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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 ||<|2> [인자] $$C$$ |||||||| [인자] $$B$$ ||<|2> 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 총편균   $$\overline{\overline{y}}$$ 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
$$F_{0}=\frac{V_{_{A}}}{V_{_{E}}}$$
기각역 : $F_{0} > F_{1-\alpha}(\nu_{_{A}},\nu_{_{E}})$
$$F_{0}=\frac{V_{_{B}}}{V_{_{E}}}$$
기각역 : $F_{0} > F_{1-\alpha}(\nu_{_{B}},\nu_{_{E}})$
$$F_{0}=\frac{V_{_{C}}}{V_{_{E}}}$$
기각역 : $F_{0} > F_{1-\alpha}(\nu_{_{C}},\nu_{_{E}})$
$$F_{0}=\frac{V_{_{A \times B}}}{V_{E}}$$
기각역 : $F_{0} > F_{1-\alpha}(\nu_{_{A \times B}},\nu_{_{E}})$
$$F_{0}=\frac{V_{_{A \times C}}}{V_{E}}$$
기각역 : $F_{0} > F_{1-\alpha}(\nu_{_{A \times C}},\nu_{_{E}})$
$$F_{0}=\frac{V_{_{B \times C}}}{V_{E}}$$
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.
(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.)
* '[인자]   $$A$$ 의 [모평균]에 관한 [추정]
'
$$i$$ [수준]에서의 [모평균]   $$\mu(A_{i})$$ 의 [점추정]값
$$\hat{\mu}(A_{i})=\widehat{\mu + a_{i}} = \overline{y}_{i..}$$
$$i$$ [수준]에서의 [모평균]   $$\mu(A_{i})$$ 의   $$100(1-\alpha) \% $$ [신뢰구간]은 아래와 같다.
$$\hat{\mu}(A_{i})= \left( \overline{y}_{i..} - t_{\alpha/2}(\nu_{E}\acute{} \ ) \sqrt{\frac{V_{E}\acute{}}{mn}} \ , \ \overline{y}_{i..} + t_{\alpha/2}(\nu_{E}\acute{} \ ) \sqrt{\frac{V_{E}\acute{}}{mn}} \right)$$
—-
* '[인자]   $$B$$ 의 [모평균]에 관한 [추정]
'
$$j$$ [수준]에서의 [모평균]   $$\mu(B_{j})$$ 의 [점추정]값
$$\hat{\mu}(B_{j})=\widehat{\mu + b_{j}} = \overline{y}_{.j.}$$
$$j$$ [수준]에서의 [모평균]   $$\mu(B_{j})$$ 의   $$100(1-\alpha) \% $$ [신뢰구간]은 아래와 같다.
$$\hat{\mu}(B_{j})= \left( \overline{y}_{.j.} - t_{\alpha/2}(\nu_{E}\acute{} \ ) \sqrt{\frac{V_{E}\acute{}}{ln}} \ , \ \overline{y}_{.j.} + t_{\alpha/2}(\nu_{E}\acute{} \ ) \sqrt{\frac{V_{E}\acute{}}{ln}} \right)$$
—-
* '[인자]   $$C$$ 의 [모평균]에 관한 [추정]
'
$$k$$ [수준]에서의 [모평균]   $$\mu(C_{k})$$ 의 [점추정]값
$$\hat{\mu}(C_{k})=\widehat{\mu + c_{k}} = \overline{y}_{..k}$$
$$k$$ [수준]에서의 [모평균]   $$\mu(C_{k})$$ 의   $$100(1-\alpha) \% $$ [신뢰구간]은 아래와 같다.
$$\hat{\mu}(C_{k})= \left( \overline{y}_{..k} - t_{\alpha/2}(\nu_{E}\acute{} \ ) \sqrt{\frac{V_{E}\acute{}}{lm}} \ , \ \overline{y}_{..k} + t_{\alpha/2}(\nu_{E}\acute{} \ ) \sqrt{\frac{V_{E}\acute{}}{lm}} \right)$$
—-
* '[인자]   $$A$$ 와   $$B$$   그리고   $$C$$ 의 [모평균]에 관한 [추정]
'
$$A$$ [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
$$\hat{\mu}(A_{i}B_{j}C_{k})=\widehat{\mu+a_{i}+b_{j}+c_{k}}=\overline{y}_{i..} + \overline{y}_{.j.} + \overline{y}_{..k} - 2 \overline{\overline{y}}$$
$$A$$ [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.
$$\hat{\mu}(A_{i}B_{j}C_{k})= \left( (\overline{y}_{i..} + \overline{y}_{.j.} + \overline{y}_{..k} - 2\overline{\overline{y}}) - t_{\alpha/2}(\nu_{E}\acute{} \ )\sqrt{\frac{V_{E}\acute{}}{n_{e}}} \ , \ (\overline{y}_{i..} + \overline{y}_{.j.} + \overline{y}_{..k} - 2\overline{\overline{y}}) - t_{\alpha/2}(\nu_{E}\acute{} \ )\sqrt{\frac{V_{E}\acute{}}{n_{e}}} \right)$$
단,   $$n_{e}$$ 는 [유효반복수]이고   $$n_{e} = \frac{lmn}{l+m+n-2}$$ 이다.