生存時間分布のパラメトリックモデル③：加速死亡時間モデル

累積ハザード関数とハザード関数の関係 $\lambda \left(t\right)=\frac{d}{dt}\mathrm{\Lambda }\left(t\right)$ から、合成関数の微分法より、 $$\begin{array}{rl}\lambda \left(t\right|\mathit{x})& =\frac{d}{dt}{\mathrm{\Lambda }}_0\{\exp (-\frac{\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }}}{\sigma })\cdot t\}\\ & ={\lambda }_0\{\exp (-\frac{\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }}}{\sigma })\cdot t\}\cdot \frac{d}{dt}\{\exp (-\frac{\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }}}{\sigma })\cdot t\}\\ & =\exp (-\frac{\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }}}{\sigma })\cdot {\lambda }_0\{\exp (-\frac{\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }}}{\sigma })\cdot t\}\end{array}$$ イベント発生の確率密度関数、ハザード関数、累積生存関数の関係より、 $$\begin{array}{rl}f\left(t\right|\mathit{x})& =\lambda \left(t\right|\mathit{x})\cdot S\left(t\right|\mathit{x})\\ & =\exp (-\frac{\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }}}{\sigma })\cdot {\lambda }_0\{\exp (-\frac{\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }}}{\sigma })\cdot t\}\cdot S_0\{\exp (-\frac{\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }}}{\sigma })\cdot t\}\end{array}$$ $◼$

累積ハザード関数の式に $t=\exp \left(\frac{y}{\sigma }\right)$ を代入すると、 $$\begin{array}{rl}\mathrm{\Lambda }\left(y\right|\mathit{x})& ={\mathrm{\Lambda }}_0\{\exp (-\frac{\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }}}{\sigma })\cdot \exp \left(\frac{y}{\sigma }\right)\}\\ & ={\mathrm{\Lambda }}_0\left[\exp \left\{\frac{y-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})}{\sigma }\right\}\right]\end{array}$$ 累積ハザード関数とハザード関数の関係 $\lambda \left(t\right)=\frac{d}{dt}\mathrm{\Lambda }\left(t\right)$ から、合成関数の微分法より、 $$\begin{array}{rl}\lambda \left(y\right|\mathit{x})& =\frac{d}{dy}{\mathrm{\Lambda }}_0\left[\exp \left\{\frac{y-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})}{\sigma }\right\}\right]\\ & ={\lambda }_0\left[\exp \left\{\frac{y-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})}{\sigma }\right\}\right]\cdot \frac{d}{dy}\left[\exp \left\{\frac{y-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})}{\sigma }\right\}\right]\cdot \frac{d}{dy}\left\{\frac{y-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})}{\sigma }\right\}\\ & =\left[\exp \left\{\frac{y-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})}{\sigma }\right\}\right]\cdot \frac{1}{\sigma }\cdot {\lambda }_0\left[\exp \left\{\frac{y-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})}{\sigma }\right\}\right]\end{array}$$ イベント発生の確率密度関数、ハザード関数、累積生存関数の関係より、 $$\begin{array}{rl}f\left(y\right|\mathit{x})& =\lambda \left(y\right|\mathit{x})\cdot S\left(y\right|\mathit{x})\\ & =\left[\exp \left\{\frac{y-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})}{\sigma }\right\}\right]\cdot \frac{1}{\sigma }\cdot {\lambda }_0\left[\exp \left\{\frac{y-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})}{\sigma }\right\}\right]\cdot S_0\left[\exp \left\{\frac{y-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})}{\sigma }\right\}\right]\end{array}$$ ここで、 $$\begin{array}{r}\epsilon =\frac{y-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})}{\sigma }\end{array}$$ とおくと、 $$\begin{array}{r}f\left(\epsilon \right)=e^{\epsilon }\cdot {\lambda }_0\left(\epsilon \right)\cdot S_0\left(\epsilon \right)\end{array}$$ $◼$

生存関数の定義より、 $$\begin{array}{rl}S\left(t\right|{\mathit{x}}_{\mathit{i}})& =P(t<t_i)\\ & =P(\log t<\log t_i)\\ & =P[\log \left\{\exp \left(\frac{y}{\sigma }\right)\right\}<\log \left\{\exp \left(\frac{y_i}{\sigma }\right)\right\}]\\ & =P(\frac{y}{\sigma }<\frac{y_i}{\sigma })\\ & =P(\frac{y}{\sigma }-\frac{\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }}}{\sigma }<\frac{y_i}{\sigma }-\frac{\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }}}{\sigma })\\ & =P[\frac{y-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})}{\sigma }<\frac{y_i-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})}{\sigma }]\\ & =P(\epsilon <{\epsilon }_i)\end{array}$$ $◼$

標準ワイブルモデルのハザード関数の式 $\lambda \left(t\right)={\gamma t}^{\gamma -1}$ に加速度因子 $\exp \{-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})\gamma \}$ をかけると、 $$\begin{array}{r}\lambda \left(t\right|\mathit{x})=\exp \{-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})\gamma \}\cdot {\gamma t}^{\gamma -1}\end{array}$$ 累積ハザード関数 $\mathrm{\Lambda }\left(t\right)={\int }_0^t\lambda \left(u\right)du$ は、 $$\begin{array}{rl}\mathrm{\Lambda }\left(t\right)& =\exp \{-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})\gamma \}{\int }_0^t\gamma u^{\gamma -1}du\\ & =\exp \{-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})\gamma \}\cdot {\left[u^{\gamma }\right]}_0^t\\ & =\exp \{-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})\gamma \}\cdot t^{\gamma }\\ & =\exp \{-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})\gamma \}\cdot \exp (\gamma \log t)\\ & =\exp \left[\{\log t-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})\}\gamma \right]\end{array}$$ 累積ハザード関数と累積生存関数の関係 $S\left(t\right)=\exp \{-\mathrm{\Lambda }\left(t\right)\}$ より、 $$\begin{array}{rl}S\left(t\right|\mathit{x})& =\exp [-\exp \left[\{\log t-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})\}\gamma \right]]\\ & =\exp [-\exp {\{\log t-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})\}}^{\gamma }]\end{array}$$ $\gamma =\frac{1}{\sigma }$ を代入すると、 $$\begin{array}{r}S\left(t\right|\mathit{x})=\exp [-\exp \left\{\frac{\log t-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})}{\sigma }\right\}]\end{array}$$ $◼$

イベント発生の確率密度関数、ハザード関数、累積生存関数の関係より、 $$\begin{array}{rl}f\left(t\right|\mathit{x})& =\lambda \left(t\right|\mathit{x})\cdot S\left(t\right|\mathit{x})\\ & =\exp \{-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})\gamma \}\cdot {\gamma t}^{\gamma -1}\cdot \exp [-\exp \left\{\frac{\log t-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})}{\sigma }\right\}]\\ & =\frac{1}{\sigma }\cdot \exp \{-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})\gamma \}\cdot \exp \{(\gamma -1)\log t\}\cdot \exp [-\exp \left\{\frac{\log t-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})}{\sigma }\right\}]\\ & =\frac{1}{\sigma }\cdot \exp [-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})\gamma +(\gamma -1)\log t-\exp \left\{\frac{\log t-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})}{\sigma }\right\}]\\ & =\frac{1}{\sigma }\cdot \exp [\frac{\log t-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})}{\sigma }-\exp \left\{\frac{\log t-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})}{\sigma }\right\}]\cdot \exp (-\log t)\end{array}$$ 対数変換の公式 $g\left(y\right)=f\left(e^y\right)\cdot e^y$ より、$y=\log t$ の確率密度関数は、 $$\begin{array}{rl}f\left(y\right|\mathit{x})& =\frac{1}{\sigma }\cdot \exp [\frac{y-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})}{\sigma }-\exp \left\{\frac{\log t-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})}{\sigma }\right\}]\cdot \exp (-y)\cdot \exp \left(y\right)\\ & =\frac{1}{\sigma }\cdot \exp [\frac{y-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})}{\sigma }-\exp \left\{\frac{\log t-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})}{\sigma }\right\}]\end{array}$$ $◼$

線形変換の公式 $Y=aX+b\iff g\left(y\right)=f\left(\frac{y-b}{a}\right)\cdot \frac{1}{\left|a\right|}$ より、 $$\begin{array}{rl}f\left(\epsilon \right)& =f\left(\epsilon \right)\cdot \frac{1}{\left|\frac{1}{\sigma }\right|}\\ & =\frac{1}{\sigma }\cdot \exp [\epsilon -\exp \left(\epsilon \right)]\cdot \frac{1}{\left|\frac{1}{\sigma }\right|}\\ & =\exp [\epsilon -\exp \left(\epsilon \right)]\end{array}$$ $◼$

指数変換の公式 $Y=e^X\iff g\left(y\right)=f(\log y)\cdot \frac{1}{y}$ より、 $$\begin{array}{rl}f\left(w\right)& =\exp [\log w-\exp (\log w)]\cdot \frac{1}{w}\\ & =\exp (\log w)\cdot \exp (-w)\cdot \frac{1}{w}\\ & =w\cdot e^{-w}\cdot \frac{1}{w}\\ & =e^{-w}\end{array}$$ $◼$

指数分布の分布関数は、 $$\begin{array}{r}F\left(w\right)=1-e^{-w}\end{array}$$ 生存関数は、 $$\begin{array}{rl}S\left(w\right)& =1-F\left(w\right)\\ & =e^{-w}\\ S\left(\epsilon \right)& =\exp [-\exp \left(\epsilon \right)]\\ S\left(y\right|\mathit{x})& =\exp [-\exp \left\{\frac{y-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})}{\sigma }\right\}]\\ S\left(t\right|\mathit{x})& =\exp [-\exp \left\{\frac{\log t-(\tilde{\alpha }+{\mathit{x}}^{\mathit{T}}\tilde{\mathit{\beta }})}{\sigma }\right\}]\end{array}$$ $◼$