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이원배치법_혼합모형_반복있음 [2012/07/26 09:54] moonrepeat [[제곱합]] |
이원배치법_혼합모형_반복있음 [2021/03/10 21:42] |
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| - | ====== 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 [[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 $( j = 1,2, \cdots ,r )$ | ||
| - | ===== 자료의 구조 ===== | ||
| - | ||<|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] |||| $$\overline{y}_{1..}$$ |||| $$\overline{y}_{2..}$$ || $$\cdots$$ |||| $$\overline{y}_{l..}$$ || || $$\overline{\overline{y}}$$ || | ||
| - | |||
| - | || $$T_{i..} = \sum_{j=1}^{m} \sum_{k=1}^{r} y_{ijk}$$ || $$\overline{y}_{i..} = \frac{T_{i..}}{mr}$$ || | ||
| - | || $$T_{.j.} = \sum_{i=1}^{l} \sum_{k=1}^{r} y_{ijk}$$ || $$\overline{y}_{.j.} = \frac{T_{.j.}}{lr}$$ || | ||
| - | || $$T_{ij.} = \sum_{k=1}^{r} y_{ijk}$$ || $$\overline{y}_{ij.} = \frac{T_{ij.}}{r}$$ || | ||
| - | || $$T = \sum_{i=1}^{l} \sum_{j=1}^{m} \sum_{k=1}^{r} y_{ijk}$$ || $$\overline{\overline{y}} = \frac{T}{lmr} = \frac{T}{N}$$ || | ||
| - | || $$N = lmr$$ || $$CT = \frac{T^{2}}{lmr} = \frac{T^{2}}{N}$$ || | ||
| - | ---- | ||
| - | ===== 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]]\\ $\rho$ | | ||
| - | | $$A$$ | $$S_{_{A}}$$ | $$\nu_{_{A}} = l - 1$$ | $$V_{_{A}} = S_{_{A}} / \nu_{_{A}}$$ | $$\sigma_{_{E}}^{ \ 2} + r \sigma_{_{A \times B}}^{ \ 2} + m r \ \sigma_{_{A}}^{ \ 2}$$ | $$V_{_{A}}/V_{_{A \times B}}$$ | $$F_{1-\alpha}(\nu_{_{A}} \ , \ \nu_{_{A \times B}})$$ | $$S_{_{A}}\acute{}$$ | $$S_{_{A}}\acute{} / S_{_{T}} $$ | | ||
| - | | $$B$$ | $$S_{_{B}}$$ | $$\nu_{_{B}} = m - 1$$ | $$V_{_{B}} = S_{_{B}} / \nu_{_{B}}$$ | $$\sigma_{_{E}}^{ \ 2} + l r\ \sigma_{_{B}}^{ \ 2}$$ | $$V_{_{B}}/V_{_{E}}$$ | $$F_{1-\alpha}(\nu_{_{B}} \ , \ \nu_{_{E}})$$ | $$S_{_{B}}\acute{}$$ | $$S_{_{B}}\acute{} / S_{_{T}} $$ | | ||
| - | | $$A \times B$$ | $$S_{_{A \times B}}$$ | $$\nu_{_{A \times B}} = (l - 1)(m - 1)$$ | $$V_{_{A \times B}} = S_{_{A \times B}} / \nu_{_{A \times B}}$$ | $$\sigma_{_{E}}^{ \ 2} + r \ \sigma_{_{A \times B}}^{ \ 2}$$ | $$V_{_{A \times B}}/V_{_{E}}$$ | $$F_{1-\alpha}(\nu_{_{A \times B}} \ , \ \nu_{_{E}})$$ | $$S_{_{A \times B}}\acute{}$$ | $$S_{_{A \times B}}\acute{} / S_{_{T}} $$ | | ||
| - | | $$E$$ | $$S_{_{E}}$$ | $$\nu_{_{E}} = lm(r - 1)$$ | $$V_{_{E}} = S_{_{E}} / \nu_{_{E}}$$ | $$\sigma_{_{E}}^{ \ 2}$$ | | | $$S_{_{E}}\acute{} = S_{_{T}} - S_{_{A}}\acute{} - S_{_{B}}\acute{} - S_{_{A \times B}}\acute{}$$ | $$S_{_{E}}\acute{} / S_{_{T}} $$ | | ||
| - | | $$T$$ | $$S_{_{T}}$$ | $$\nu_{_{T}} = lmr - 1$$ | | | | | $$S_{_{T}}$$ | $$1$$ | | ||
| - | |||
| - | ---- | ||
| - | * [[실험계획법]] | ||
| - | * [[이원배치법]] | ||