이원배치법 (혼합모형) (반복있음)

데이터 구조

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 측정값오차 ( $e_{ijk} \sim N(0, \sigma_{E}^{ \ 2})$ 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 $( j = 1,2, \cdots ,r )$

자료의 구조

인자
$B$
인자 $A$ 합계 평균
$$A_{1}$$ $$A_{2}$$ $$\cdots$$ $$A_{l}$$
$$B_{1}$$ $$y_{111}$$ $$T_{11.}$$ $$y_{211}$$ $$T_{21.}$$ $$\cdots$$ $$y_{l11}$$ $$T_{l1.}$$ $$T_{.1.}$$ $$\overline{y}_{.1.}$$
$$y_{112}$$ $$y_{212}$$ $$y_{l12}$$
$$\vdots$$ $$\overline{y}_{11.}$$ $$\vdots$$ $$\overline{y}_{21.}$$ $$\vdots$$ $$\overline{y}_{l1.}$$
$$y_{11r}$$ $$y_{21r}$$ $$y_{l1r}$$
$$B_{2}$$ $$y_{121}$$ $$T_{12.}$$ $$y_{221}$$ $$T_{22.}$$ $$\cdots$$ $$y_{l21}$$ $$T_{l2.}$$ $$T_{.2.}$$ $$\overline{y}_{.2.}$$
$$y_{122}$$ $$y_{222}$$ $$y_{l22}$$
$$\vdots$$ $$\overline{y}_{12.}$$ $$\vdots$$ $$\overline{y}_{22.}$$ $$\vdots$$ $$\overline{y}_{l2.}$$
$$y_{12r}$$ $$y_{22r}$$ $$y_{l2r}$$
$$\vdots$$ $$\vdots$$ $$\vdots$$ $$\vdots$$ $$\vdots$$ $$\vdots$$ $$\vdots$$
$$B_{m}$$ $$y_{1m1}$$ $$T_{1m.}$$ $$y_{2m1}$$ $$T_{2m.}$$ $$\cdots$$ $$y_{lm1}$$ $$T_{lm.}$$ $$T_{.m.}$$ $$\overline{y}_{.m.}$$
$$y_{1m2}$$ $$y_{2m2}$$ $$y_{lm2}$$
$$\vdots$$ $$\overline{y}_{1m.}$$ $$\vdots$$ $$\overline{y}_{2m.}$$ $$\vdots$$ $$\overline{y}_{lm.}$$
$$y_{1mr}$$ $$y_{2mr}$$ $$y_{lmr}$$
합계 $$T_{1..}$$ $$T_{2..}$$ $$\cdots$$ $$T_{l..}$$ $$T$$
평균 $$\overline{y}_{1..}$$ $$\overline{y}_{2..}$$ $$\cdots$$ $$\overline{y}_{l..}$$ $$\overline{\overline{y}}$$
$$T_{i..} = \sum_{j=1}^{m} \sum_{k=1}^{r} y_{ijk}$$ $$\overline{y}_{i..} = \frac{T_{i..}}{mr}$$
$$T_{.j.} = \sum_{i=1}^{l} \sum_{k=1}^{r} y_{ijk}$$ $$\overline{y}_{.j.} = \frac{T_{.j.}}{lr}$$
$$T_{ij.} = \sum_{k=1}^{r} y_{ijk}$$ $$\overline{y}_{ij.} = \frac{T_{ij.}}{r}$$
$$T = \sum_{i=1}^{l} \sum_{j=1}^{m} \sum_{k=1}^{r} y_{ijk}$$ $$\overline{\overline{y}} = \frac{T}{lmr} = \frac{T}{N}$$
$$N = lmr$$ $$CT = \frac{T^{2}}{lmr} = \frac{T^{2}}{N}$$

제곱합

개개의 데이터 $y_{ijk}$와 총평균 $\overline{\overline{y}}$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.

$$\begin{displaymath}\begin{split} S_{T} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(y_{ijk}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}y_{ijk}^{ \ 2} - CT \end{split}\end{displaymath}$$ $$\begin{displaymath}\begin{split} S_{A} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(y_{i..}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\frac{T_{i..}^{ \ 2}}{mr}-CT \end{split}\end{displaymath}$$ $$\begin{displaymath}\begin{split} S_{B} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(y_{.j.}-\overline{\overline{y}})^{2} \\ &= \sum_{j=1}^{m}\frac{T_{.j.}^{ \ 2}}{lr}-CT \end{split}\end{displaymath}$$ $$\begin{displaymath}\begin{split} S_{A \times B} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(\overline{y}_{ij.}-\overline{y}_{i..}-\overline{y}_{.j.}+\overline{\overline{y}})^{2} \\ &= S_{AB} - S_{A} - S_{B} \end{split}\end{displaymath}$$ $$\begin{displaymath}\begin{split} S_{AB} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(\overline{y}_{ij.}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m} \frac{T_{ij.}^{ \ 2}}{r} -CT \end{split}\end{displaymath}$$ $$\begin{displaymath}\begin{split} S_{E} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(y_{ijk}-\overline{y}_{ij.})^{2} \\ &= S_{T}-S_{AB} \end{split}\end{displaymath}$$ ===== 자유도 ===== $$\nu_{_{A}} = l-1$$ $$\nu_{_{B}} = m-1$$ $$\nu_{_{A \times B}} = \nu_{_{AB}} - \nu_{_{A}} - \nu_{_{B}} = (l-1)(m-1)$$ $$\nu_{_{AB}} = lm-1$$ $$\nu_{_{E}} = \nu_{_{T}} - \nu_{_{AB}}=lm(r-1)$$ $$\nu_{_{T}} = lmr-1=N-1$$ ===== 평균제곱 ===== $$V_{A} = \frac{S_{A}}{\nu_{_{A}}}$$ $$V_{B} = \frac{S_{B}}{\nu_{_{B}}}$$ $$V_{A \times B} = \frac{S_{A \times B}}{\nu_{_{A \times B}}}$$ $$V_{AB} = \frac{S_{AB}}{\nu_{_{AB}}}$$ $$V_{E} = \frac{S_{E}}{\nu_{_{E}}}$$ ===== 분산분석표 ===== ^ [[요인]] ^ [[제곱합]]\\ $SS$ ^ [[자유도]]\\ $DF$ ^ [[평균제곱]]\\ $MS$ ^ $E(MS)$ ^ $F_{0}$ ^ 기각치 ^ [[순변동]]\\ $S\acute{}$ ^ [[기여율]]\\ $\rho$ |

$$A$$ $$S_{_{A}}$$ $$\nu_{_{A}} = l - 1$$ $$V_{_{A}} = S_{_{A}} / \nu_{_{A}}$$ $$\sigma_{_{E}}^{ \ 2} + r \sigma_{_{A \times B}}^{ \ 2} + m r \ \sigma_{_{A}}^{ \ 2}$$ $$V_{_{A}}/V_{_{A \times B}}$$ $$F_{1-\alpha}(\nu_{_{A}} \ , \ \nu_{_{A \times B}})$$ $$S_{_{A}}\acute{}$$ $$S_{_{A}}\acute{} / S_{_{T}} $$
$$B$$ $$S_{_{B}}$$ $$\nu_{_{B}} = m - 1$$ $$V_{_{B}} = S_{_{B}} / \nu_{_{B}}$$ $$\sigma_{_{E}}^{ \ 2} + l r\ \sigma_{_{B}}^{ \ 2}$$ $$V_{_{B}}/V_{_{E}}$$ $$F_{1-\alpha}(\nu_{_{B}} \ , \ \nu_{_{E}})$$ $$S_{_{B}}\acute{}$$ $$S_{_{B}}\acute{} / S_{_{T}} $$
$$A \times B$$ $$S_{_{A \times B}}$$ $$\nu_{_{A \times B}} = (l - 1)(m - 1)$$ $$V_{_{A \times B}} = S_{_{A \times B}} / \nu_{_{A \times B}}$$ $$\sigma_{_{E}}^{ \ 2} + r \ \sigma_{_{A \times B}}^{ \ 2}$$ $$V_{_{A \times B}}/V_{_{E}}$$ $$F_{1-\alpha}(\nu_{_{A \times B}} \ , \ \nu_{_{E}})$$ $$S_{_{A \times B}}\acute{}$$ $$S_{_{A \times B}}\acute{} / S_{_{T}} $$
$$E$$ $$S_{_{E}}$$ $$\nu_{_{E}} = lm(r - 1)$$ $$V_{_{E}} = S_{_{E}} / \nu_{_{E}}$$ $$\sigma_{_{E}}^{ \ 2}$$ $$S_{_{E}}\acute{} = S_{_{T}} - S_{_{A}}\acute{} - S_{_{B}}\acute{} - S_{_{A \times B}}\acute{}$$ $$S_{_{E}}\acute{} / S_{_{T}} $$
$$T$$ $$S_{_{T}}$$ $$\nu_{_{T}} = lmr - 1$$ $$S_{_{T}}$$ $$1$$