{2(\frac{x_{12}-x_{21}}{2})^2}{x_{12}+x_{21}}+\frac{2(\frac{-x_{12}+x_{21}}{2})^2}{x_{12}+x_{21}} \\ \\ &=& \frac{4(\frac{x_{12}-x_{21}}{2})^2}{x_{12}+x_{21}} \\ \\ &=& \frac{(x_{12}-x_{21})^2}{x_{12}+x_{21}} \end{eqnarray} と導出できる。

統計学の行間埋め 第５章

統計学の行間埋め　第５章