自由度

ここでは『カテゴリカルデータの連関モデル』では詳細に説明されていない，自由度の求め方を示す．

第2章　2元表の連関モデル

行変数は$A$カテゴリは$i = 1,2,\dots,I$，列変数は$B$でカテゴリは$j = 1,2,\dots,J$.

式（2.4）：$O$モデル

\[\begin{align} df &= \overbrace{IJ}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B}]\\ & = IJ - I - J + 1\\ & = (I - 1)(J - 1) \end{align}\]

式（2.6）：$FI$モデル

\[\begin{align} df & = \overbrace{IJ}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B}+ \overbrace{(I - 1) \times (J - 1)}^{\lambda_{ij}^{AB}}]\\ &= IJ - IJ \\ & = 0 \end{align}\]

式（2.7）：$U$モデル

式（2.9）：$R$モデル

\[\begin{align} df &=\overbrace{IJ}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(I - 1)}^{\tau_i^A}]\\ &= IJ - 2I - J + 2\\ &= (I - 1)(J - 2) \end{align}\]

式（2.11）：$C$モデル

\[\begin{align} df &=\overbrace{IJ}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(J - 1)}^{\tau_j^B}]\\ &= IJ - I - 2J + 2\\ &= (I - 2)(J - 1) \end{align}\]

式（2.13）：$R+C$モデル

\[\begin{align} df &=\overbrace{IJ}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(I - 1)}^{\tau_i^A} + \overbrace{(J - 2)}^{\tau_j^B}] \\ &= IJ - 2I - 2J + 4\\ &= (I - 2)(J - 2) \end{align}\]

\[\begin{align} df &=\overbrace{IJ}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(I - 2)}^{\tau_i^A} + \overbrace{(J - 1)}^{\tau_j^B}] \\ &= IJ - 2I - 2J + 4\\ &= (I - 2)(J - 2) \end{align}\]

式（2.14）：$R+C$モデル

\[\begin{align} df &=\overbrace{IJ}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{1}^{\beta} + \overbrace{(I - 2)}^{\tau_i^A} + \overbrace{(J - 2)}^{\tau_j^B}] \\ &= IJ - 2I - 2J + 4\\ &= (I - 2)(J - 2) \end{align}\]

式（2.14）で$I=J$について$\tau_i^A=\tau_j^B$の時

\[\begin{align} df &= \overbrace{I^2}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(I - 1)}^{\lambda_j^B} + \overbrace{1}^{\beta} + \overbrace{(I - 2)}^{\tau_i^A}]\\ &= I^2 - 3I + 2\\ &= (I - 1)(I - 2) \end{align}\]

式（2.16）：$RC$モデル

\[\begin{align} df &= \overbrace{IJ}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{1}^{\phi} + \overbrace{(I - 2)}^{\mu_i} + \overbrace{(J - 2)}^{\nu_j}]\\ &= IJ - 2I -2J + 4 \\ &= (I - 2)(I - 2) \end{align}\]

式（2.20）：$U+RC$モデル

式（2.22）：$R+RC$モデル

\[\begin{align} df &=\overbrace{IJ}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(I - 2)}^{\tau_i^A} + \overbrace{1}^{\phi_1} + \overbrace{(I - 2)}^{\mu_i} + \overbrace{(J - 2)}^{\nu_j}]\\ &= IJ - 3I - 2J + 6\\ &= (I - 2)(J - 3) \end{align}\]

式（2.24）：$C+RC$モデル

\[\begin{align} df &=\overbrace{IJ}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(J - 2)}^{\tau_j^B} + \overbrace{1}^{\phi_1} + \overbrace{(I - 2)}^{\mu_i} + \overbrace{(J - 2)}^{\nu_j}]\\ &= IJ - 2I - 3J + 6\\ &= (I - 3)(J - 2) \end{align}\]

式（2.26）：$R+C+RC$モデル

\[\begin{align} df &= \overbrace{IJ}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(I - 3)}^{\tau_i^A} + \overbrace{(J - 2)}^{\tau_j^B}+ \overbrace{1}^{\phi_1} + \overbrace{(I - 2)}^{\mu_i} + \overbrace{(J - 2)}^{\nu_j}]\\ &= IJ - 3I - 3J + 9\\ &= (I - 3)(J - 3) \end{align}\]

\[\begin{align} df &= \overbrace{IJ}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(I - 2)}^{\tau_i^A} + \overbrace{(J - 3)}^{\tau_j^B}+ \overbrace{1}^{\phi_1} + \overbrace{(I - 2)}^{\mu_i} + \overbrace{(J - 2)}^{\nu_j}]\\ &= IJ - 3I - 3J + 9\\ &= (I - 3)(J - 3) \end{align}\]

式（2.28）：$RC(2)$モデル

\[\begin{align} df &= \overbrace{IJ}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B}+ \overbrace{1}^{\phi_1} + \overbrace{(I - 2)}^{\mu_{i1}} + \overbrace{(J - 2)}^{\nu_{j1}} + \overbrace{1}^{\phi_2} + \overbrace{(I - 2)}^{\mu_{i2}} + \overbrace{(J - 2)}^{\nu_{j2}}- \overbrace{2}^{次元間制約}]\\ &= IJ - 3I - 3J + 9 \\ &= (I - 3)(J - 3) \end{align}\]

式（2.31）：$RC(M)$モデル

次元間制約については$M (M - 1)/2 \times 2 = M (M - 1)$を引けばよい．$M (M - 1)/2$は次元の組み合わせであり，2は$\mu$と$\nu$の2種類あるからである．

\[\begin{align} df &= \overbrace{IJ}^{セルの数} - \{\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + [\overbrace{M}^{\phi_m} + \overbrace{M(I - 2)}^{\mu_{im}} + \overbrace{M(J - 2)}^{\nu_{jm}} - \overbrace{M(M - 1)}^{次元間制約}] \} \\ &= IJ - I - J + 1 + 2M - MI - MJ + M^2 \\ & = I(J - M - 1) - J + 1 + 2M - MJ + M^2\\ & = I(J - M - 1) - M(J - M - 1) - (J - M - 1) \\ & = (I - M - 1)(J - M - 1) \end{align}\]

$M = 2$の時 \[\begin{align} df &= (I - M - 1)(J - M - 1)\\ &= (I - 2 - 1)(J - 2 - 1) \\ &= (I - 3)(J - 3) \end{align}\] となり，式（2.28）の$RC(2)$の自由度となる．

第3章　3元表に対する部分連関モデル

行変数は$A$カテゴリは$i = 1,2,\dots,I$，列変数は$B$でカテゴリは$j = 1,2,\dots,J$，列変数は$C$でカテゴリは$k = 1,2,\dots,K$.

式（3.5）：$CI$モデル

\[\begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C}+ \overbrace{(I - 1)(J - 1)}^{\lambda_{ij}^{AB}} + \overbrace{(I - 1)(K - 1)}^{\lambda_{ik}^{AC}}]\\ & = IJK - IJ - IK + I \\ & = I(JK - J - K + 1) \\ & = I(J - 1)(K - 1) \end{align}\]

式（3.7）：$CIA$モデル

\[\begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C} \\ &\quad- \{\overbrace{M_1}^{\phi_r^{AB}} + \overbrace{M_1(I - 2)}^{\mu_{ir}} + \overbrace{M_1(J - 2)}^{\nu_{ir}} - \overbrace{M_1(M_1 - 1)}^{次元間制約} \} \\ &\quad - \{\overbrace{M_2}^{\phi_r^{AC}} + \overbrace{M_2(I - 2)}^{\mu^*_{ir}} + \overbrace{M_2(K - 2)}^{\eta_{kr}} - \overbrace{M_2(M_2 - 1)}^{次元間制約} \}] \\ &= IJK - I - J - K + 2 - M_1 (I + J - M_1 - 2) - M_2 (I + K - M_2 - 2) \end{align}\]

式（3.7）で$M_1 = M_2 = 1$の時

\[\begin{align} df &= IJK - I - J - K + 2 - M_1 (I + J - M_1 - 2) - M_2 (I + K - M_2 - 2)\\ & = IJK - I - J - K + 2 - (I + J - 1 - 2) - (I + K - 1 - 2)\\ & = IJK - 3I - 2J - 2K + 8 \end{align}\]

式（3.7）で$M_1 = M_2 = 1$かつ$\mu_i = \mu^*_i$の時

\[\begin{align} df &= IJK - I - J - K + 2 - M_1 (I + J - M_1 - 2) - M_2 (K - M_2)\\ & = IJK - I - J - K + 2 - (I + J - 1 - 2) - (K - 1)\\ & = IJK - 2I - 2J - 2K + 6 \end{align}\]

式（3.9）

\[\begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C}+ \overbrace{(I - 1)(J - 1)}^{\lambda_{ij}^{AB}} + \overbrace{(I - 1)(K - 1)}^{\lambda_{ik}^{AC}} + \overbrace{(J - 1)(K - 1)}^{\lambda_{jk}^{BC}} ]\\ & = IJK + I + J + K - IJ - IK - JK - 1\\ & = I(JK - J - K + 1) - (JK - J - K + 1)\\ & = I(J - 1)(K - 1) - (J - 1)(K - 1)\\ & = (I - 1)(J - 1)(K - 1) \end{align}\]

式（3.10）

\[\begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C}\\ & \quad + \overbrace{1}^{\phi_{1}^{AB}} + \overbrace{(I - 2)}^{\mu_{i}} + \overbrace{(J - 2)}^{\nu_{j}} + \overbrace{1}^{\phi_{1}^{AC}} + \overbrace{(I - 2)}^{\mu_{i}} + \overbrace{(K - 2)}^{\eta_{j}} + \overbrace{1}^{\phi_{1}^{BC}} + \overbrace{(J - 2)}^{\nu_{j}} + \overbrace{(K - 2)}^{\eta_{k}} ] \\ & = IJK - I - J - K + 2 - [3 - 2(I - 2) -2(J-2) -2(K -2)]\\ & = IJK - 3I - 3J - 3K + 11 \end{align}\]

式（3.11）

式（3.12）

\[\begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C}\\ & \quad + \overbrace{M_1}^{\phi_{m}^{AB}} + \overbrace{M_1(I - 2)}^{\mu_{im}} + \overbrace{M_1(J - 2)}^{\nu_{jm}} - \overbrace{M_1(M_1 - 1)}^{次元間制約} \\ & \quad + \overbrace{M_2}^{\phi_{m}^{AC}} + \overbrace{M_2(I - 2)}^{\mu^*_{im}} + \overbrace{M_2(K - 2)}^{\eta_{jm}} - \overbrace{M_2(M_2 - 1)}^{次元間制約} \\ & \quad + \overbrace{M_3}^{\phi_{m}^{BC}} + \overbrace{M_3(J - 2)}^{\nu^*_{jm}} + \overbrace{M_3(K - 2)}^{\eta^*_{km}} - \overbrace{M_3(M_3 - 1)}^{次元間制約}] \\ & = IJK - I - J - K + 2 - M_1(I + J - M_1 - 2) - M_2(I + K - M_2 - 2)- M_3(J + K - M_3 - 2) \end{align}\]

式（3.12）で$M_1 = M_2 = M_3 = M$

\[\begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C}\\ & \quad + \overbrace{M}^{\phi_{m}^{AB}} + \overbrace{M(I - 2)}^{\mu_{im}} + \overbrace{M(J - 2)}^{\nu_{jm}} - \overbrace{M(M - 1)}^{次元間制約} \\ & \quad + \overbrace{M}^{\phi_{m}^{AC}} + \overbrace{M(I - 2)}^{\mu^*_{im}} + \overbrace{M(K - 2)}^{\eta_{km}} - \overbrace{M(M - 1)}^{次元間制約} \\ & \quad + \overbrace{M}^{\phi_{m}^{BC}} + \overbrace{M(J - 2)}^{\nu^*_{jm}} + \overbrace{M(K - 2)}^{\eta^*_{km}} - \overbrace{M(M - 1)}^{次元間制約}] \\ & = IJK - I - J - K + 2 - M(I + J - M - 2) - M(I + J - M - 2)- M(I + J - M - 2) \\ & = IJK - I - J - K + 2 - M(2I + 2J + 2K - 3M - 6) \end{align}\]

式（3.12）で$M_1 = M_2 = M_3 = M$かつ一貫したスコア

必要な次元間制約は$3M(M-1)/2$ではなく$M(M-1)/2$である．

第4章：3元表に対する条件付き連関モデル

式（4.1）：$CI$モデル

\[\begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C}+ \overbrace{(I - 1)(K - 1)}^{\lambda_{ik}^{AC}} + \overbrace{(J - 1)(K - 1)}^{\lambda_{jk}^{BC}}]\\ & = IJK - IK - JK + K \\ & = (IJ - I - J + 1)K \\ & = (I - 1)(J - 1)K \end{align}\]

式（4.3）：$FI$モデル

\[\begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C} \\ & \quad + \overbrace{(I - 1)(K - 1)}^{\lambda_{ik}^{AC}} + \overbrace{(J - 1)(K - 1)}^{\lambda_{jk}^{BC}} + \overbrace{(I - 1)(J - 1)}^{\lambda_{ij}^{AB}}]\\ & = IJK - IK - JK - IJ + I + J + K - 1 \\ & = I(JK - K - J + 1) - (JK - K - J + 1)\\ & = (I - 1)(JK - K - J + 1)\\ & = (I - 1)(J - 1 )(K- 1) \end{align}\]

式（4.5）：飽和モデル

$FI$モデルの自由度$(I - 1)(J - 1 )(K- 1)$を使い，そこから3元交互作用のパラメータ数$(I - 1)(J - 1 )(K- 1)$を引く．

\[\begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C} \\ & \quad + \overbrace{(I - 1)(K - 1)}^{\lambda_{ik}^{AC}} + \overbrace{(J - 1)(K - 1)}^{\lambda_{jk}^{BC}} + \overbrace{(I - 1)(J - 1)}^{\lambda_{ij}^{AB}} + \overbrace{(I - 1)(J - 1)(K - 1)}^{\lambda_{ijk}^{ABC}} ] \\ & = (I - 1)(J - 1 )(K- 1) - (I - 1)(J - 1)(K - 1) \\ & = 0 \end{align}\]

式（4.6）：$LL_1$

$FI$モデルの自由度$(I - 1)(J - 1 )(K- 1)$を使い，そこから$(K - 1)$を引く．

\[\begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C} \\ & \quad + \overbrace{(I - 1)(K - 1)}^{\lambda_{ik}^{AC}} + \overbrace{(J - 1)(K - 1)}^{\lambda_{jk}^{BC}} + \overbrace{(I - 1)(J - 1)}^{\lambda_{ij}^{AB}} + \overbrace{(K - 1)}^{\beta_k}]\\ & = (I - 1)(J - 1 )(K- 1) - (K - 1)\\ & = (IJ - I - J)(K- 1) \end{align}\]

式（4.10）：$LL_2$

式（4.14）：$LL_3$

$\phi_{k}$については$\phi_{1} = 0$，$\phi_{K} = 1$とするなど2つの制約が必要である．

\[\begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C} \\ & \quad + \overbrace{(I - 1)(K - 1)}^{\lambda_{ik}^{AC}} + \overbrace{(J - 1)(K - 1)}^{\lambda_{jk}^{BC}} + \overbrace{(I - 1)(J - 1)}^{\lambda_{ij}^{AB}}]\\ & \quad + \overbrace{(K - 2)}^{\phi_{k}} + \overbrace{(I - 1)(J - 1)}^{\psi_{ij}}]\\ & = (I - 1)(J - 1)(K- 1) - (K - 2) - (I - 1)(J - 1) \\ & = (I - 1)(J - 1)(K- 2) - (K - 2) \\ & = (IJ - I - J)(K- 2) \end{align}\]

式（4.21）：等質$(R+C)-L$

\[\begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C}+ \overbrace{(I - 1)(K - 1)}^{\lambda_{ik}^{AC}} + \overbrace{(J - 1)(K - 1)}^{\lambda_{jk}^{BC}} \\ & + \quad \overbrace{1}^{\phi} + \overbrace{(I-2)}^{\tau_i^A} + \overbrace{(J-2)}^{\tau_j^B}]\\ & = (I-1)(J-1)K - (I-2) - (J - 2) -1\\ & = IJK - IK - JK + K - I - J + 3\\ & = IJK - K(I + J - 1) - (I + J - 1) + 2\\ & = IJK - (K+1)(I + J - 1) + 2\\ \end{align}\]

式（4.24）：異質$(R+C)-L$

\[\begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C}+ \overbrace{(I - 1)(K - 1)}^{\lambda_{ik}^{AC}} + \overbrace{(J - 1)(K - 1)}^{\lambda_{jk}^{BC}} \\ & + \quad \overbrace{K}^{\phi_k} + \overbrace{(I-2)K}^{\tau_{ik}^{AC}} + \overbrace{(J-2)K}^{\tau_{jk}^{BC}}]\\ & = (I-1)(J-1)K - (I-2)K - (J-2)K - K \end{align}\]

式（4.27）：部分異質$(R+C)-L$

\[\begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C}+ \overbrace{(I - 1)(K - 1)}^{\lambda_{ik}^{AC}} + \overbrace{(J - 1)(K - 1)}^{\lambda_{jk}^{BC}} \\ & + \quad \overbrace{K}^{\phi_k} + \overbrace{(I-2)}^{\tau_{i}^{AC}} + \overbrace{(J-2)}^{\tau_{j}^{BC}}]\\ & = (I-1)(J-1)K - (I-2) - (J-2) - K\\ & = IJK - IK - JK - I - J + 4\\ & = IJK - (I + J)(K + 1) + 4 \end{align}\]

式（4.30）：部分異質$(R+C)-L$

\[\begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C}+ \overbrace{(I - 1)(K - 1)}^{\lambda_{ik}^{AC}} + \overbrace{(J - 1)(K - 1)}^{\lambda_{jk}^{BC}} \\ & + \quad \overbrace{K}^{\phi_k} + \overbrace{(I-2)K}^{\tau_{ik}^{AC}} + \overbrace{(J-2)}^{\tau_{j}^{BC}}]\\ & = (I-1)(J-1)K - (I-2)K - (J-2) - K\\ & = IJK - IK - JK + K - IK + 2K - J + 2 - K\\ & = IJK - 2IK - JK + 2K - J + 2\\ & = J(IK - K - 1) - 2(IK - K - 1)\\ & = (J - 2)(IK - K - 1) \end{align}\]

式（4.31）：部分異質$(R+C)-L$

\[\begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C}+ \overbrace{(I - 1)(K - 1)}^{\lambda_{ik}^{AC}} + \overbrace{(J - 1)(K - 1)}^{\lambda_{jk}^{BC}} \\ & + \quad \overbrace{K}^{\phi_k} + \overbrace{(I-2)}^{\tau_{i}^{AC}} + \overbrace{(J-2)K}^{\tau_{jk}^{BC}}]\\ & = (I-1)(J-1)K - (I-2) - (J-2)K - K\\ & = IJK - IK - JK + K - I + 2 - JK + 2K - K\\ & = IJK - IK - 2JK + 2K - I + 2\\ & = I(JK - K - 1) - 2(JK - K - 1)\\ & = (I - 2)(JK - K - 1) \end{align}\]

式（4.39）：同質$RC(M)-L$

\[\begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C}+ \overbrace{(I - 1)(K - 1)}^{\lambda_{ik}^{AC}} + \overbrace{(J - 1)(K - 1)}^{\lambda_{jk}^{BC}} \\ & + \quad \overbrace{M}^{\phi_m} + \overbrace{(I-2)M}^{\mu_{im}} + \overbrace{(J-2)M}^{\nu_{jm}} - \overbrace{M(M-1)}^{次元間制約}] \\ & = (I-1)(J-1)K - M(I + J - M - 2) \end{align}\]

式（4.42）：異質$RC(M)-L$

\[\begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C}+ \overbrace{(I - 1)(K - 1)}^{\lambda_{ik}^{AC}} + \overbrace{(J - 1)(K - 1)}^{\lambda_{jk}^{BC}} \\ & + \quad \overbrace{KM}^{\phi_{mk}} + \overbrace{(I-2)KM}^{\mu_{imk}} + \overbrace{(J-2)KM}^{\nu_{jmk}} - \overbrace{KM(M-1)}^{次元間制約}] \\ & = K(I-1)(J-1) - KM(I - 1) - KM(J - 1) - KM^2\\ & = K(I-1)[(J-1) - M] - KM[(J - 1) - M]\\ & = K(I - 1 - M)(J - 1 - M) \end{align}\]

式（4.45）

\[\begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C}+ \overbrace{(I - 1)(K - 1)}^{\lambda_{ik}^{AC}} + \overbrace{(J - 1)(K - 1)}^{\lambda_{jk}^{BC}} \\ & + \quad \overbrace{K}^{\phi_k} + \overbrace{(I-2)}^{\mu_{i}} + \overbrace{(J-2)K}^{\mu_{jk}}]\\ & = IJK - IK - JK + K - K - I + 2 - JK + 2K\\ & = I(JK - K - 1) -2(JK - K - 1)\\ & = (I - 2)(JK - K - 1) \end{align}\]

式（4.48）

\[\begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C}+ \overbrace{(I - 1)(K - 1)}^{\lambda_{ik}^{AC}} + \overbrace{(J - 1)(K - 1)}^{\lambda_{jk}^{BC}} \\ & + \quad \overbrace{K}^{\phi_k} + \overbrace{(I-2)K}^{\mu_{ik}} + \overbrace{(J-2)}^{\mu_{j}}]\\ & \quad = IJK - IK - JK + K - K - IK + 2K - J + 2\\ & \quad = J(IK - K - 1) -2(IK - K - 1)\\ & \quad = (J - 2)(IK - K - 1) \end{align}\]

式（4.51）

\[\begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C}+ \overbrace{(I - 1)(K - 1)}^{\lambda_{ik}^{AC}} + \overbrace{(J - 1)(K - 1)}^{\lambda_{jk}^{BC}} \\ & + \quad \overbrace{K}^{\phi_k} + \overbrace{(I-2)}^{\mu_{i}} + \overbrace{(J-2)}^{\mu_{j}}]\\ & \quad = (IJK - IK - JK + K) - K - I + 2 - J + 2\\ & \quad = IJK - K(I + J) - (I + J) + 4\\ & \quad = IJK - (I + J)(K+1) + 4 \end{align}\]

# 自由度 {-} ここでは『カテゴリカルデータの連関モデル』では詳細に説明されていない，自由度の求め方を示す． ## 第2章　2元表の連関モデル行変数は$A$カテゴリは$i = 1,2,\dots,I$，列変数は$B$でカテゴリは$j = 1,2,\dots,J$. **式（2.4）：$O$モデル** \begin{align} df &= \overbrace{IJ}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B}]\\ & = IJ - I - J + 1\\ & = (I - 1)(J - 1) \end{align} **式（2.6）：$FI$モデル** \begin{align} df & = \overbrace{IJ}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B}+ \overbrace{(I - 1) \times (J - 1)}^{\lambda_{ij}^{AB}}]\\ &= IJ - IJ \\ & = 0 \end{align} **式（2.7）：$U$モデル** \begin{align} df &=\overbrace{IJ}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{1}^{\beta}]\\ &= IJ - I - J \end{align} **式（2.9）：$R$モデル** \begin{align} df &=\overbrace{IJ}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(I - 1)}^{\tau_i^A}]\\ &= IJ - 2I - J + 2\\ &= (I - 1)(J - 2) \end{align} **式（2.11）：$C$モデル** \begin{align} df &=\overbrace{IJ}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(J - 1)}^{\tau_j^B}]\\ &= IJ - I - 2J + 2\\ &= (I - 2)(J - 1) \end{align} **式（2.13）：$R+C$モデル** \begin{align} df &=\overbrace{IJ}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(I - 1)}^{\tau_i^A} + \overbrace{(J - 2)}^{\tau_j^B}] \\ &= IJ - 2I - 2J + 4\\ &= (I - 2)(J - 2) \end{align} \begin{align} df &=\overbrace{IJ}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(I - 2)}^{\tau_i^A} + \overbrace{(J - 1)}^{\tau_j^B}] \\ &= IJ - 2I - 2J + 4\\ &= (I - 2)(J - 2) \end{align} **式（2.14）：$R+C$モデル** \begin{align} df &=\overbrace{IJ}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{1}^{\beta} + \overbrace{(I - 2)}^{\tau_i^A} + \overbrace{(J - 2)}^{\tau_j^B}] \\ &= IJ - 2I - 2J + 4\\ &= (I - 2)(J - 2) \end{align} **式（2.14）で$I=J$について$\tau_i^A=\tau_j^B$の時** \begin{align} df &= \overbrace{I^2}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(I - 1)}^{\lambda_j^B} + \overbrace{1}^{\beta} + \overbrace{(I - 2)}^{\tau_i^A}]\\ &= I^2 - 3I + 2\\ &= (I - 1)(I - 2) \end{align} **式（2.16）：$RC$モデル** \begin{align} df &= \overbrace{IJ}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{1}^{\phi} + \overbrace{(I - 2)}^{\mu_i} + \overbrace{(J - 2)}^{\nu_j}]\\ &= IJ - 2I -2J + 4 \\ &= (I - 2)(I - 2) \end{align} **式（2.20）：$U+RC$モデル** \begin{align} df &=\overbrace{IJ}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{1}^{\phi_1} + \overbrace{1}^{\phi_2}+ \overbrace{(I - 2)}^{\mu_i} + \overbrace{(J - 2)}^{\nu_j}]\\ &= IJ - 2I - 2J + 3 \end{align} **式（2.22）：$R+RC$モデル** \begin{align} df &=\overbrace{IJ}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(I - 2)}^{\tau_i^A} + \overbrace{1}^{\phi_1} + \overbrace{(I - 2)}^{\mu_i} + \overbrace{(J - 2)}^{\nu_j}]\\ &= IJ - 3I - 2J + 6\\ &= (I - 2)(J - 3) \end{align} **式（2.24）：$C+RC$モデル** \begin{align} df &=\overbrace{IJ}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(J - 2)}^{\tau_j^B} + \overbrace{1}^{\phi_1} + \overbrace{(I - 2)}^{\mu_i} + \overbrace{(J - 2)}^{\nu_j}]\\ &= IJ - 2I - 3J + 6\\ &= (I - 3)(J - 2) \end{align} **式（2.26）：$R+C+RC$モデル** \begin{align} df &= \overbrace{IJ}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(I - 3)}^{\tau_i^A} + \overbrace{(J - 2)}^{\tau_j^B}+ \overbrace{1}^{\phi_1} + \overbrace{(I - 2)}^{\mu_i} + \overbrace{(J - 2)}^{\nu_j}]\\ &= IJ - 3I - 3J + 9\\ &= (I - 3)(J - 3) \end{align} \begin{align} df &= \overbrace{IJ}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(I - 2)}^{\tau_i^A} + \overbrace{(J - 3)}^{\tau_j^B}+ \overbrace{1}^{\phi_1} + \overbrace{(I - 2)}^{\mu_i} + \overbrace{(J - 2)}^{\nu_j}]\\ &= IJ - 3I - 3J + 9\\ &= (I - 3)(J - 3) \end{align} **式（2.28）：$RC(2)$モデル** \begin{align} df &= \overbrace{IJ}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B}+ \overbrace{1}^{\phi_1} + \overbrace{(I - 2)}^{\mu_{i1}} + \overbrace{(J - 2)}^{\nu_{j1}} + \overbrace{1}^{\phi_2} + \overbrace{(I - 2)}^{\mu_{i2}} + \overbrace{(J - 2)}^{\nu_{j2}}- \overbrace{2}^{次元間制約}]\\ &= IJ - 3I - 3J + 9 \\ &= (I - 3)(J - 3) \end{align} **式（2.31）：$RC(M)$モデル** 次元間制約については$M (M - 1)/2 \times 2 = M (M - 1)$を引けばよい．$M (M - 1)/2$は次元の組み合わせであり，2は$\mu$と$\nu$の2種類あるからである． \begin{align} df &= \overbrace{IJ}^{セルの数} - \{\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + [\overbrace{M}^{\phi_m} + \overbrace{M(I - 2)}^{\mu_{im}} + \overbrace{M(J - 2)}^{\nu_{jm}} - \overbrace{M(M - 1)}^{次元間制約}] \} \\ &= IJ - I - J + 1 + 2M - MI - MJ + M^2 \\ & = I(J - M - 1) - J + 1 + 2M - MJ + M^2\\ & = I(J - M - 1) - M(J - M - 1) - (J - M - 1) \\ & = (I - M - 1)(J - M - 1) \end{align} $M = 2$の時 \begin{align} df &= (I - M - 1)(J - M - 1)\\ &= (I - 2 - 1)(J - 2 - 1) \\ &= (I - 3)(J - 3) \end{align} となり，式（2.28）の$RC(2)$の自由度となる． ## 第3章　3元表に対する部分連関モデル行変数は$A$カテゴリは$i = 1,2,\dots,I$，列変数は$B$でカテゴリは$j = 1,2,\dots,J$，列変数は$C$でカテゴリは$k = 1,2,\dots,K$. **式（3.5）：$CI$モデル** \begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C}+ \overbrace{(I - 1)(J - 1)}^{\lambda_{ij}^{AB}} + \overbrace{(I - 1)(K - 1)}^{\lambda_{ik}^{AC}}]\\ & = IJK - IJ - IK + I \\ & = I(JK - J - K + 1) \\ & = I(J - 1)(K - 1) \end{align} **式（3.7）：$CIA$モデル** \begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C} \\ &\quad- \{\overbrace{M_1}^{\phi_r^{AB}} + \overbrace{M_1(I - 2)}^{\mu_{ir}} + \overbrace{M_1(J - 2)}^{\nu_{ir}} - \overbrace{M_1(M_1 - 1)}^{次元間制約} \} \\ &\quad - \{\overbrace{M_2}^{\phi_r^{AC}} + \overbrace{M_2(I - 2)}^{\mu^*_{ir}} + \overbrace{M_2(K - 2)}^{\eta_{kr}} - \overbrace{M_2(M_2 - 1)}^{次元間制約} \}] \\ &= IJK - I - J - K + 2 - M_1 (I + J - M_1 - 2) - M_2 (I + K - M_2 - 2) \end{align} **式（3.7）で$M_1 = M_2 = 1$の時** \begin{align} df &= IJK - I - J - K + 2 - M_1 (I + J - M_1 - 2) - M_2 (I + K - M_2 - 2)\\ & = IJK - I - J - K + 2 - (I + J - 1 - 2) - (I + K - 1 - 2)\\ & = IJK - 3I - 2J - 2K + 8 \end{align} **式（3.7）で$M_1 = M_2 = 1$かつ$\mu_i = \mu^*_i$の時** \begin{align} df &= IJK - I - J - K + 2 - M_1 (I + J - M_1 - 2) - M_2 (K - M_2)\\ & = IJK - I - J - K + 2 - (I + J - 1 - 2) - (K - 1)\\ & = IJK - 2I - 2J - 2K + 6 \end{align} **式（3.9）** \begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C}+ \overbrace{(I - 1)(J - 1)}^{\lambda_{ij}^{AB}} + \overbrace{(I - 1)(K - 1)}^{\lambda_{ik}^{AC}} + \overbrace{(J - 1)(K - 1)}^{\lambda_{jk}^{BC}} ]\\ & = IJK + I + J + K - IJ - IK - JK - 1\\ & = I(JK - J - K + 1) - (JK - J - K + 1)\\ & = I(J - 1)(K - 1) - (J - 1)(K - 1)\\ & = (I - 1)(J - 1)(K - 1) \end{align} **式（3.10）** \begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C}\\ & \quad + \overbrace{1}^{\phi_{1}^{AB}} + \overbrace{(I - 2)}^{\mu_{i}} + \overbrace{(J - 2)}^{\nu_{j}} + \overbrace{1}^{\phi_{1}^{AC}} + \overbrace{(I - 2)}^{\mu_{i}} + \overbrace{(K - 2)}^{\eta_{j}} + \overbrace{1}^{\phi_{1}^{BC}} + \overbrace{(J - 2)}^{\nu_{j}} + \overbrace{(K - 2)}^{\eta_{k}} ] \\ & = IJK - I - J - K + 2 - [3 - 2(I - 2) -2(J-2) -2(K -2)]\\ & = IJK - 3I - 3J - 3K + 11 \end{align} **式（3.11）** \begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C}\\ & \quad + \overbrace{1}^{\phi_{1}^{AB}} + \overbrace{1}^{\phi_{1}^{AC}} + \overbrace{1}^{\phi_{1}^{BC}} + \overbrace{(I - 2)}^{\mu_{i}} + \overbrace{(J - 2)}^{\nu_{j}} + \overbrace{(K - 2)}^{\eta_{j}} + ] \\ & = IJK - I - J - K + 2 - [3 - (I - 2) -(J-2) - (K -2)]\\ & = IJK - 2I - 2J - 2K + 5 \end{align} **式（3.12）** \begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C}\\ & \quad + \overbrace{M_1}^{\phi_{m}^{AB}} + \overbrace{M_1(I - 2)}^{\mu_{im}} + \overbrace{M_1(J - 2)}^{\nu_{jm}} - \overbrace{M_1(M_1 - 1)}^{次元間制約} \\ & \quad + \overbrace{M_2}^{\phi_{m}^{AC}} + \overbrace{M_2(I - 2)}^{\mu^*_{im}} + \overbrace{M_2(K - 2)}^{\eta_{jm}} - \overbrace{M_2(M_2 - 1)}^{次元間制約} \\ & \quad + \overbrace{M_3}^{\phi_{m}^{BC}} + \overbrace{M_3(J - 2)}^{\nu^*_{jm}} + \overbrace{M_3(K - 2)}^{\eta^*_{km}} - \overbrace{M_3(M_3 - 1)}^{次元間制約}] \\ & = IJK - I - J - K + 2 - M_1(I + J - M_1 - 2) - M_2(I + K - M_2 - 2)- M_3(J + K - M_3 - 2) \end{align} **式（3.12）で$M_1 = M_2 = M_3 = M$** \begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C}\\ & \quad + \overbrace{M}^{\phi_{m}^{AB}} + \overbrace{M(I - 2)}^{\mu_{im}} + \overbrace{M(J - 2)}^{\nu_{jm}} - \overbrace{M(M - 1)}^{次元間制約} \\ & \quad + \overbrace{M}^{\phi_{m}^{AC}} + \overbrace{M(I - 2)}^{\mu^*_{im}} + \overbrace{M(K - 2)}^{\eta_{km}} - \overbrace{M(M - 1)}^{次元間制約} \\ & \quad + \overbrace{M}^{\phi_{m}^{BC}} + \overbrace{M(J - 2)}^{\nu^*_{jm}} + \overbrace{M(K - 2)}^{\eta^*_{km}} - \overbrace{M(M - 1)}^{次元間制約}] \\ & = IJK - I - J - K + 2 - M(I + J - M - 2) - M(I + J - M - 2)- M(I + J - M - 2) \\ & = IJK - I - J - K + 2 - M(2I + 2J + 2K - 3M - 6) \end{align} **式（3.12）で$M_1 = M_2 = M_3 = M$かつ一貫したスコア** 必要な次元間制約は$3M(M-1)/2$ではなく$M(M-1)/2$である． \begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C}\\ & \quad + \overbrace{M}^{\phi_{m}^{AB}} + \overbrace{M}^{\phi_{m}^{AC}} + \overbrace{M}^{\phi_{m}^{BC}} + \overbrace{M(I - 2)}^{\mu_{im}} + \overbrace{M(J - 2)}^{\nu_{jm}} + \overbrace{M(K - 2)}^{\eta_{km}} - \overbrace{M(M - 1)/2}^{次元間制約}] \\ & = IJK - I - J - K + 2 - M(I + J + K - 3) + M(M - 1)/2 \end{align} ## 第4章：3元表に対する条件付き連関モデル **式（4.1）：$CI$モデル** \begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C}+ \overbrace{(I - 1)(K - 1)}^{\lambda_{ik}^{AC}} + \overbrace{(J - 1)(K - 1)}^{\lambda_{jk}^{BC}}]\\ & = IJK - IK - JK + K \\ & = (IJ - I - J + 1)K \\ & = (I - 1)(J - 1)K \end{align} **式（4.3）：$FI$モデル** \begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C} \\ & \quad + \overbrace{(I - 1)(K - 1)}^{\lambda_{ik}^{AC}} + \overbrace{(J - 1)(K - 1)}^{\lambda_{jk}^{BC}} + \overbrace{(I - 1)(J - 1)}^{\lambda_{ij}^{AB}}]\\ & = IJK - IK - JK - IJ + I + J + K - 1 \\ & = I(JK - K - J + 1) - (JK - K - J + 1)\\ & = (I - 1)(JK - K - J + 1)\\ & = (I - 1)(J - 1 )(K- 1) \end{align} **式（4.5）：飽和モデル** $FI$モデルの自由度$(I - 1)(J - 1 )(K- 1)$を使い，そこから3元交互作用のパラメータ数$(I - 1)(J - 1 )(K- 1)$を引く． \begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C} \\ & \quad + \overbrace{(I - 1)(K - 1)}^{\lambda_{ik}^{AC}} + \overbrace{(J - 1)(K - 1)}^{\lambda_{jk}^{BC}} + \overbrace{(I - 1)(J - 1)}^{\lambda_{ij}^{AB}} + \overbrace{(I - 1)(J - 1)(K - 1)}^{\lambda_{ijk}^{ABC}} ] \\ & = (I - 1)(J - 1 )(K- 1) - (I - 1)(J - 1)(K - 1) \\ & = 0 \end{align} **式（4.6）：$LL_1$** $FI$モデルの自由度$(I - 1)(J - 1 )(K- 1)$を使い，そこから$(K - 1)$を引く． \begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C} \\ & \quad + \overbrace{(I - 1)(K - 1)}^{\lambda_{ik}^{AC}} + \overbrace{(J - 1)(K - 1)}^{\lambda_{jk}^{BC}} + \overbrace{(I - 1)(J - 1)}^{\lambda_{ij}^{AB}} + \overbrace{(K - 1)}^{\beta_k}]\\ & = (I - 1)(J - 1 )(K- 1) - (K - 1)\\ & = (IJ - I - J)(K- 1) \end{align} **式（4.10）：$LL_2$** \begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C} \\ & \quad + \overbrace{(I - 1)(K - 1)}^{\lambda_{ik}^{AC}} + \overbrace{(J - 1)(K - 1)}^{\lambda_{jk}^{BC}} + \overbrace{(K - 1)}^{\phi_{k}} + \overbrace{(I - 1)(J - 1)}^{\psi_{ij}}]\\ & = (I - 1)(J - 1 )(K- 1) - (K - 1) \\ & = (IJ - I - J)(K - 1) \end{align} **式（4.14）：$LL_3$** $\phi_{k}$については$\phi_{1} = 0$，$\phi_{K} = 1$とするなど2つの制約が必要である． \begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C} \\ & \quad + \overbrace{(I - 1)(K - 1)}^{\lambda_{ik}^{AC}} + \overbrace{(J - 1)(K - 1)}^{\lambda_{jk}^{BC}} + \overbrace{(I - 1)(J - 1)}^{\lambda_{ij}^{AB}}]\\ & \quad + \overbrace{(K - 2)}^{\phi_{k}} + \overbrace{(I - 1)(J - 1)}^{\psi_{ij}}]\\ & = (I - 1)(J - 1)(K- 1) - (K - 2) - (I - 1)(J - 1) \\ & = (I - 1)(J - 1)(K- 2) - (K - 2) \\ & = (IJ - I - J)(K- 2) \end{align} **式（4.21）：等質$(R+C)-L$** \begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C}+ \overbrace{(I - 1)(K - 1)}^{\lambda_{ik}^{AC}} + \overbrace{(J - 1)(K - 1)}^{\lambda_{jk}^{BC}} \\ & + \quad \overbrace{1}^{\phi} + \overbrace{(I-2)}^{\tau_i^A} + \overbrace{(J-2)}^{\tau_j^B}]\\ & = (I-1)(J-1)K - (I-2) - (J - 2) -1\\ & = IJK - IK - JK + K - I - J + 3\\ & = IJK - K(I + J - 1) - (I + J - 1) + 2\\ & = IJK - (K+1)(I + J - 1) + 2\\ \end{align} **式（4.24）：異質$(R+C)-L$** \begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C}+ \overbrace{(I - 1)(K - 1)}^{\lambda_{ik}^{AC}} + \overbrace{(J - 1)(K - 1)}^{\lambda_{jk}^{BC}} \\ & + \quad \overbrace{K}^{\phi_k} + \overbrace{(I-2)K}^{\tau_{ik}^{AC}} + \overbrace{(J-2)K}^{\tau_{jk}^{BC}}]\\ & = (I-1)(J-1)K - (I-2)K - (J-2)K - K \end{align} **式（4.27）：部分異質$(R+C)-L$** \begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C}+ \overbrace{(I - 1)(K - 1)}^{\lambda_{ik}^{AC}} + \overbrace{(J - 1)(K - 1)}^{\lambda_{jk}^{BC}} \\ & + \quad \overbrace{K}^{\phi_k} + \overbrace{(I-2)}^{\tau_{i}^{AC}} + \overbrace{(J-2)}^{\tau_{j}^{BC}}]\\ & = (I-1)(J-1)K - (I-2) - (J-2) - K\\ & = IJK - IK - JK - I - J + 4\\ & = IJK - (I + J)(K + 1) + 4 \end{align} **式（4.30）：部分異質$(R+C)-L$** \begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C}+ \overbrace{(I - 1)(K - 1)}^{\lambda_{ik}^{AC}} + \overbrace{(J - 1)(K - 1)}^{\lambda_{jk}^{BC}} \\ & + \quad \overbrace{K}^{\phi_k} + \overbrace{(I-2)K}^{\tau_{ik}^{AC}} + \overbrace{(J-2)}^{\tau_{j}^{BC}}]\\ & = (I-1)(J-1)K - (I-2)K - (J-2) - K\\ & = IJK - IK - JK + K - IK + 2K - J + 2 - K\\ & = IJK - 2IK - JK + 2K - J + 2\\ & = J(IK - K - 1) - 2(IK - K - 1)\\ & = (J - 2)(IK - K - 1) \end{align} **式（4.31）：部分異質$(R+C)-L$** \begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C}+ \overbrace{(I - 1)(K - 1)}^{\lambda_{ik}^{AC}} + \overbrace{(J - 1)(K - 1)}^{\lambda_{jk}^{BC}} \\ & + \quad \overbrace{K}^{\phi_k} + \overbrace{(I-2)}^{\tau_{i}^{AC}} + \overbrace{(J-2)K}^{\tau_{jk}^{BC}}]\\ & = (I-1)(J-1)K - (I-2) - (J-2)K - K\\ & = IJK - IK - JK + K - I + 2 - JK + 2K - K\\ & = IJK - IK - 2JK + 2K - I + 2\\ & = I(JK - K - 1) - 2(JK - K - 1)\\ & = (I - 2)(JK - K - 1) \end{align} **式（4.39）：同質$RC(M)-L$** \begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C}+ \overbrace{(I - 1)(K - 1)}^{\lambda_{ik}^{AC}} + \overbrace{(J - 1)(K - 1)}^{\lambda_{jk}^{BC}} \\ & + \quad \overbrace{M}^{\phi_m} + \overbrace{(I-2)M}^{\mu_{im}} + \overbrace{(J-2)M}^{\nu_{jm}} - \overbrace{M(M-1)}^{次元間制約}] \\ & = (I-1)(J-1)K - M(I + J - M - 2) \end{align} **式（4.42）：異質$RC(M)-L$** \begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C}+ \overbrace{(I - 1)(K - 1)}^{\lambda_{ik}^{AC}} + \overbrace{(J - 1)(K - 1)}^{\lambda_{jk}^{BC}} \\ & + \quad \overbrace{KM}^{\phi_{mk}} + \overbrace{(I-2)KM}^{\mu_{imk}} + \overbrace{(J-2)KM}^{\nu_{jmk}} - \overbrace{KM(M-1)}^{次元間制約}] \\ & = K(I-1)(J-1) - KM(I - 1) - KM(J - 1) - KM^2\\ & = K(I-1)[(J-1) - M] - KM[(J - 1) - M]\\ & = K(I - 1 - M)(J - 1 - M) \end{align} **式（4.45）** \begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C}+ \overbrace{(I - 1)(K - 1)}^{\lambda_{ik}^{AC}} + \overbrace{(J - 1)(K - 1)}^{\lambda_{jk}^{BC}} \\ & + \quad \overbrace{K}^{\phi_k} + \overbrace{(I-2)}^{\mu_{i}} + \overbrace{(J-2)K}^{\mu_{jk}}]\\ & = IJK - IK - JK + K - K - I + 2 - JK + 2K\\ & = I(JK - K - 1) -2(JK - K - 1)\\ & = (I - 2)(JK - K - 1) \end{align} **式（4.48）** \begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C}+ \overbrace{(I - 1)(K - 1)}^{\lambda_{ik}^{AC}} + \overbrace{(J - 1)(K - 1)}^{\lambda_{jk}^{BC}} \\ & + \quad \overbrace{K}^{\phi_k} + \overbrace{(I-2)K}^{\mu_{ik}} + \overbrace{(J-2)}^{\mu_{j}}]\\ & \quad = IJK - IK - JK + K - K - IK + 2K - J + 2\\ & \quad = J(IK - K - 1) -2(IK - K - 1)\\ & \quad = (J - 2)(IK - K - 1) \end{align} **式（4.51）** \begin{align} df &= \overbrace{IJK}^{セルの数} - [\overbrace{1}^{\lambda} + \overbrace{(I - 1)}^{\lambda_i^A} + \overbrace{(J - 1)}^{\lambda_j^B} + \overbrace{(K - 1)}^{\lambda_k^C}+ \overbrace{(I - 1)(K - 1)}^{\lambda_{ik}^{AC}} + \overbrace{(J - 1)(K - 1)}^{\lambda_{jk}^{BC}} \\ & + \quad \overbrace{K}^{\phi_k} + \overbrace{(I-2)}^{\mu_{i}} + \overbrace{(J-2)}^{\mu_{j}}]\\ & \quad = (IJK - IK - JK + K) - K - I + 2 - J + 2\\ & \quad = IJK - K(I + J) - (I + J) + 4\\ & \quad = IJK - (I + J)(K+1) + 4 \end{align}

第2章 2元表の連関モデル

第3章 3元表に対する部分連関モデル

第4章：3元表に対する条件付き連関モデル

第2章　2元表の連関モデル

第3章　3元表に対する部分連関モデル