이원배치법 (혼합모형) (반복있음)

[인자] $$A$$ 는 [모수인자]

[인자] $$B$$ 는 [변량인자]

$$ y_{ijk} = \mu + a_{i} + b_{j} + (ab)_{ij} + e_{ijk} $$

 $$y_{ijk}$$  :  $$A_{i}$$ 와 $$B_{j}$$ 에서 얻은 $$k$$ 번째 [측정값]

 $$\mu$$  : 실험전체의 [모평균]

 $$a_{i}$$  :  $$A_{i}$$ 가 주는 효과

 $$b_{j}$$  :  $$B_{j}$$ 가 주는 효과 ( $$b_{j} \sim N(0, \sigma_{B}^{ \ 2})$$ 이고 서로 [독립])

 $$(ab)_{ij}$$  :  $$A_{i}$$ 와 $$B_{j}$$ 의 [교호작용] 효과 $$\left( \sum_{i=1}^{l}(ab)_{ij}=0 \ , \ \sum_{j=1}^{m}(ab)_{ij} \neq 0 \right)$$

 $$e_{ijk}$$  :  $$A_{i}$$ 와 $$B_{j}$$ 에서 얻은 $$k$$ 번째 [측정값]의 [오차]  ( $$e_{ijk} \sim N(0, \sigma_{E}^{ \ 2})$$ 이고 서로 [독립])

  $$i$$  : 인자 $$A$$ 의 [수준] 수 $$( i = 1,2, \cdots ,l )$$

  $$j$$  : 인자 $$B$$ 의 [수준] 수 $$( j = 1,2, \cdots ,m )$$

  $$k$$  : 실험의 [반복] 수 $$( j = 1,2, \cdots ,r )$$

—-

||<|2> [인자] $$B$$ |||||||||||||| [인자] $$A$$ ||<|2> 합계 ||<|2> [평균] || |||| $$A_{1}$$ |||| $$A_{2}$$ || $$\cdots$$ |||| $$A_{l}$$ || |||||||||||||||||||| || ||<|4> $$B_{1}$$ || $$y_{111}$$ ||<|2> $$T_{11.}$$ || $$y_{211}$$ ||<|2> $$T_{21.}$$ ||<|4> $$\cdots$$ || $$y_{l11}$$ ||<|2> $$T_{l1.}$$ ||<|4> $$T_{.1.}$$ ||<|4> $$\overline{y}_{.1.}$$ || || $$y_{112}$$ || $$y_{212}$$ || $$y_{l12}$$ || || $$\vdots$$ ||<|2> $$\overline{y}_{11.}$$ || $$\vdots$$ ||<|2> $$\overline{y}_{21.}$$ || $$\vdots$$ ||<|2> $$\overline{y}_{l1.}$$ || || $$y_{11r}$$ || $$y_{21r}$$ || $$y_{l1r}$$ || ||<|4> $$B_{2}$$ || $$y_{121}$$ ||<|2> $$T_{12.}$$ || $$y_{221}$$ ||<|2> $$T_{22.}$$ ||<|4> $$\cdots$$ || $$y_{l21}$$ ||<|2> $$T_{l2.}$$ ||<|4> $$T_{.2.}$$ ||<|4> $$\overline{y}_{.2.}$$ || || $$y_{122}$$ || $$y_{222}$$ || $$y_{l22}$$ || || $$\vdots$$ ||<|2> $$\overline{y}_{12.}$$ || $$\vdots$$ ||<|2> $$\overline{y}_{22.}$$ || $$\vdots$$ ||<|2> $$\overline{y}_{l2.}$$ || || $$y_{12r}$$ || $$y_{22r}$$ || $$y_{l2r}$$ || || $$\vdots$$ |||| $$\vdots$$ |||| $$\vdots$$ || $$\vdots$$ |||| $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || ||<|4> $$B_{m}$$ || $$y_{1m1}$$ ||<|2> $$T_{1m.}$$ || $$y_{2m1}$$ ||<|2> $$T_{2m.}$$ ||<|4> $$\cdots$$ || $$y_{lm1}$$ ||<|2> $$T_{lm.}$$ ||<|4> $$T_{.m.}$$ ||<|4> $$\overline{y}_{.m.}$$ || || $$y_{1m2}$$ || $$y_{2m2}$$ || $$y_{lm2}$$ || || $$\vdots$$ ||<|2> $$\overline{y}_{1m.}$$ || $$\vdots$$ ||<|2> $$\overline{y}_{2m.}$$ || $$\vdots$$ ||<|2> $$\overline{y}_{lm.}$$ || || $$y_{1mr}$$ || $$y_{2mr}$$ || $$y_{lmr}$$ || |||||||||||||||||||| || || 합계 |||| $$T_{1..}$$ |||| $$T_{2..}$$ || $$\cdots$$ |||| $$T_{l..}$$ || $$T$$ || || || [평균] |||| $$\overline{y}_{1..}$$ |||| $$\overline{y}_{2..}$$ || $$\cdots$$ |||| $$\overline{y}_{l..}$$ || || $$\overline{\overline{y}}$$ ||

|| $$T_{i..} = \sum_{j=1}^{m} \sum_{k=1}^{r} y_{ijk}$$ || $$\overline{y}_{i..} = \frac{T_{i..}}{mr}$$ ||
|| $$T_{.j.} = \sum_{i=1}^{l} \sum_{k=1}^{r} y_{ijk}$$ || $$\overline{y}_{.j.} = \frac{T_{.j.}}{lr}$$ ||
|| $$T_{ij.} = \sum_{k=1}^{r} y_{ijk}$$ || $$\overline{y}_{ij.} = \frac{T_{ij.}}{r}$$ ||
|| $$T = \sum_{i=1}^{l} \sum_{j=1}^{m} \sum_{k=1}^{r} y_{ijk}$$ || $$\overline{\overline{y}} = \frac{T}{lmr} = \frac{T}{N}$$ ||
|| $$N = lmr$$ || $$CT = \frac{T^{2}}{lmr} = \frac{T^{2}}{N}$$ ||

—-

개개의 데이터 $$y_{ijk}$$ 와 총편균 $$\overline{\overline{y}}$$ 의 차이는 다음과 같이 네 부분으로 나뉘어진다.

$$(y_{ijk}-\overline{\overline{y}})=(y_{i..}-\overline{\overline{y}})+(y_{.j.}-\overline{\overline{y}})+(\overline{y}_{ij.}-\overline{y}_{i..}-\overline{y}_{.j.}+\overline{\overline{y}})+(y_{ijk}-\overline{y}_{ij.})$$

양변을 제곱한 후에 모든 $$i, \ j, \ k$$ 에 대하여 합하면 아래의 등식을 얻을 수 있다.

$$\begin{displaymath}\begin{split} \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(y_{ijk}-\overline{\overline{y}})^{2} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(y_{i..}-\overline{\overline{y}})^{2}+\sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(y_{.j.}-\overline{\overline{y}})^{2}+\sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(\overline{y}_{ij.}-\overline{y}_{i..}-\overline{y}_{.j.}+\overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(y_{ijk}-\overline{y}_{ij.})^{2} \end{split}\end{displaymath}$$

위 식에서 왼쪽 항은 총변동 $$S_{T}$$ 이고, 오른쪽 항은 차례대로 $$A$$ 의 [변동], $$B$$ 의 [변동], $$A, \ B$$ 의 [교호작용]의 변동 [오차변동]인 $$S_{A}$$ , $$S_{B}$$ , $$S_{A \times B}$$ , $$S_{E}$$ 가 된다.

$$\begin{displaymath}\begin{split} S_{T} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(y_{ijk}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}y_{ijk}^{ \ 2} - CT \end{split}\end{displaymath}$$

$$\begin{displaymath}\begin{split} S_{A} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(y_{i..}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\frac{T_{i..}^{ \ 2}}{mr}-CT \end{split}\end{displaymath}$$

$$\begin{displaymath}\begin{split} S_{B} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(y_{.j.}-\overline{\overline{y}})^{2} \\ &= \sum_{j=1}^{m}\frac{T_{.j.}^{ \ 2}}{lr}-CT \end{split}\end{displaymath}$$

$$\begin{displaymath}\begin{split} S_{A \times B} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(\overline{y}_{ij.}-\overline{y}_{i..}-\overline{y}_{.j.}+\overline{\overline{y}})^{2} \\ &= S_{AB} - S_{A} - S_{B} \end{split}\end{displaymath}$$

$$\begin{displaymath}\begin{split} S_{AB} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(\overline{y}_{ij.}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m} \frac{T_{ij.}^{ \ 2}}{r} -CT \end{split}\end{displaymath}$$

$$\begin{displaymath}\begin{split} S_{E} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(y_{ijk}-\overline{y}_{ij.})^{2} \\ &= S_{T}-S_{AB} \end{split}\end{displaymath}$$

—-

$$\nu_{_{A}} = l-1$$

$$\nu_{_{B}} = m-1$$

$$\nu_{_{A \times B}} = \nu_{_{AB}} - \nu_{_{A}} - \nu_{_{B}} = (l-1)(m-1)$$

$$\nu_{_{AB}} = lm-1$$

$$\nu_{_{E}} = \nu_{_{T}} - \nu_{_{AB}}=lm(r-1)$$

$$\nu_{_{T}} = lmr-1=N-1$$

$$V_{A} = \frac{S_{A}}{\nu_{_{A}}}$$

$$V_{B} = \frac{S_{B}}{\nu_{_{B}}}$$

$$V_{A \times B} = \frac{S_{A \times B}}{\nu_{_{A \times B}}}$$

$$V_{AB} = \frac{S_{AB}}{\nu_{_{AB}}}$$

$$V_{E} = \frac{S_{E}}{\nu_{_{E}}}$$

|| '[요인]' || '[제곱합]' $$SS$$ || '[자유도]' $$DF$$ || '[평균제곱]' $$MS$$ || $$E(MS)$$ || $$F_{0}$$ || '기각치' || '[순변동]' $$ S\acute{} $$ || '[기여율]' $$\rho$$ || |||||||||||||||||| || || $$A$$ || $$S_{_{A}}$$ || $$\nu_{_{A}} = l - 1$$ || $$V_{_{A}} = S_{_{A}} / \nu_{_{A}}$$ || $$\sigma_{_{E}}^{ \ 2} + r \sigma_{_{A \times B}}^{ \ 2} + m r \ \sigma_{_{A}}^{ \ 2}$$ || $$V_{_{A}}/V_{_{A \times B}}$$ || $$F_{1-\alpha}(\nu_{_{A}} \ , \ \nu_{_{A \times B}})$$ || $$S_{_{A}}\acute{}$$ || $$S_{_{A}}\acute{} / S_{_{T}} $$ || || $$B$$ || $$S_{_{B}}$$ || $$\nu_{_{B}} = m - 1$$ || $$V_{_{B}} = S_{_{B}} / \nu_{_{B}}$$ || $$\sigma_{_{E}}^{ \ 2} + l r\ \sigma_{_{B}}^{ \ 2}$$ || $$V_{_{B}}/V_{_{E}}$$ || $$F_{1-\alpha}(\nu_{_{B}} \ , \ \nu_{_{E}})$$ || $$S_{_{B}}\acute{}$$ || $$S_{_{B}}\acute{} / S_{_{T}} $$ || || $$A \times B$$ || $$S_{_{A \times B}}$$ || $$\nu_{_{A \times B}} = (l - 1)(m - 1)$$ || $$V_{_{A \times B}} = S_{_{A \times B}} / \nu_{_{A \times B}}$$ || $$\sigma_{_{E}}^{ \ 2} + r \ \sigma_{_{A \times B}}^{ \ 2}$$ || $$V_{_{A \times B}}/V_{_{E}}$$ || $$F_{1-\alpha}(\nu_{_{A \times B}} \ , \ \nu_{_{E}})$$ || $$S_{_{A \times B}}\acute{}$$ || $$S_{_{A \times B}}\acute{} / S_{_{T}} $$ || || $$E$$ || $$S_{_{E}}$$ || $$\nu_{_{E}} = lm(r - 1)$$ || $$V_{_{E}} = S_{_{E}} / \nu_{_{E}}$$ || $$\sigma_{_{E}}^{ \ 2}$$ || || || $$S_{_{E}}\acute{} = S_{_{T}} - S_{_{A}}\acute{} - S_{_{B}}\acute{} - S_{_{A \times B}}\acute{}$$ || $$S_{_{E}}\acute{} / S_{_{T}} $$ || |||||||||||||||||| || || $$T$$ || $$S_{_{T}}$$ || $$\nu_{_{T}} = lmr - 1$$ || || || || || $$S_{_{T}}$$ || $$1$$ ||

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