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삼원배치법_모수모형_반복없음 [2012/07/25 22:14] moonrepeat [자료의 구조] |
삼원배치법_모수모형_반복없음 [2021/03/10 21:42] |
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| - | ====== 삼원배치법 (모수모형) (반복없음) ====== | ||
| - | ===== 데이터 구조 ===== | ||
| - | [[요인]] $A$는 [[모수인자]] | ||
| - | [[요인]] $B$는 [[모수인자]] | ||
| - | |||
| - | [[요인]] $C$는 [[모수인자]] | ||
| - | |||
| - | $$ y_{ijk} = \mu + a_{i} + b_{j} + c_{k} + (ab)_{ij} + (ac)_{ik} + (bc)_{jk} + e_{ijk} $$ | ||
| - | |||
| - | * $y_{ijk}$ : $A_{i}$ 와 $B_{j}$ , 그리고 $C_{k}$ 에서 얻은 [[측정값]] | ||
| - | * $\mu$ : 실험전체의 [[모평균]] | ||
| - | * $a_{i}$ : $A_{i}$ 가 주는 효과 | ||
| - | * $b_{j}$ : $B_{j}$ 가 주는 효과 | ||
| - | * $c_{k}$ : $C_{k}$ 가 주는 효과 | ||
| - | * $(ab)_{ij}$ : $A_{i}$ 와 $B_{j}$ 의 [[교호작용]] 효과 | ||
| - | * $(ac)_{ik}$ : $A_{i}$ 와 $C_{k}$ 의 [[교호작용]] 효과 | ||
| - | * $(bc)_{jk}$ : $B_{j}$ 와 $C_{k}$ 의 [[교호작용]] 효과 | ||
| - | * $e_{ijk}$ : $A_{i}$ 와 $B_{j}$ , 그리고 $C_{k}$ 에서 얻은 [[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 $( j = 1,2, \cdots ,m )$ | ||
| - | * $k$ : [[인자]] $C$ 의 [[수준]] 수 $( k = 1,2, \cdots ,n )$ | ||
| - | ===== 자료의 구조 ===== | ||
| - | | <|2> [[인자]] $$B$$ | <|2> [[인자]] $$C$$ | | | | [[인자]] $$A$$ | | ||
| - | | $$A_{1}$$ | $$A_{2}$$ | $$\cdots$$ | $$A_{l}$$ | | ||
| - | | <|4> $$B_{1}$$ | $$C_{1}$$ | $$y_{111}$$ | $$y_{211}$$ | $$\cdots$$ | $$y_{l11}$$ | | ||
| - | | $$C_{2}$$ | $$y_{112}$$ | $$y_{212}$$ | $$\cdots$$ | $$y_{l12}$$ | | ||
| - | | $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | | $$\vdots$$ | | ||
| - | | $$C_{n}$$ | $$y_{11n}$$ | $$y_{21n}$$ | $$\cdots$$ | $$y_{l1n}$$ | | ||
| - | | <|4> $$B_{2}$$ | $$C_{1}$$ | $$y_{121}$$ | $$y_{221}$$ | $$\cdots$$ | $$y_{l21}$$ | | ||
| - | | $$C_{2}$$ | $$y_{122}$$ | $$y_{222}$$ | $$\cdots$$ | $$y_{l22}$$ | | ||
| - | | $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | | $$\vdots$$ | | ||
| - | | $$C_{n}$$ | $$y_{12n}$$ | $$y_{22n}$$ | $$\cdots$$ | $$y_{l2n}$$ | | ||
| - | | | $$\vdots$$ | | | | $$\vdots$$ | | ||
| - | | <|4> $$B_{m}$$ | $$C_{1}$$ | $$y_{1m1}$$ | $$y_{2m1}$$ | $$\cdots$$ | $$y_{lm1}$$ | | ||
| - | | $$C_{2}$$ | $$y_{1m2}$$ | $$y_{2m2}$$ | $$\cdots$$ | $$y_{lm2}$$ | | ||
| - | | $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | | $$\vdots$$ | | ||
| - | | $$C_{n}$$ | $$y_{1mn}$$ | $$y_{2mn}$$ | $$\cdots$$ | $$y_{lmn}$$ | | ||
| - | |||
| - | $$AB$$ 2원표 | ||
| - | | <|2> [[인자]] $$B$$ | | | | [[인자]] $$A$$ | <|2> 합계 | | ||
| - | | $$A_{1}$$ | $$A_{2}$$ | $$\cdots$$ | $$A_{l}$$ | | ||
| - | | $$B_{1}$$ | $$T_{11.}$$ | $$T_{21.}$$ | $$\cdots$$ | $$T_{l1.}$$ | $$T_{.1.}$$ | | ||
| - | | $$B_{2}$$ | $$T_{12.}$$ | $$T_{22.}$$ | $$\cdots$$ | $$T_{l2.}$$ | $$T_{.2.}$$ | | ||
| - | | $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | | $$\vdots$$ | $$\vdots$$ | | ||
| - | | $$B_{m}$$ | $$T_{1m.}$$ | $$T_{2m.}$$ | $$\cdots$$ | $$T_{lm.}$$ | $$T_{.m.}$$ | | ||
| - | | 합계 | $$T_{1..}$$ | $$T_{2..}$$ | $$\cdots$$ | $$T_{l..}$$ | $$T$$ | | ||
| - | |||
| - | $$AC$$ 2원표 | ||
| - | | <|2> [[인자]] $$C$$ | | | | [[인자]] $$A$$ | <|2> 합계 | | ||
| - | | $$A_{1}$$ | $$A_{2}$$ | $$\cdots$$ | $$A_{l}$$ | | ||
| - | | $$C_{1}$$ | $$T_{1.1}$$ | $$T_{2.1}$$ | $$\cdots$$ | $$T_{l.1}$$ | $$T_{..1}$$ | | ||
| - | | $$C_{2}$$ | $$T_{1.2}$$ | $$T_{2.2}$$ | $$\cdots$$ | $$T_{l.2}$$ | $$T_{..2}$$ | | ||
| - | | $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | | $$\vdots$$ | $$\vdots$$ | | ||
| - | | $$C_{n}$$ | $$T_{1.n}$$ | $$T_{2.n}$$ | $$\cdots$$ | $$T_{l.n}$$ | $$T_{..n}$$ | | ||
| - | | 합계 | $$T_{1..}$$ | $$T_{2..}$$ | $$\cdots$$ | $$T_{l..}$$ | $$T$$ | | ||
| - | |||
| - | $BC$ 2원표 | ||
| - | ^ [[인자]]\\ $C$ ^ [[인자]] $B$ ^^^^ 합계 | | ||
| - | ^:::^ $$B_{1}$$ ^ $$B_{2}$$ ^ $$\cdots$$ ^ $$B_{m}$$ ^:::| | ||
| - | | $$C_{1}$$ | $$T_{.11}$$ | $$T_{.21}$$ | $$\cdots$$ | $$T_{.m1}$$ | $$T_{..1}$$ | | ||
| - | | $$C_{2}$$ | $$T_{.12}$$ | $$T_{.22}$$ | $$\cdots$$ | $$T_{.m2}$$ | $$T_{..2}$$ | | ||
| - | | $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | | $$\vdots$$ | $$\vdots$$ | | ||
| - | | $$C_{n}$$ | $$T_{.1n}$$ | $$T_{.2n}$$ | $$\cdots$$ | $$T_{.mn}$$ | $$T_{..n}$$ | | ||
| - | ^ 합계 ^ $$T_{.1.}$$ ^ $$T_{.2.}$$ ^ $$\cdots$$ ^ $$T_{.m.}$$ ^ $$T$$ | | ||
| - | |||
| - | | $$T_{i..} = \sum_{j=1}^{m} \sum_{k=1}^{n} y_{ijk}$$ | $$\overline{y}_{i..} = \frac{T_{i..}}{mn}$$ | | ||
| - | | $$T_{.j.} = \sum_{i=1}^{l} \sum_{k=1}^{n} y_{ijk}$$ | $$\overline{y}_{.j.} = \frac{T_{.j.}}{ln}$$ | | ||
| - | | $$T_{..k} = \sum_{i=1}^{l} \sum_{j=1}^{m} y_{ijk}$$ | $$\overline{y}_{..k} = \frac{T_{..k}}{lm}$$ | | ||
| - | | $$T_{ij.} = \sum_{k=1}^{n} y_{ijk}$$ | $$\overline{y}_{ij.} = \frac{T_{ij.}}{n}$$ | | ||
| - | | $$T_{i.k} = \sum_{j=1}^{m} y_{ijk}$$ | $$\overline{y}_{i.k} = \frac{T_{i.k}}{m}$$ | | ||
| - | | $$T_{.jk} = \sum_{i=1}^{l} y_{ijk}$$ | $$\overline{y}_{.jk} = \frac{T_{.jk}}{l}$$ | | ||
| - | | $$T = \sum_{i=1}^{l} \sum_{j=1}^{m} \sum_{k=1}^{n} y_{ijk}$$ | $$\overline{\overline{y}} = \frac{T}{lmn} = \frac{T}{N}$$ | | ||
| - | | $$N = lmn$$ | $$CT = \frac{T^{2}}{lmn} = \frac{T^{2}}{N}$$ | | ||
| - | ===== 제곱합 ===== | ||
| - | 개개의 데이터 $y_{ijk}$와 총평균 $\overline{\overline{y}}$의 차이는 다음과 같이 7부분으로 나뉘어진다. | ||
| - | |||
| - | $$\begin{displaymath}\begin{split} (y_{ijk}-\overline{\overline{y}}) &= (\overline{y}_{i..} - \overline{\overline{y}}) + (\overline{y}_{.j.} - \overline{\overline{y}}) + (\overline{y}_{..k} - \overline{\overline{y}}) \\ &+ (\overline{y}_{ij.} - \overline{y}_{i..} - \overline{y}_{.j.} + \overline{\overline{y}}) + (\overline{y}_{i.k} - \overline{y}_{i..} - \overline{y}_{..k} + \overline{\overline{y}}) + (\overline{y}_{.jk} - \overline{y}_{.j.} - \overline{y}_{..k} + \overline{\overline{y}}) \\ &+ (y_{ijk} - \overline{y}_{ij.} - \overline{y}_{i.k} - \overline{y}_{.jk} + \overline{y}_{i..} + \overline{y}_{.j.} + \overline{y}_{..k} - \overline{\overline{y}}) \end{split}\end{displaymath}$$ | ||
| - | |||
| - | 양변을 제곱한 후에 모든 $i, \ j, \ k$에 대하여 합하면 아래의 등식을 얻을 수 있다. | ||
| - | |||
| - | $$\begin{displaymath}\begin{split} \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(y_{ijk}-\overline{\overline{y}})^{2} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{i..} - \overline{\overline{y}})^{2} + \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{.j.} - \overline{\overline{y}})^{2} + \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{..k} - \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{ij.} - \overline{y}_{i..} - \overline{y}_{.j.} + \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{i.k} - \overline{y}_{i..} - \overline{y}_{..k} + \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{.jk} - \overline{y}_{.j.} - \overline{y}_{..k} + \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(y_{ijk} - \overline{y}_{ij.} - \overline{y}_{i.k} - \overline{y}_{.jk} + \overline{y}_{i..} + \overline{y}_{.j.} + \overline{y}_{..k} - \overline{\overline{y}})^{2} \end{split}\end{displaymath}$$ | ||
| - | |||
| - | 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. | ||
| - | |||
| - | $$\begin{displaymath}\begin{split} S_{T} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(y_{ijk}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}y_{ijk}^{ \ 2} - CT \end{split}\end{displaymath}$$ | ||
| - | |||
| - | $$\begin{displaymath}\begin{split} S_{A} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(y_{i..}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\frac{T_{i..}^{ \ 2}}{mn}-CT \end{split}\end{displaymath}$$ | ||
| - | |||
| - | $$\begin{displaymath}\begin{split} S_{B} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(y_{.j.}-\overline{\overline{y}})^{2} \\ &= \sum_{j=1}^{m}\frac{T_{.j.}^{ \ 2}}{ln}-CT \end{split}\end{displaymath}$$ | ||
| - | |||
| - | $$\begin{displaymath}\begin{split} S_{C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(y_{..k}-\overline{\overline{y}})^{2} \\ &= \sum_{k=1}^{n}\frac{T_{..k}^{ \ 2}}{lm}-CT \end{split}\end{displaymath}$$ | ||
| - | |||
| - | $$\begin{displaymath}\begin{split} S_{A \times B} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{ij.}-\overline{y}_{i..}-\overline{y}_{.j.}+\overline{\overline{y}})^{2} \\ &= S_{AB} - S_{A} - S_{B} \end{split}\end{displaymath}$$ | ||
| - | |||
| - | $$\begin{displaymath}\begin{split} S_{AB} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{ij.}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m} \frac{T_{ij.}^{ \ 2}}{n} -CT \end{split}\end{displaymath}$$ | ||
| - | |||
| - | $$\begin{displaymath}\begin{split} S_{A \times C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{i.k}-\overline{y}_{i..}-\overline{y}_{..k}+\overline{\overline{y}})^{2} \\ &= S_{AC} - S_{A} - S_{C} \end{split}\end{displaymath}$$ | ||
| - | |||
| - | $$\begin{displaymath}\begin{split} S_{AC} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{i.k}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{k=1}^{n} \frac{T_{i.k}^{ \ 2}}{m} -CT \end{split}\end{displaymath}$$ | ||
| - | |||
| - | $$\begin{displaymath}\begin{split} S_{B \times C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{.jk}-\overline{y}_{.j.}-\overline{y}_{..k}+\overline{\overline{y}})^{2} \\ &= S_{BC} - S_{B} - S_{C} \end{split}\end{displaymath}$$ | ||
| - | |||
| - | $$\begin{displaymath}\begin{split} S_{BC} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(\overline{y}_{.jk}-\overline{\overline{y}})^{2} \\ &= \sum_{j=1}^{m}\sum_{k=1}^{n} \frac{T_{.jk}^{ \ 2}}{l} -CT \end{split}\end{displaymath}$$ | ||
| - | |||
| - | $$\begin{displaymath}\begin{split} S_{E} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}(y_{ijk}-\overline{y}_{ij.}-\overline{y}_{i.k}-\overline{y}_{.jk}+\overline{y}_{i..}+\overline{y}_{.j.}+\overline{y}_{..k}-\overline{\overline{y}})^{2} \\ &= S_{T}-(S_{A}+S_{B}+S_{C}+S_{A \times B}+S_{A \times C}+S_{B \times C}) \end{split}\end{displaymath}$$ | ||
| - | ===== 자유도 ===== | ||
| - | $$\nu_{A}=l-1$$ | ||
| - | |||
| - | $$\nu_{B}=m-1$$ | ||
| - | |||
| - | $$\nu_{C}=n-1$$ | ||
| - | |||
| - | $$\nu_{A \times B}=\nu_{A} \times \nu_{B}=(l-1)(m-1)$$ | ||
| - | |||
| - | $$\nu_{A \times C}=\nu_{A} \times \nu_{C}=(l-1)(n-1)$$ | ||
| - | |||
| - | $$\nu_{B \times C}=\nu_{B} \times \nu_{C}=(m-1)(n-1)$$ | ||
| - | |||
| - | $$\nu_{E}=\nu_{T}-(\nu_{A}+\nu_{B}+\nu_{C}+\nu_{A \times B}+\nu_{A \times C}+\nu_{B \times C})=(l-1)(m-1)(n-1)$$ | ||
| - | |||
| - | $$\nu_{T}=lmn-1=N-1$$ | ||
| - | ===== 평균제곱 ===== | ||
| - | $$V_{A}=\frac{S_{A}}{\nu_{A}}$$ | ||
| - | |||
| - | $$V_{B}=\frac{S_{B}}{\nu_{B}}$$ | ||
| - | |||
| - | $$V_{C}=\frac{S_{C}}{\nu_{C}}$$ | ||
| - | |||
| - | $$V_{A \times B}=\frac{S_{A \times B}}{\nu_{A \times B}}$$ | ||
| - | |||
| - | $$V_{AB}=\frac{S_{AB}}{\nu_{AB}}$$ | ||
| - | |||
| - | $$V_{A \times C}=\frac{S_{A \times C}}{\nu_{A \times C}}$$ | ||
| - | |||
| - | $$V_{AC}=\frac{S_{AC}}{\nu_{AC}}$$ | ||
| - | |||
| - | $$V_{B \times C}=\frac{S_{B \times C}}{\nu_{B \times C}}$$ | ||
| - | |||
| - | $$V_{BC}=\frac{S_{BC}}{\nu_{BC}}$$ | ||
| - | |||
| - | $$V_{E}=\frac{S_{E}}{\nu_{E}}$$ | ||
| - | ===== 평균제곱의 기대값 ===== | ||
| - | $$E(V_{A})=\sigma_{E}^{ \ 2} +mn \sigma_{A}^{ \ 2}$$ | ||
| - | |||
| - | $$E(V_{B})=\sigma_{E}^{ \ 2} +ln \sigma_{B}^{ \ 2}$$ | ||
| - | |||
| - | $$E(V_{C})=\sigma_{E}^{ \ 2} +lm \sigma_{C}^{ \ 2}$$ | ||
| - | |||
| - | $$E(V_{A \times B})=\sigma_{E}^{ \ 2} +n \sigma_{A \times B}^{ \ 2}$$ | ||
| - | |||
| - | $$E(V_{A \times C})=\sigma_{E}^{ \ 2} +m \sigma_{A \times C}^{ \ 2}$$ | ||
| - | |||
| - | $$E(V_{B \times C})=\sigma_{E}^{ \ 2} +l \sigma_{A \times B}^{ \ 2}$$ | ||
| - | |||
| - | $$E(V_{E})=\sigma_{E}^{ \ 2}$$ | ||
| - | ===== 분산분석표 ===== | ||
| - | ^ [[요인]] ^ [[제곱합]]\\ $SS$ ^ [[자유도]]\\ $DF$ ^ [[평균제곱]]\\ $MS$ ^ $E(MS)$ ^ $F_{0}$ ^ [[기각치]] ^ [[순변동]]\\ $S\acute{}$ ^ [[기여율]]\\ $\rho$ | | ||
| - | | $$A$$ | $$S_{_{A}}$$ | $$\nu_{_{A}}=l-1$$ | $$V_{_{A}}=S_{_{A}}/\nu_{_{A}}$$ | $$\sigma_{_{E}}^{ \ 2}+mn \ \sigma_{_{A}}^{2}$$ | $$V_{_{A}}/V_{_{E}}$$ | $$F_{1-\alpha}(\nu_{_{A}} \ , \ \nu_{_{E}})$$ | $$S_{_{A}}\acute{}$$ | $$S_{_{A}}\acute{}/S_{_{T}}$$ | | ||
| - | | $$B$$ | $$S_{_{B}}$$ | $$\nu_{_{B}}=m-1$$ | $$V_{_{B}}=S_{_{B}}/\nu_{_{B}}$$ | $$\sigma_{_{E}}^{ \ 2}+ln \ \sigma_{_{B}}^{2}$$ | $$V_{_{B}}/V_{_{E}}$$ | $$F_{1-\alpha}(\nu_{_{B}} \ , \ \nu_{_{E}})$$ | $$S_{_{B}}\acute{}$$ | $$S_{_{B}}\acute{}/S_{_{T}}$$ | | ||
| - | | $$C$$ | $$S_{_{C}}$$ | $$\nu_{_{C}}=n-1$$ | $$V_{_{C}}=S_{_{C}}/\nu_{_{C}}$$ | $$\sigma_{_{E}}^{ \ 2}+lm \ \sigma_{_{C}}^{2}$$ | $$V_{_{C}}/V_{_{E}}$$ | $$F_{1-\alpha}(\nu_{_{C}} \ , \ \nu_{_{E}})$$ | $$S_{_{C}}\acute{}$$ | $$S_{_{C}}\acute{}/S_{_{T}}$$ | | ||
| - | | $$A \times B$$ | $$S_{_{A \times B}}$$ | $$\nu_{_{A \times B}}=(l-1)(m-1)$$ | $$V_{_{A \times B}}=S_{_{A \times B}}/\nu_{_{A \times B}}$$ | $$\sigma_{_{E}}^{ \ 2}+n \ \sigma_{_{A \times B}}^{2}$$ | $$V_{_{A \times B}}/V_{_{E}}$$ | $$F_{1-\alpha}(\nu_{_{A \times B}} \ , \ \nu_{_{E}})$$ | $$S_{_{A \times B}}\acute{}$$ | $$S_{_{A \times B}}\acute{}/S_{_{T}}$$ | | ||
| - | | $$A \times C$$ | $$S_{_{A \times C}}$$ | $$\nu_{_{A \times C}}=(l-1)(n-1)$$ | $$V_{_{A \times C}}=S_{_{A \times C}}/\nu_{_{A \times C}}$$ | $$\sigma_{_{E}}^{ \ 2}+m \ \sigma_{_{A \times C}}^{2}$$ | $$V_{_{A \times C}}/V_{_{E}}$$ | $$F_{1-\alpha}(\nu_{_{A \times C}} \ , \ \nu_{_{E}})$$ | $$S_{_{A \times C}}\acute{}$$ | $$S_{_{A \times C}}\acute{}/S_{_{T}}$$ | | ||
| - | | $$B \times C$$ | $$S_{_{B \times C}}$$ | $$\nu_{_{B \times C}}=(m-1)(n-1)$$ | $$V_{_{B \times C}}=S_{_{B \times C}}/\nu_{_{B \times C}}$$ | $$\sigma_{_{E}}^{ \ 2}+l \ \sigma_{_{B \times C}}^{2}$$ | $$V_{_{B \times C}}/V_{_{E}}$$ | $$F_{1-\alpha}(\nu_{_{B \times C}} \ , \ \nu_{_{E}})$$ | $$S_{_{B \times C}}\acute{}$$ | $$S_{_{B \times C}}\acute{}/S_{_{T}}$$ | | ||
| - | | $$E$$ | $$S_{_{E}}$$ | $$\nu_{_{E}}=(l-1)(m-1)(n-1)$$ | $$V_{_{E}}=S_{_{E}}/\nu_{_{E}}$$ | $$\sigma_{_{E}}^{ \ 2}$$ | | | $$S_{_{E}}\acute{}$$ | $$S_{_{E}}\acute{}/S_{_{T}}$$ | | ||
| - | | $$T$$ | $$S_{_{T}}$$ | $$\nu_{_{T}}=lmn-1$$ | | | | | $$S_{_{T}}$$ | $$1$$ | | ||
| - | ===== 분산분석 ===== | ||
| - | [[인자]] $A$에 대한 [[분산분석]] | ||
| - | |||
| - | $$F_{0}=\frac{V_{_{A}}}{V_{_{E}}}$$ | ||
| - | |||
| - | [[기각역]] : $F_{0} > F_{1-\alpha}(\nu_{_{A}},\nu_{_{E}})$ | ||
| - | |||
| - | ---- | ||
| - | [[인자]] $B$에 대한 [[분산분석]] | ||
| - | |||
| - | $$F_{0}=\frac{V_{_{B}}}{V_{_{E}}}$$ | ||
| - | |||
| - | [[기각역]] : $F_{0} > F_{1-\alpha}(\nu_{_{B}},\nu_{_{E}})$ | ||
| - | |||
| - | ---- | ||
| - | [[인자]] $C$에 대한 [[분산분석]] | ||
| - | |||
| - | $$F_{0}=\frac{V_{_{C}}}{V_{_{E}}}$$ | ||
| - | |||
| - | [[기각역]] : $F_{0} > F_{1-\alpha}(\nu_{_{C}},\nu_{_{E}})$ | ||
| - | |||
| - | ---- | ||
| - | [[인자]] $A , \ B$의 [[교호작용]] 대한 [[분산분석]] | ||
| - | |||
| - | $$F_{0}=\frac{V_{_{A \times B}}}{V_{E}}$$ | ||
| - | |||
| - | [[기각역]] : $F_{0} > F_{1-\alpha}(\nu_{_{A \times B}},\nu_{_{E}})$ | ||
| - | |||
| - | ---- | ||
| - | [[인자]] $A , \ C$의 [[교호작용]] 대한 [[분산분석]] | ||
| - | |||
| - | $$F_{0}=\frac{V_{_{A \times C}}}{V_{E}}$$ | ||
| - | |||
| - | [[기각역]] : $F_{0} > F_{1-\alpha}(\nu_{_{A \times C}},\nu_{_{E}})$ | ||
| - | |||
| - | ---- | ||
| - | [[인자]] $B , \ C$의 [[교호작용]] 대한 [[분산분석]] | ||
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| - | $$F_{0}=\frac{V_{_{B \times C}}}{V_{E}}$$ | ||
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| - | [[기각역]] : $F_{0} > F_{1-\alpha}(\nu_{_{B \times C}},\nu_{_{E}})$ | ||
| - | ===== 각 수준의 모평균의 추정 (주효과만이 유의한 경우) ===== | ||
| - | 주효과인 인자 $A, B, C$만이 유의한 경우 [[교호작용]]들이 모두 오차항에 [[풀링]]되어 버린다. | ||
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| - | (단, $S_{E}\acute{}=S_{E}+S_{A \times B}+S_{A \times C}+S_{B \times C}, \ \nu_{E}\acute{}=\nu_{E}+\nu_{A \times B}+\nu_{A \times C}+\nu_{B \times C}, \ V_{E}\acute{}=S_{E}\acute{}/\nu_{E}\acute{}$이다.) | ||
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| - | [[인자]] $A$의 [[모평균]]에 관한 [[추정]] | ||
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| - | $i$ [[수준]]에서의 [[모평균]] $\mu(A_{i})$의 [[점추정]]값 | ||
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| - | $$\hat{\mu}(A_{i})=\widehat{\mu + a_{i}} = \overline{y}_{i..}$$ | ||
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| - | $i$ [[수준]]에서의 [[모평균]] $\mu(A_{i})$의 $100(1-\alpha) \% $ [[신뢰구간]]은 아래와 같다. | ||
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| - | $\hat{\mu}(A_{i})= \left( \overline{y}_{i..} - t_{\alpha/2}(\nu_{E}\acute{} \ ) \sqrt{\frac{V_{E}\acute{}}{mn}} \ , \ \overline{y}_{i..} + t_{\alpha/2}(\nu_{E}\acute{} \ ) \sqrt{\frac{V_{E}\acute{}}{mn}} \right)$ | ||
| - | |||
| - | ---- | ||
| - | [[인자]] $B$의 [[모평균]]에 관한 [[추정]] | ||
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| - | $j$ [[수준]]에서의 [[모평균]] $\mu(B_{j})$의 [[점추정]]값 | ||
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| - | $\hat{\mu}(B_{j})=\widehat{\mu + b_{j}} = \overline{y}_{.j.}$ | ||
| - | |||
| - | $j$ [[수준]]에서의 [[모평균]] $\mu(B_{j})$의 $100(1-\alpha) \% $ [[신뢰구간]]은 아래와 같다. | ||
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| - | $\hat{\mu}(B_{j})= \left( \overline{y}_{.j.} - t_{\alpha/2}(\nu_{E}\acute{} \ ) \sqrt{\frac{V_{E}\acute{}}{ln}} \ , \ \overline{y}_{.j.} + t_{\alpha/2}(\nu_{E}\acute{} \ ) \sqrt{\frac{V_{E}\acute{}}{ln}} \right)$ | ||
| - | |||
| - | ---- | ||
| - | [[인자]] $C$의 [[모평균]]에 관한 [[추정]] | ||
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| - | $k$ [[수준]]에서의 [[모평균]] $\mu(C_{k})$의 [[점추정]]값 | ||
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| - | $$\hat{\mu}(C_{k})=\widehat{\mu + c_{k}} = \overline{y}_{..k}$$ | ||
| - | |||
| - | $k$ [[수준]]에서의 [[모평균]] $\mu(C_{k})$의 $100(1-\alpha) \% $ [[신뢰구간]]은 아래와 같다. | ||
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| - | $$\hat{\mu}(C_{k})= \left( \overline{y}_{..k} - t_{\alpha/2}(\nu_{E}\acute{} \ ) \sqrt{\frac{V_{E}\acute{}}{lm}} \ , \ \overline{y}_{..k} + t_{\alpha/2}(\nu_{E}\acute{} \ ) \sqrt{\frac{V_{E}\acute{}}{lm}} \right)$$ | ||
| - | |||
| - | ---- | ||
| - | [[인자]] $A$와 $B$ 그리고 $C$의 [[모평균]]에 관한 [[추정]] | ||
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| - | $A$ [[인자]]의 $i$ [[수준]]과 $B$ [[인자]]의 $j$ [[수준]], $C$ [[인자]]의 $k$ [[수준]]에서의 [[모평균]] $\mu(A_{i}B_{j}C_{k})$의 [[점추정]]값 | ||
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| - | $\hat{\mu}(A_{i}B_{j}C_{k})=\widehat{\mu+a_{i}+b_{j}+c_{k}}=\overline{y}_{i..} + \overline{y}_{.j.} + \overline{y}_{..k} - 2 \overline{\overline{y}}$ | ||
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| - | $A$ [[인자]]의 $i$ [[수준]]과 $B$ [[인자]]의 $j$ [[수준]], $C$ [[인자]]의 $k$ [[수준]]에서의 [[모평균]] $\mu(A_{i}B_{j}C_{k})$$ 의   $$100(1-\alpha) \% $ [[신뢰구간]]은 아래와 같다. | ||
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| - | $\hat{\mu}(A_{i}B_{j}C_{k})= \left( (\overline{y}_{i..} + \overline{y}_{.j.} + \overline{y}_{..k} - 2\overline{\overline{y}}) - t_{\alpha/2}(\nu_{E}\acute{} \ )\sqrt{\frac{V_{E}\acute{}}{n_{e}}} \ , \ (\overline{y}_{i..} + \overline{y}_{.j.} + \overline{y}_{..k} - 2\overline{\overline{y}}) - t_{\alpha/2}(\nu_{E}\acute{} \ )\sqrt{\frac{V_{E}\acute{}}{n_{e}}} \right)$ | ||
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| - | 단, $n_{e}$는 [[유효반복수]]이고 $n_{e} = \frac{lmn}{l+m+n-2}$이다. | ||
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| - | ---- | ||
| - | * [[실험계획법]] | ||
| - | * [[삼원배치법]] | ||