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삼원배치법_모수모형_반복없음 [2012/07/25 21:53] moonrepeat [각 수준의 모평균의 추정 (주효과만이 유의한 경우)] |
삼원배치법_모수모형_반복없음 [2021/03/10 21:42] (현재) |
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| * $k$ : [[인자]] $C$ 의 [[수준]] 수 $( k = 1,2, \cdots ,n )$ | * $k$ : [[인자]] $C$ 의 [[수준]] 수 $( k = 1,2, \cdots ,n )$ | ||
| ===== 자료의 구조 ===== | ===== 자료의 구조 ===== | ||
| - | ||<|2> [인자] $$B$$ ||<|2> [인자] $$C$$ |||||||| [인자] $$A$$ || | + | ^ [[인자]]\\ $B$ ^ [[인자]]\\ $C$ ^ [[인자]] $A$ |||| |
| - | || $$A_{1}$$ || $$A_{2}$$ || $$\cdots$$ || $$A_{l}$$ || | + | ^:::^:::^ $$A_{1}$$ ^ $$A_{2}$$ ^ $$\cdots$$ ^ $$A_{l}$$ | |
| - | |||||||||||| || | + | ^ $$B_{1}$$ ^ $$C_{1}$$ | $$y_{111}$$ | $$y_{211}$$ | $$\cdots$$ | $$y_{l11}$$ | |
| - | ||<|4> $$B_{1}$$ || $$C_{1}$$ || $$y_{111}$$ || $$y_{211}$$ || $$\cdots$$ || $$y_{l11}$$ || | + | ^:::^ $$C_{2}$$ | $$y_{112}$$ | $$y_{212}$$ | $$\cdots$$ | $$y_{l12}$$ | |
| - | || $$C_{2}$$ || $$y_{112}$$ || $$y_{212}$$ || $$\cdots$$ || $$y_{l12}$$ || | + | ^:::^ $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | | $$\vdots$$ | |
| - | || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || || $$\vdots$$ || | + | ^:::^ $$C_{n}$$ | $$y_{11n}$$ | $$y_{21n}$$ | $$\cdots$$ | $$y_{l1n}$$ | |
| - | || $$C_{n}$$ || $$y_{11n}$$ || $$y_{21n}$$ || $$\cdots$$ || $$y_{l1n}$$ || | + | ^ $$B_{2}$$ ^ $$C_{1}$$ | $$y_{121}$$ | $$y_{221}$$ | $$\cdots$$ | $$y_{l21}$$ | |
| - | ||<|4> $$B_{2}$$ || $$C_{1}$$ || $$y_{121}$$ || $$y_{221}$$ || $$\cdots$$ || $$y_{l21}$$ || | + | ^:::^ $$C_{2}$$ | $$y_{122}$$ | $$y_{222}$$ | $$\cdots$$ | $$y_{l22}$$ | |
| - | || $$C_{2}$$ || $$y_{122}$$ || $$y_{222}$$ || $$\cdots$$ || $$y_{l22}$$ || | + | ^:::^ $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | | $$\vdots$$ | |
| - | || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || || $$\vdots$$ || | + | ^:::^ $$C_{n}$$ | $$y_{12n}$$ | $$y_{22n}$$ | $$\cdots$$ | $$y_{l2n}$$ | |
| - | || $$C_{n}$$ || $$y_{12n}$$ || $$y_{22n}$$ || $$\cdots$$ || $$y_{l2n}$$ || | + | ^ $$\vdots$$ || $$\vdots$$ |||| |
| - | |||| $$\vdots$$ |||||||| $$\vdots$$ || | + | ^ $$B_{m}$$ ^ $$C_{1}$$ | $$y_{1m1}$$ | $$y_{2m1}$$ | $$\cdots$$ | $$y_{lm1}$$ | |
| - | ||<|4> $$B_{m}$$ || $$C_{1}$$ || $$y_{1m1}$$ || $$y_{2m1}$$ || $$\cdots$$ || $$y_{lm1}$$ || | + | ^:::^ $$C_{2}$$ | $$y_{1m2}$$ | $$y_{2m2}$$ | $$\cdots$$ | $$y_{lm2}$$ | |
| - | || $$C_{2}$$ || $$y_{1m2}$$ || $$y_{2m2}$$ || $$\cdots$$ || $$y_{lm2}$$ || | + | ^:::^ $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | | $$\vdots$$ | |
| - | || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || || $$\vdots$$ || | + | ^:::^ $$C_{n}$$ | $$y_{1mn}$$ | $$y_{2mn}$$ | $$\cdots$$ | $$y_{lmn}$$ | |
| - | || $$C_{n}$$ || $$y_{1mn}$$ || $$y_{2mn}$$ || $$\cdots$$ || $$y_{lmn}$$ || | + | |
| - | $$AB$$ 2원표 | + | $AB$ 2원표 |
| - | ||<|2> [인자] $$B$$ |||||||| [인자] $$A$$ ||<|2> 합계 || | + | ^ [[인자]]\\ $B$ ^ [[인자]] $A$ ^^^^ 합계 | |
| - | || $$A_{1}$$ || $$A_{2}$$ || $$\cdots$$ || $$A_{l}$$ || | + | ^:::^ $$A_{1}$$ ^ $$A_{2}$$ ^ $$\cdots$$ ^ $$A_{l}$$ ^:::| |
| - | |||||||||||| || | + | | $$B_{1}$$ | $$T_{11.}$$ | $$T_{21.}$$ | $$\cdots$$ | $$T_{l1.}$$ | $$T_{.1.}$$ | |
| - | || $$B_{1}$$ || $$T_{11.}$$ || $$T_{21.}$$ || $$\cdots$$ || $$T_{l1.}$$ || $$T_{.1.}$$ || | + | | $$B_{2}$$ | $$T_{12.}$$ | $$T_{22.}$$ | $$\cdots$$ | $$T_{l2.}$$ | $$T_{.2.}$$ | |
| - | || $$B_{2}$$ || $$T_{12.}$$ || $$T_{22.}$$ || $$\cdots$$ || $$T_{l2.}$$ || $$T_{.2.}$$ || | + | | $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | | $$\vdots$$ | $$\vdots$$ | |
| - | || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || || $$\vdots$$ || $$\vdots$$ || | + | | $$B_{m}$$ | $$T_{1m.}$$ | $$T_{2m.}$$ | $$\cdots$$ | $$T_{lm.}$$ | $$T_{.m.}$$ | |
| - | || $$B_{m}$$ || $$T_{1m.}$$ || $$T_{2m.}$$ || $$\cdots$$ || $$T_{lm.}$$ || $$T_{.m.}$$ || | + | ^ 합계 ^ $$T_{1..}$$ ^ $$T_{2..}$$ ^ $$\cdots$$ ^ $$T_{l..}$$ ^ $$T$$ | |
| - | |||||||||||| || | + | |
| - | || 합계 || $$T_{1..}$$ || $$T_{2..}$$ || $$\cdots$$ || $$T_{l..}$$ || $$T$$ || | + | |
| - | $$AC$$ 2원표 | + | $AC$ 2원표 |
| - | ||<|2> [인자] $$C$$ |||||||| [인자] $$A$$ ||<|2> 합계 || | + | ^ [[인자]]\\ $C$ ^ [[인자]] $A$ ^^^^ 합계 | |
| - | || $$A_{1}$$ || $$A_{2}$$ || $$\cdots$$ || $$A_{l}$$ || | + | ^:::^ $$A_{1}$$ ^ $$A_{2}$$ ^ $$\cdots$$ ^ $$A_{l}$$ ^:::| |
| - | |||||||||||| || | + | | $$C_{1}$$ | $$T_{1.1}$$ | $$T_{2.1}$$ | $$\cdots$$ | $$T_{l.1}$$ | $$T_{..1}$$ | |
| - | || $$C_{1}$$ || $$T_{1.1}$$ || $$T_{2.1}$$ || $$\cdots$$ || $$T_{l.1}$$ || $$T_{..1}$$ || | + | | $$C_{2}$$ | $$T_{1.2}$$ | $$T_{2.2}$$ | $$\cdots$$ | $$T_{l.2}$$ | $$T_{..2}$$ | |
| - | || $$C_{2}$$ || $$T_{1.2}$$ || $$T_{2.2}$$ || $$\cdots$$ || $$T_{l.2}$$ || $$T_{..2}$$ || | + | | $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | | $$\vdots$$ | $$\vdots$$ | |
| - | || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || || $$\vdots$$ || $$\vdots$$ || | + | | $$C_{n}$$ | $$T_{1.n}$$ | $$T_{2.n}$$ | $$\cdots$$ | $$T_{l.n}$$ | $$T_{..n}$$ | |
| - | || $$C_{n}$$ || $$T_{1.n}$$ || $$T_{2.n}$$ || $$\cdots$$ || $$T_{l.n}$$ || $$T_{..n}$$ || | + | ^ 합계 ^ $$T_{1..}$$ ^ $$T_{2..}$$ ^ $$\cdots$$ ^ $$T_{l..}$$ ^ $$T$$ | |
| - | |||||||||||| || | + | |
| - | || 합계 || $$T_{1..}$$ || $$T_{2..}$$ || $$\cdots$$ || $$T_{l..}$$ || $$T$$ || | + | |
| - | $$BC$$ 2원표 | + | $BC$ 2원표 |
| - | ||<|2> [인자] $$C$$ |||||||| [인자] $$B$$ ||<|2> 합계 || | + | ^ [[인자]]\\ $C$ ^ [[인자]] $B$ ^^^^ 합계 | |
| - | || $$B_{1}$$ || $$B_{2}$$ || $$\cdots$$ || $$B_{m}$$ || | + | ^:::^ $$B_{1}$$ ^ $$B_{2}$$ ^ $$\cdots$$ ^ $$B_{m}$$ ^:::| |
| - | |||||||||||| || | + | | $$C_{1}$$ | $$T_{.11}$$ | $$T_{.21}$$ | $$\cdots$$ | $$T_{.m1}$$ | $$T_{..1}$$ | |
| - | || $$C_{1}$$ || $$T_{.11}$$ || $$T_{.21}$$ || $$\cdots$$ || $$T_{.m1}$$ || $$T_{..1}$$ || | + | | $$C_{2}$$ | $$T_{.12}$$ | $$T_{.22}$$ | $$\cdots$$ | $$T_{.m2}$$ | $$T_{..2}$$ | |
| - | || $$C_{2}$$ || $$T_{.12}$$ || $$T_{.22}$$ || $$\cdots$$ || $$T_{.m2}$$ || $$T_{..2}$$ || | + | | $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | | $$\vdots$$ | $$\vdots$$ | |
| - | || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || || $$\vdots$$ || $$\vdots$$ || | + | | $$C_{n}$$ | $$T_{.1n}$$ | $$T_{.2n}$$ | $$\cdots$$ | $$T_{.mn}$$ | $$T_{..n}$$ | |
| - | || $$C_{n}$$ || $$T_{.1n}$$ || $$T_{.2n}$$ || $$\cdots$$ || $$T_{.mn}$$ || $$T_{..n}$$ || | + | ^ 합계 ^ $$T_{.1.}$$ ^ $$T_{.2.}$$ ^ $$\cdots$$ ^ $$T_{.m.}$$ ^ $$T$$ | |
| - | |||||||||||| || | + | |
| - | || 합계 || $$T_{.1.}$$ || $$T_{.2.}$$ || $$\cdots$$ || $$T_{.m.}$$ || $$T$$ || | + | |
| - | || $$T_{i..} = \sum_{j=1}^{m} \sum_{k=1}^{n} y_{ijk}$$ || $$\overline{y}_{i..} = \frac{T_{i..}}{mn}$$ || | + | | $$T_{i..} = \sum_{j=1}^{m} \sum_{k=1}^{n} y_{ijk}$$ | $$\overline{y}_{i..} = \frac{T_{i..}}{mn}$$ | |
| - | || $$T_{.j.} = \sum_{i=1}^{l} \sum_{k=1}^{n} y_{ijk}$$ || $$\overline{y}_{.j.} = \frac{T_{.j.}}{ln}$$ || | + | | $$T_{.j.} = \sum_{i=1}^{l} \sum_{k=1}^{n} y_{ijk}$$ | $$\overline{y}_{.j.} = \frac{T_{.j.}}{ln}$$ | |
| - | || $$T_{..k} = \sum_{i=1}^{l} \sum_{j=1}^{m} y_{ijk}$$ || $$\overline{y}_{..k} = \frac{T_{..k}}{lm}$$ || | + | | $$T_{..k} = \sum_{i=1}^{l} \sum_{j=1}^{m} y_{ijk}$$ | $$\overline{y}_{..k} = \frac{T_{..k}}{lm}$$ | |
| - | || $$T_{ij.} = \sum_{k=1}^{n} y_{ijk}$$ || $$\overline{y}_{ij.} = \frac{T_{ij.}}{n}$$ || | + | | $$T_{ij.} = \sum_{k=1}^{n} y_{ijk}$$ | $$\overline{y}_{ij.} = \frac{T_{ij.}}{n}$$ | |
| - | || $$T_{i.k} = \sum_{j=1}^{m} y_{ijk}$$ || $$\overline{y}_{i.k} = \frac{T_{i.k}}{m}$$ || | + | | $$T_{i.k} = \sum_{j=1}^{m} y_{ijk}$$ | $$\overline{y}_{i.k} = \frac{T_{i.k}}{m}$$ | |
| - | || $$T_{.jk} = \sum_{i=1}^{l} y_{ijk}$$ || $$\overline{y}_{.jk} = \frac{T_{.jk}}{l}$$ || | + | | $$T_{.jk} = \sum_{i=1}^{l} y_{ijk}$$ | $$\overline{y}_{.jk} = \frac{T_{.jk}}{l}$$ | |
| - | || $$T = \sum_{i=1}^{l} \sum_{j=1}^{m} \sum_{k=1}^{n} y_{ijk}$$ || $$\overline{\overline{y}} = \frac{T}{lmn} = \frac{T}{N}$$ || | + | | $$T = \sum_{i=1}^{l} \sum_{j=1}^{m} \sum_{k=1}^{n} y_{ijk}$$ | $$\overline{\overline{y}} = \frac{T}{lmn} = \frac{T}{N}$$ | |
| - | || $$N = lmn$$ || $$CT = \frac{T^{2}}{lmn} = \frac{T^{2}}{N}$$ || | + | | $$N = lmn$$ | $$CT = \frac{T^{2}}{lmn} = \frac{T^{2}}{N}$$ | |
| ===== 제곱합 ===== | ===== 제곱합 ===== | ||
| - | 개개의 데이터   $$y_{ijk}$$ 와 총편균   $$\overline{\overline{y}}$$ 의 차이는 다음과 같이 7부분으로 나뉘어진다. | + | 개개의 데이터 $y_{ijk}$와 총평균 $\overline{\overline{y}}$의 차이는 다음과 같이 7부분으로 나뉘어진다. |
| - | $$\begin{displaymath}\begin{split} (y_{ijk}-\overline{\overline{y}}) &= (\overline{y}_{i..} - \overline{\overline{y}}) + (\overline{y}_{.j.} - \overline{\overline{y}}) + (\overline{y}_{..k} - \overline{\overline{y}}) \\ &+ (\overline{y}_{ij.} - \overline{y}_{i..} - \overline{y}_{.j.} + \overline{\overline{y}}) + (\overline{y}_{i.k} - \overline{y}_{i..} - \overline{y}_{..k} + \overline{\overline{y}}) + (\overline{y}_{.jk} - \overline{y}_{.j.} - \overline{y}_{..k} + \overline{\overline{y}}) \\ &+ (y_{ijk} - \overline{y}_{ij.} - \overline{y}_{i.k} - \overline{y}_{.jk} + \overline{y}_{i..} + \overline{y}_{.j.} + \overline{y}_{..k} - \overline{\overline{y}}) \end{split}\end{displaymath}$$ | + | $$\begin{displaymath}\begin{split} (y_{ijk}-\overline{\overline{y}}) &= (\overline{y}_{i..} - \overline{\overline{y}}) + (\overline{y}_{.j.} - \overline{\overline{y}}) + (\overline{y}_{..k} - \overline{\overline{y}}) \\ &+ (\overline{y}_{ij.} - \overline{y}_{i..} - \overline{y}_{.j.} + \overline{\overline{y}}) + (\overline{y}_{i.k} - \overline{y}_{i..} - \overline{y}_{..k} + \overline{\overline{y}}) + (\overline{y}_{.jk} - \overline{y}_{.j.} - \overline{y}_{..k} + \overline{\overline{y}}) \\ &+ (y_{ijk} - \overline{y}_{ij.} - \overline{y}_{i.k} - \overline{y}_{.jk} + \overline{y}_{i..} + \overline{y}_{.j.} + \overline{y}_{..k} - \overline{\overline{y}}) \end{split}\end{displaymath}$$ |
| - | 양변을 제곱한 후에 모든   $$i, \ j, \ k$$ 에 대하여 합하면 아래의 등식을 얻을 수 있다. | + | 양변을 제곱한 후에 모든 $i, \ j, \ k$에 대하여 합하면 아래의 등식을 얻을 수 있다. |
| - | $$\begin{displaymath}\begin{split} \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(y_{ijk}-\overline{\overline{y}})^{2} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{i..} - \overline{\overline{y}})^{2} + \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{.j.} - \overline{\overline{y}})^{2} + \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{..k} - \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{ij.} - \overline{y}_{i..} - \overline{y}_{.j.} + \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{i.k} - \overline{y}_{i..} - \overline{y}_{..k} + \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{.jk} - \overline{y}_{.j.} - \overline{y}_{..k} + \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(y_{ijk} - \overline{y}_{ij.} - \overline{y}_{i.k} - \overline{y}_{.jk} + \overline{y}_{i..} + \overline{y}_{.j.} + \overline{y}_{..k} - \overline{\overline{y}})^{2} \end{split}\end{displaymath}$$ | + | $$\begin{displaymath}\begin{split} \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(y_{ijk}-\overline{\overline{y}})^{2} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{i..} - \overline{\overline{y}})^{2} + \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{.j.} - \overline{\overline{y}})^{2} + \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{..k} - \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{ij.} - \overline{y}_{i..} - \overline{y}_{.j.} + \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{i.k} - \overline{y}_{i..} - \overline{y}_{..k} + \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{.jk} - \overline{y}_{.j.} - \overline{y}_{..k} + \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(y_{ijk} - \overline{y}_{ij.} - \overline{y}_{i.k} - \overline{y}_{.jk} + \overline{y}_{i..} + \overline{y}_{.j.} + \overline{y}_{..k} - \overline{\overline{y}})^{2} \end{split}\end{displaymath}$$ |
| - | 위 식에서 왼쪽 항은 총변동 $$S_{T}$$ 이고, 오른쪽 항은 차례대로   $$A$$ 의 [변동],   $$B$$ 의 [변동],   $$C$$ 의 [변동],   $$A, \ B$$ 의 [교호작용]의 변동,   $$A, \ C$$ 의 [교호작용]의 변동,   $$B, \ C$$ 의 [교호작용]의 변동, [오차변동]인   $$S_{A}$$ , $$S_{B}$$ , $$S_{C}$$ , $$S_{A \times B}$$ , $$S_{A \times C}$$ , $$S_{B \times C}$$ , $$S_{E}$$ 가 된다. | + | 위 식에서 왼쪽 항은 총변동 $S_{T}$이고, 오른쪽 항은 차례대로 $A$의 [[변동]], $B$의 [[변동]], $C$의 [[변동]], $A, \ B$의 [[교호작용]]의 변동, $A, \ C$의 [[교호작용]]의 변동, $B, \ C$의 [[교호작용]]의 변동, [[오차변동]]인 $S_{A}$, $S_{B}$, $S_{C}$, $S_{A \times B}$, $S_{A \times C}$, $S_{B \times C}$, $S_{E}$가 된다. |
| + | $$\begin{displaymath}\begin{split} S_{T} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(y_{ijk}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}y_{ijk}^{ \ 2} - CT \end{split}\end{displaymath}$$ | ||
| - | $$\begin{displaymath}\begin{split} S_{T} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(y_{ijk}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}y_{ijk}^{ \ 2} - CT \end{split}\end{displaymath}$$ | + | $$\begin{displaymath}\begin{split} S_{A} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(y_{i..}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\frac{T_{i..}^{ \ 2}}{mn}-CT \end{split}\end{displaymath}$$ |
| - | $$\begin{displaymath}\begin{split} S_{A} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(y_{i..}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\frac{T_{i..}^{ \ 2}}{mn}-CT \end{split}\end{displaymath}$$ | + | $$\begin{displaymath}\begin{split} S_{B} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(y_{.j.}-\overline{\overline{y}})^{2} \\ &= \sum_{j=1}^{m}\frac{T_{.j.}^{ \ 2}}{ln}-CT \end{split}\end{displaymath}$$ |
| - | $$\begin{displaymath}\begin{split} S_{B} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(y_{.j.}-\overline{\overline{y}})^{2} \\ &= \sum_{j=1}^{m}\frac{T_{.j.}^{ \ 2}}{ln}-CT \end{split}\end{displaymath}$$ | + | $$\begin{displaymath}\begin{split} S_{C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(y_{..k}-\overline{\overline{y}})^{2} \\ &= \sum_{k=1}^{n}\frac{T_{..k}^{ \ 2}}{lm}-CT \end{split}\end{displaymath}$$ |
| - | $$\begin{displaymath}\begin{split} S_{C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(y_{..k}-\overline{\overline{y}})^{2} \\ &= \sum_{k=1}^{n}\frac{T_{..k}^{ \ 2}}{lm}-CT \end{split}\end{displaymath}$$ | + | $$\begin{displaymath}\begin{split} S_{A \times B} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{ij.}-\overline{y}_{i..}-\overline{y}_{.j.}+\overline{\overline{y}})^{2} \\ &= S_{AB} - S_{A} - S_{B} \end{split}\end{displaymath}$$ |
| - | $$\begin{displaymath}\begin{split} S_{A \times B} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{ij.}-\overline{y}_{i..}-\overline{y}_{.j.}+\overline{\overline{y}})^{2} \\ &= S_{AB} - S_{A} - S_{B} \end{split}\end{displaymath}$$ | + | $$\begin{displaymath}\begin{split} S_{AB} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{ij.}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m} \frac{T_{ij.}^{ \ 2}}{n} -CT \end{split}\end{displaymath}$$ |
| - | $$\begin{displaymath}\begin{split} S_{AB} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{ij.}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m} \frac{T_{ij.}^{ \ 2}}{n} -CT \end{split}\end{displaymath}$$ | + | $$\begin{displaymath}\begin{split} S_{A \times C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{i.k}-\overline{y}_{i..}-\overline{y}_{..k}+\overline{\overline{y}})^{2} \\ &= S_{AC} - S_{A} - S_{C} \end{split}\end{displaymath}$$ |
| - | $$\begin{displaymath}\begin{split} S_{A \times C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{i.k}-\overline{y}_{i..}-\overline{y}_{..k}+\overline{\overline{y}})^{2} \\ &= S_{AC} - S_{A} - S_{C} \end{split}\end{displaymath}$$ | + | $$\begin{displaymath}\begin{split} S_{AC} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{i.k}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{k=1}^{n} \frac{T_{i.k}^{ \ 2}}{m} -CT \end{split}\end{displaymath}$$ |
| - | $$\begin{displaymath}\begin{split} S_{AC} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{i.k}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{k=1}^{n} \frac{T_{i.k}^{ \ 2}}{m} -CT \end{split}\end{displaymath}$$ | + | $$\begin{displaymath}\begin{split} S_{B \times C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{.jk}-\overline{y}_{.j.}-\overline{y}_{..k}+\overline{\overline{y}})^{2} \\ &= S_{BC} - S_{B} - S_{C} \end{split}\end{displaymath}$$ |
| - | $$\begin{displaymath}\begin{split} S_{B \times C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{.jk}-\overline{y}_{.j.}-\overline{y}_{..k}+\overline{\overline{y}})^{2} \\ &= S_{BC} - S_{B} - S_{C} \end{split}\end{displaymath}$$ | + | $$\begin{displaymath}\begin{split} S_{BC} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{.jk}-\overline{\overline{y}})^{2} \\ &= \sum_{j=1}^{m}\sum_{k=1}^{n} \frac{T_{.jk}^{ \ 2}}{l} -CT \end{split}\end{displaymath}$$ |
| - | $$\begin{displaymath}\begin{split} S_{BC} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{.jk}-\overline{\overline{y}})^{2} \\ &= \sum_{j=1}^{m}\sum_{k=1}^{n} \frac{T_{.jk}^{ \ 2}}{l} -CT \end{split}\end{displaymath}$$ | + | $$\begin{displaymath}\begin{split} S_{E} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(y_{ijk}-\overline{y}_{ij.}-\overline{y}_{i.k}-\overline{y}_{.jk}+\overline{y}_{i..}+\overline{y}_{.j.}+\overline{y}_{..k}-\overline{\overline{y}})^{2} \\ &= S_{T}-(S_{A}+S_{B}+S_{C}+S_{A \times B}+S_{A \times C}+S_{B \times C}) \end{split}\end{displaymath}$$ |
| - | + | ||
| - | $$\begin{displaymath}\begin{split} S_{E} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(y_{ijk}-\overline{y}_{ij.}-\overline{y}_{i.k}-\overline{y}_{.jk}+\overline{y}_{i..}+\overline{y}_{.j.}+\overline{y}_{..k}-\overline{\overline{y}})^{2} \\ &= S_{T}-(S_{A}+S_{B}+S_{C}+S_{A \times B}+S_{A \times C}+S_{B \times C}) \end{split}\end{displaymath}$$ | + | |
| ===== 자유도 ===== | ===== 자유도 ===== | ||
| $$\nu_{A}=l-1$$ | $$\nu_{A}=l-1$$ | ||