Title 項目反応理論を用いたプロ野球選手の評価について ( 統計的モデルの新たな展望とそれに関連する話題 ) Author(s) 時光, 順平 ; 鳥越, 規央 Citation 数理解析研究所講究録 (2012), 1804: 21-29 Issue Date 2012-08 URL http://hdl.handle.net/2433/194387 Right Type Departmental Bulletin Paper Textversion publisher Kyoto University
$\omega_{j}$ 1804 2012 21-29 21 ( ) ( ) 1. TOEFL ( [2],[3]) 2. 2.1 2 (Item Response Theory, IRT) TOFLE 2 2 1,0 $U$ $N\cross n$ $N$ $n$ $ $ $s_{i}$ $u_{ij}$ u, 1, 0 ( ) $N$ $k_{j}$ # $P_{j}=k_{j}/N$ (2.1) $u_{/}$ 0-1 $P_{j}= \frac{1}{n}\sum_{i=1}^{n}u_{ij}$ (2.2) $y$ $y_{i}$ $\mathcal{y}=(\mathcal{y}_{1}, \mathcal{y}_{2}, \cdots \mathcal{y}_{i}\ldots \mathcal{y}_{n}) $ (2.3) $y_{i}=$ $=$ 1 $\omega_{j}u_{ij}$ (2.4) $r_{j}$ $u_{j}$ $y$
$\theta$, 22 $r_{j}=\rho(u_{j}, y)$ (2.5) 2.2 $\emptyset(z)$ $\Phi(f(\theta))=\int_{-\infty}^{f(\theta)}\phi(z)dz$ (2.6) $f(\theta)$ $\theta$ (2.6) $\int_{-\infty}^{f(\theta)}\varphi(z\rangle iz\cong\frac{1}{1+\exp\{-df(\theta)\}}$ (2.7) $D$ $D=1.7$ $p(\theta)$ (Item Characteristic Curve,ICC) [1] 1 $\theta$ $p_{j}(\theta)$ $p_{j}( \theta)=\frac{1}{1+\exp\{-da(\theta-b_{j})\}}$ (2.8) $b_{j}$ $a$ 2.1 1- ICC $(a=1,b_{j}=0,1,2)$ 2.1 1- ICC $a=1$ $b_{j}$ (2.8) $a$ 2 0,1,2 $p_{j}( \theta)=\frac{1}{1+\exp\{-da_{j}(\theta-b_{j})\}}$ (2.9) $a_{j}$
$b_{\overline{\gamma}}0$ $c,\cdot$ 23 2.2 2 $a_{j}$ $\theta=bj$ (2.9) $c_{j}$ 3 ICC 1,2 $p_{j}( \theta)=c_{j}+\frac{1-c_{j}}{1+\exp\{-da_{j}(\theta-b_{j})\}}$ (2.10) $C_{j}$ 2.3 3- $a_{j}=1,b_{\overline{\gamma}}0$ 0.0,0.3,0.5 ICC 3. $u_{j}$ $u_{j}=c$ $u_{j}=0,1,2, \cdots C-1$ (3.1) $C$ $\theta$ $p_{jc}(\theta)$ $p_{j}$ $(\theta).=p(u_{j}=c \theta)=p_{jc}*(\theta)-p_{j(c+1)}*(\theta)$ (3.2) $P_{j }^{*}(\theta)$ (Boundary Characteristic Curve,
24 BCC) $p_{jc}( \theta)=\frac{\prime 1}{1+\exp\vdash Da_{j}(\theta-b_{jc})\}}$ (3.3) $\theta$ $p_{jc}(\theta)=1 p_{jc}(\theta)=0$ (3.4) (3.2) (Item Response Category Characteristic Curve, IRCCC) $u_{ij}=0$ $u_{ij}=c-1$ $p_{j0}=0.5$ $p_{j(c-1)}=0.5$ $p_{j0}$ $p_{jc-1}$ $b_{j0}=b_{j1} b_{j(c-1)}=b_{j(c-1)}$ (3.5) $b_{j_{\mathcal{c}}}= \frac{b_{jc}^{*}+b_{j(c+1)}^{l}}{2}$ (3.6) 3.1 $m$ $1\cross n$ $u_{c}^{m}j$ $m$ $c$ $\theta_{i}$ $u_{c}^{m}j$ $m$ $c$ $u_{c}^{m}j=1$ $u_{c}^{m}j=0$ 6 $m$ $p( m \theta_{i})=\prod_{j=1}^{n}\prod_{c=0}^{c-1}p_{jc}*(\theta_{i})^{u_{cj}^{m}}$ (3.7) $\{\begin{array}{ll}a=(a_{1},a_{2},\cdots,a_{j},\cdots,a_{n}) (3.8)b=(b_{10},b_{11},\cdots,b_{1(c-1)},\cdots,b_{20},b_{21},\cdots,b_{2(c-1)},b_{10},b_{11},\cdots,b_{j(c-1)}) (3.9)0=(\theta_{1},\theta_{2},\cdots,\theta_{j},\cdots,\theta_{N}) \end{array}$ (3.10) $n$ $N$ $m$ $p( m \theta,a,b)=\prod_{i=1}^{n}p(m \theta_{i},a,b)=\prod_{i=1}^{n}\prod_{j=1}^{n}\prod_{c=0}^{c-1}p(m \theta_{i},a_{j},b_{jc})$ (3.11) $m$ $\theta,a,b$ $L(O,a,b)=P(m \Theta,a,b) (312)$
25 $\log L(\theta,a,b)=\sum_{ =1}^{n}\sum_{c=0}^{C-1}u_{cj}^{m}l^{*}ogp_{jc}(\theta_{i})$ (3.13) $g(\theta)$ $m$ $p(m)$ $p( m)=\int_{-\infty}^{\infty}p(m \theta)g(\theta\cross\theta$ (3.14) $N_{m}$ $m$ $L(a,b)$ $L( a,b)=\frac{n!}{m}\prod^{m}\{p(m)\}^{n_{m}}$ $\prod_{m=1}n_{m}!^{m=1}$ (3.15) $\log L=\log N.-\log\sum_{m=1}^{M}N_{m}!+\sum_{m=1}^{M}N_{m}\log p(m)$ (3.16) 3.2 $U_{i}$ $C\cross n$ $C$ $n$ $u_{c}^{i}j$ $u_{cj}^{i}=0$ $U_{i}$ $C$ $u_{c}^{i}j$ $c$ uci$j^{\cdot}=1$ 10 3.1 $U_{i}=\{\begin{array}{llllllllll}0 0 0 1 0 0 0 0 0 00 1 0 0 0 0 1 0 0 01 0 1 0 1 1 0 1 1 1\end{array}\}$ (3.17)
$\theta_{i}$ $\theta_{i}$ 26 2 $U_{i}$ $p(u_{i} \theta_{i})=\prod_{j=1}^{n}\prod_{c=0}^{c-1}p_{jc}(\theta_{i})^{u_{q}^{i}}$ (3.18) $L(\theta_{i})$ $L(\theta_{i})=p(U_{i} \theta_{i})$ (3.19) $\log jj$ (3.20) $\theta_{i}$ $0$ 4. 2011 1 2 2 ( ) ( ) 3 24 24 8 4 12 2 12 1 2 12 2 3 3 $1\cdot 3$ $2^{-}3$ 2011 10 66 4.1 4.1
27 ( ) ( ) 2011 276 241 2 3 ( ) $5O$
28 5.2 5.3, 5.4 5.3 24 5.4 12 5.5, 5.6 5.5 24
29 6. 1 3 [1] (2010) $ ^{}-$ 17, $33^{-}47$ [2] (2002) [ ], [3] (2002) [ ],