$$ y_{ijkp} = \mu + a_{i} + b_{j} + c_{k} + (ab)_{ij} + (ac)_{ik} + (bc)_{jk} + (abc)_{ijk} + e_{ijkp} $$
인자 $B$ | 인자 $C$ | 인자 $A$ | |||
---|---|---|---|---|---|
$$A_{1}$$ | $$A_{2}$$ | $$\cdots$$ | $$A_{l}$$ | ||
$$B_{1}$$ | $$C_{1}$$ | $$y_{1111}$$ | $$y_{2111}$$ | $$\cdots$$ | $$y_{l111}$$ |
$$\vdots$$ | $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | ||
$$y_{111r}$$ | $$y_{211r}$$ | $$\cdots$$ | $$y_{l11r}$$ | ||
$$C_{2}$$ | $$y_{1121}$$ | $$y_{2121}$$ | $$\cdots$$ | $$y_{l121}$$ | |
$$\vdots$$ | $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | ||
$$y_{112r}$$ | $$y_{212r}$$ | $$\cdots$$ | $$y_{l12r}$$ | ||
$$\vdots$$ | $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | ||
$$C_{n}$$ | $$y_{11n1}$$ | $$y_{21n1}$$ | $$\cdots$$ | $$y_{l1n1}$$ | |
$$\vdots$$ | $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | ||
$$y_{11nr}$$ | $$y_{21nr}$$ | $$\cdots$$ | $$y_{l1nr}$$ | ||
$$B_{2}$$ | $$C_{1}$$ | $$y_{1211}$$ | $$y_{2211}$$ | $$\cdots$$ | $$y_{l211}$$ |
$$\vdots$$ | $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | ||
$$y_{121r}$$ | $$y_{221r}$$ | $$\cdots$$ | $$y_{l21r}$$ | ||
$$C_{2}$$ | $$y_{1221}$$ | $$y_{2221}$$ | $$\cdots$$ | $$y_{l221}$$ | |
$$\vdots$$ | $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | ||
$$y_{122r}$$ | $$y_{222r}$$ | $$\cdots$$ | $$y_{l22r}$$ | ||
$$\vdots$$ | $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | ||
$$C_{n}$$ | $$y_{12n1}$$ | $$y_{22n1}$$ | $$\cdots$$ | $$y_{l2n1}$$ | |
$$\vdots$$ | $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | ||
$$y_{12nr}$$ | $$y_{22nr}$$ | $$\cdots$$ | $$y_{l2nr}$$ | ||
$$\vdots$$ | $$\vdots$$ | ||||
$$B_{m}$$ | $$C_{1}$$ | $$y_{1m11}$$ | $$y_{2m11}$$ | $$\cdots$$ | $$y_{lm11}$$ |
$$\vdots$$ | $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | ||
$$y_{1m1r}$$ | $$y_{2m1r}$$ | $$\cdots$$ | $$y_{lm1r}$$ | ||
$$C_{2}$$ | $$y_{1m21}$$ | $$y_{2m21}$$ | $$\cdots$$ | $$y_{lm21}$$ | |
$$\vdots$$ | $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | ||
$$y_{1m2r}$$ | $$y_{2m2r}$$ | $$\cdots$$ | $$y_{lm2r}$$ | ||
$$\vdots$$ | $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | ||
$$C_{n}$$ | $$y_{1mn1}$$ | $$y_{2mn1}$$ | $$\cdots$$ | $$y_{lmn1}$$ | |
$$\vdots$$ | $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | ||
$$y_{1mnr}$$ | $$y_{2mnr}$$ | $$\cdots$$ | $$y_{lmnr}$$ |
$AB$ 2원표
인자 $B$ | 인자 $A$ | 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
인자 $C$ | 인자 $A$ | 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 | $$T_{.1..}$$ | $$T_{.2..}$$ | $$\cdots$$ | $$T_{.m..}$$ | $$T$$ |
$$T_{i...} = \sum_{j=1}^{m} \sum_{k=1}^{n} \sum_{p=1}^{r} y_{ijkp}$$ | $$\overline{y}_{i...} = \frac{T_{i...}}{mnr}$$ |
$$T_{.j..} = \sum_{i=1}^{l} \sum_{k=1}^{n} \sum_{p=1}^{r} y_{ijkp}$$ | $$\overline{y}_{.j..} = \frac{T_{.j..}}{lnr}$$ |
$$T_{..k.} = \sum_{i=1}^{l} \sum_{j=1}^{m} \sum_{p=1}^{r} y_{ijkp}$$ | $$\overline{y}_{..k.} = \frac{T_{..k.}}{lmr}$$ |
$$T_{ij..} = \sum_{k=1}^{n} \sum_{p=1}^{r} y_{ijkp}$$ | $$\overline{y}_{ij..} = \frac{T_{ij..}}{nr}$$ |
$$T_{i.k.} = \sum_{j=1}^{m} \sum_{p=1}^{r} y_{ijkp}$$ | $$\overline{y}_{i.k.} = \frac{T_{i.k.}}{mr}$$ |
$$T_{.jk.} = \sum_{i=1}^{l} \sum_{p=1}^{r} y_{ijkp}$$ | $$\overline{y}_{.jk.} = \frac{T_{.jk.}}{lr}$$ |
$$T_{ijk.} = \sum_{p=1}^{r} y_{ijkp}$$ | $$\overline{y}_{ijk.} = \frac{T_{ijk.}}{r}$$ |
$$T = \sum_{i=1}^{l} \sum_{j=1}^{m} \sum_{k=1}^{n} \sum_{p=1}^{r} y_{ijkp}$$ | $$\overline{\overline{y}} = \frac{T}{lmnr} = \frac{T}{N}$$ |
$$N = lmnr$$ | $$CT = \frac{T^{2}}{lmnr} = \frac{T^{2}}{N}$$ |
개개의 데이터 $y_{ijkp}$와 총평균 $\overline{\overline{y}}$의 차이는 다음과 같이 8부분으로 나뉘어진다.
$$\begin{displaymath}\begin{split} (y_{ijkp}-\overline{\overline{y}}) &= (\overline{y}_{i...} - \overline{\overline{y}}) + (\overline{y}_{.j..} - \overline{\overline{y}}) + (\overline{y}_{..k.} - \overline{\overline{y}}) \\ &+ (\overline{y}_{ij..} - \overline{y}_{i...} - \overline{y}_{.j..} + \overline{\overline{y}}) + (\overline{y}_{i.k.} - \overline{y}_{i...} - \overline{y}_{..k.} + \overline{\overline{y}}) + (\overline{y}_{.jk.} - \overline{y}_{.j..} - \overline{y}_{..k.} + \overline{\overline{y}}) \\ &+ (y_{ijk.} - \overline{y}_{ij..} - \overline{y}_{i.k.} - \overline{y}_{.jk.} + \overline{y}_{i...} + \overline{y}_{.j..} + \overline{y}_{..k.} - \overline{\overline{y}}) \\ &+ (y_{ijkp}-\overline{y}_{ijk.}) \end{split}\end{displaymath}$$ 양변을 제곱한 후에 모든 $i, \ j, \ k, \ p$에 대하여 합하면 아래의 등식을 얻을 수 있다.
$$\begin{displaymath}\begin{split} \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijkp}-\overline{\overline{y}})^{2} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{i...} - \overline{\overline{y}})^{2} + \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{.j..} - \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{..k.} - \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{ij..} - \overline{y}_{i...} - \overline{y}_{.j..} + \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{i.k.} - \overline{y}_{i...} - \overline{y}_{..k.} + \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{.jk.} - \overline{y}_{.j..} - \overline{y}_{..k.} + \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijk.} - \overline{y}_{ij..} - \overline{y}_{i.k.} - \overline{y}_{.jk.} + \overline{y}_{i...} + \overline{y}_{.j..} + \overline{y}_{..k.} - \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijkp}-\overline{y}_{ijk.})^{2} \end{split}\end{displaymath}$$ 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]], $A, \ B, \ C$의 [[교호작용]]의 변동, [[오차변동]]인 $S_{A}$, $S_{B}$, $S_{C}$, $S_{A \times B}$, $S_{A \times C}$, $S_{B \times C}$, $S_{A \times B \times C}$, $S_{E}$가 된다.
$$\begin{displaymath}\begin{split} S_{T} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijkp}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}y_{ijkp}^{ \ 2} - CT \end{split}\end{displaymath}$$ $$\begin{displaymath}\begin{split} S_{A} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{i...}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\frac{T_{i...}^{ \ 2}}{mnr}-CT \end{split}\end{displaymath}$$ $$\begin{displaymath}\begin{split} S_{B} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{.j..}-\overline{\overline{y}})^{2} \\ &= \sum_{j=1}^{m}\frac{T_{.j..}^{ \ 2}}{lnr}-CT \end{split}\end{displaymath}$$ $$\begin{displaymath}\begin{split} S_{C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{..k.}-\overline{\overline{y}})^{2} \\ &= \sum_{k=1}^{n}\frac{T_{..k.}^{ \ 2}}{lmr}-CT \end{split}\end{displaymath}$$ $$\begin{displaymath}\begin{split} S_{A \times B} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{ij..}-\overline{y}_{i...}-\overline{y}_{.j..}+\overline{\overline{y}})^{2} \\ &= S_{AB} - S_{A} - S_{B} \end{split}\end{displaymath}$$ $$\begin{displaymath}\begin{split} S_{AB} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{ij..}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m} \frac{T_{ij..}^{ \ 2}}{nr} -CT \end{split}\end{displaymath}$$ $$\begin{displaymath}\begin{split} S_{A \times C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{i.k.}-\overline{y}_{i...}-\overline{y}_{..k.}+\overline{\overline{y}})^{2} \\ &= S_{AC} - S_{A} - S_{C} \end{split}\end{displaymath}$$ $$\begin{displaymath}\begin{split} S_{AC} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{i.k.}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{k=1}^{n} \frac{T_{i.k.}^{ \ 2}}{mr} -CT \end{split}\end{displaymath}$$ $$\begin{displaymath}\begin{split} S_{B \times C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{.jk.}-\overline{y}_{.j..}-\overline{y}_{..k.}+\overline{\overline{y}})^{2} \\ &= S_{BC} - S_{B} - S_{C} \end{split}\end{displaymath}$$ $$\begin{displaymath}\begin{split} S_{BC} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{.jk.}-\overline{\overline{y}})^{2} \\ &= \sum_{j=1}^{m}\sum_{k=1}^{n} \frac{T_{.jk.}^{ \ 2}}{lr} -CT \end{split}\end{displaymath}$$ $$\begin{displaymath}\begin{split} S_{A \times B \times C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijk.}-\overline{y}_{ij..}-\overline{y}_{i.k.}-\overline{y}_{.jk.}+\overline{y}_{i...}+\overline{y}_{.j..}+\overline{y}_{..k.}-\overline{\overline{y}})^{2} \\ &= S_{ABC}-(S_{A}+S_{B}+S_{C}+S_{A \times B}+S_{A \times C}+S_{B \times C}) \end{split}\end{displaymath}$$ $$\begin{displaymath}\begin{split} S_{ABC} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijk.}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\frac{T_{ijk.}^{ \ 2}}{r} -CT \end{split}\end{displaymath}$$ $$\begin{displaymath}\begin{split} S_{E} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijkp}-\overline{\overline{y}})^{2} \\ &= S_{T} - S_{ABC} \end{split}\end{displaymath}$$ ===== 자유도 ===== $$\nu_{A}=l-1$$ $$\nu_{B}=m-1$$ $$\nu_{C}=n-1$$ $$\nu_{A \times B}=\nu_{A} \times \nu_{B}=(l-1)(m-1)$$ $$\nu_{A \times C}=\nu_{A} \times \nu_{C}=(l-1)(n-1)$$ $$\nu_{B \times C}=\nu_{B} \times \nu_{C}=(m-1)(n-1)$$ $$\nu_{A \times B \times C}=\nu_{A} \times \nu_{B} \times \nu_{C} =(l-1)(m-1)(n-1)$$ $$\nu_{E}=\nu_{T}-(\nu_{A}+\nu_{B}+\nu_{C}+\nu_{A \times B}+\nu_{A \times C}+\nu_{B \times C}+\nu_{A \times B \times C})=lmn(r-1)$$ $$\nu_{T}=lmnr-1=N-1$$ ===== 평균제곱 ===== $$V_{A}=\frac{S_{A}}{\nu_{A}}$$ $$V_{B}=\frac{S_{B}}{\nu_{B}}$$ $$V_{C}=\frac{S_{C}}{\nu_{C}}$$ $$V_{A \times B}=\frac{S_{A \times B}}{\nu_{A \times B}}$$ $$V_{A \times C}=\frac{S_{A \times C}}{\nu_{A \times C}}$$ $$V_{B \times C}=\frac{S_{B \times C}}{\nu_{B \times C}}$$ $$V_{A \times B \times C}=\frac{S_{A \times B \times C}}{\nu_{A \times B \times C}}$$ $$V_{E}=\frac{S_{E}}{\nu_{E}}$$ ===== 평균제곱의 기대값 ===== $$E(V_{A})=\sigma_{E}^{ \ 2} +mnr \sigma_{A}^{ \ 2}$$ $$E(V_{B})=\sigma_{E}^{ \ 2} +lnr \sigma_{B}^{ \ 2}$$ $$E(V_{C})=\sigma_{E}^{ \ 2} +lmr \sigma_{C}^{ \ 2}$$ $$E(V_{A \times B})=\sigma_{E}^{ \ 2} +nr \sigma_{A \times B}^{ \ 2}$$ $$E(V_{A \times C})=\sigma_{E}^{ \ 2} +mr \sigma_{A \times C}^{ \ 2}$$ $$E(V_{B \times C})=\sigma_{E}^{ \ 2} +lr \sigma_{A \times B}^{ \ 2}$$ $$E(V_{A \times B \times C})=\sigma_{E}^{ \ 2} +r \sigma_{A \times B \times C}^{ \ 2}$$ $$E(V_{E})=\sigma_{E}^{ \ 2}$$ ===== 분산분석표 ===== ^ [[요인]] ^ [[제곱합]]\\ $SS$ ^ [[자유도]]\\ $DF$ ^ [[평균제곱]]\\ $MS$ ^ $E(MS)$ ^ $F_{0}$ ^ [[기각치]] ^ [[순변동]]\\ $S\acute{}$ ^ [[기여율]]\\ $\rho$ |
$$A$$ | $$S_{_{A}}$$ | $$\nu_{_{A}}=l-1$$ | $$V_{_{A}}=S_{_{A}}/\nu_{_{A}}$$ | $$\sigma_{_{E}}^{ \ 2}+mnr \ \sigma_{_{A}}^{2}$$ | $$V_{_{A}}/V_{_{E}}$$ | $$F_{1-\alpha}(\nu_{_{A}} \ , \ \nu_{_{E}})$$ | $$S_{_{A}}\acute{}$$ | $$S_{_{A}}\acute{}/S_{_{T}}$$ |
$$B$$ | $$S_{_{B}}$$ | $$\nu_{_{B}}=m-1$$ | $$V_{_{B}}=S_{_{B}}/\nu_{_{B}}$$ | $$\sigma_{_{E}}^{ \ 2}+lnr \ \sigma_{_{B}}^{2}$$ | $$V_{_{B}}/V_{_{E}}$$ | $$F_{1-\alpha}(\nu_{_{B}} \ , \ \nu_{_{E}})$$ | $$S_{_{B}}\acute{}$$ | $$S_{_{B}}\acute{}/S_{_{T}}$$ |
$$C$$ | $$S_{_{C}}$$ | $$\nu_{_{C}}=n-1$$ | $$V_{_{C}}=S_{_{C}}/\nu_{_{C}}$$ | $$\sigma_{_{E}}^{ \ 2}+lmr \ \sigma_{_{C}}^{2}$$ | $$V_{_{C}}/V_{_{E}}$$ | $$F_{1-\alpha}(\nu_{_{C}} \ , \ \nu_{_{E}})$$ | $$S_{_{C}}\acute{}$$ | $$S_{_{C}}\acute{}/S_{_{T}}$$ |
$$A \times B$$ | $$S_{_{A \times B}}$$ | $$\nu_{_{A \times B}}=(l-1)(m-1)$$ | $$V_{_{A \times B}}=S_{_{A \times B}}/\nu_{_{A \times B}}$$ | $$\sigma_{_{E}}^{ \ 2}+nr \ \sigma_{_{A \times B}}^{2}$$ | $$V_{_{A \times B}}/V_{_{E}}$$ | $$F_{1-\alpha}(\nu_{_{A \times B}} \ , \ \nu_{_{E}})$$ | $$S_{_{A \times B}}\acute{}$$ | $$S_{_{A \times B}}\acute{}/S_{_{T}}$$ |
$$A \times C$$ | $$S_{_{A \times C}}$$ | $$\nu_{_{A \times C}}=(l-1)(n-1)$$ | $$V_{_{A \times C}}=S_{_{A \times C}}/\nu_{_{A \times C}}$$ | $$\sigma_{_{E}}^{ \ 2}+mr \ \sigma_{_{A \times C}}^{2}$$ | $$V_{_{A \times C}}/V_{_{E}}$$ | $$F_{1-\alpha}(\nu_{_{A \times C}} \ , \ \nu_{_{E}})$$ | $$S_{_{A \times C}}\acute{}$$ | $$S_{_{A \times C}}\acute{}/S_{_{T}}$$ |
$$B \times C$$ | $$S_{_{B \times C}}$$ | $$\nu_{_{B \times C}}=(m-1)(n-1)$$ | $$V_{_{B \times C}}=S_{_{B \times C}}/\nu_{_{B \times C}}$$ | $$\sigma_{_{E}}^{ \ 2}+lr \ \sigma_{_{B \times C}}^{2}$$ | $$V_{_{B \times C}}/V_{_{E}}$$ | $$F_{1-\alpha}(\nu_{_{B \times C}} \ , \ \nu_{_{E}})$$ | $$S_{_{B \times C}}\acute{}$$ | $$S_{_{B \times C}}\acute{}/S_{_{T}}$$ |
$$A \times B \times C$$ | $$S_{_{A \times B \times C}}$$ | $$\nu_{_{A \times B \times C}}=(l-1)(m-1)(n-1)$$ | $$V_{_{A \times B \times C}}=S_{_{A \times B \times C}}/\nu_{_{A \times B \times C}}$$ | $$\sigma_{_{E}}^{ \ 2}+r \ \sigma_{_{A \times B \times C}}^{ \ 2}$$ | $$V_{_{A \times B \times C}}/V_{_{E}}$$ | $$F_{1-\alpha}(\nu_{_{A \times B \times C}} \ , \ \nu_{_{E}})$$ | $$S_{_{A \times B \times C}}\acute{}$$ | $$S_{_{A \times B \times C}}\acute{}/S_{_{T}}$$ |
$$E$$ | $$S_{_{E}}$$ | $$\nu_{_{E}}=lmn(r-1)$$ | $$V_{_{E}}=S_{_{E}}/\nu_{_{E}}$$ | $$\sigma_{_{E}}^{ \ 2}$$ | $$S_{_{E}}\acute{}$$ | $$S_{_{E}}\acute{}/S_{_{T}}$$ | ||
$$T$$ | $$S_{_{T}}$$ | $$\nu_{_{T}}=lmnr-1$$ | $$S_{_{T}}$$ | $$1$$ |
기각역 : $F_{0} > F_{1-\alpha}(\nu_{_{A}},\nu_{_{E}})$
기각역 : $F_{0} > F_{1-\alpha}(\nu_{_{B}},\nu_{_{E}})$
기각역 : $F_{0} > F_{1-\alpha}(\nu_{_{C}},\nu_{_{E}})$
기각역 : $F_{0} > F_{1-\alpha}(\nu_{_{A \times B}},\nu_{_{E}})$
기각역 : $F_{0} > F_{1-\alpha}(\nu_{_{A \times C}},\nu_{_{E}})$
기각역 : $F_{0} > F_{1-\alpha}(\nu_{_{B \times C}},\nu_{_{E}})$
인자 $A , \ B , \ C$의 교호작용에 대한 분산분석
기각역 : $F_{0} > F_{1-\alpha}(\nu_{A \times B \times C},\nu_{_{E}})$
주효과인 인자 $A, B, C$만이 유의한 경우 교호작용들이 모두 오차항에 풀링되어 버린다.
(단, $S_{E}\acute{}=S_{E}+S_{A \times B}+S_{A \times C}+S_{B \times C}+S_{A \times B \times C}, \ \nu_{E}\acute{}=\nu_{E}+\nu_{A \times B}+\nu_{A \times C}+\nu_{B \times C}+\nu_{A \times B \times C}, \ V_{E}\acute{}=S_{E}\acute{}/\nu_{E}\acute{}$이다.)
$i$ 수준에서의 모평균 $\mu(A_{i})$의 점추정값
$i$ 수준에서의 모평균 $\mu(A_{i})$의 $100(1-\alpha) \% $ 신뢰구간은 아래와 같다.
$j$ 수준에서의 모평균 $\mu(B_{j})$의 점추정값
$j$ 수준에서의 모평균 $\mu(B_{j})$의 $100(1-\alpha) \%$ 신뢰구간은 아래와 같다.
$k$ 수준에서의 모평균 $\mu(C_{k})$의 점추정값
$k$ 수준에서의 모평균 $\mu(C_{k})$의 $100(1-\alpha) \% $ 신뢰구간은 아래와 같다.
인자 $A$와 $B$ 그리고 $C$의 모평균에 관한 추정
$A$ 인자의 $i$ 수준과 $B$ 인자의 $j$ 수준, $C$ 인자의 $k$ 수준에서의 모평균 $\mu(A_{i}B_{j}C_{k})$의 점추정값
$A$ 인자의 $i$ 수준과 $B$ 인자의 $j$ 수준, $C$ 인자의 $k$ 수준에서의 모평균 $\mu(A_{i}B_{j}C_{k})$의 $100(1-\alpha) \% $ 신뢰구간은 아래와 같다.
단, $n_{e}$는 유효반복수이고 $n_{e} = \frac{lmnr}{l+m+n-2}$이다.