삼원배치법 (모수모형) (반복있음)

[요인]&nbsp&nbsp $$A$$ 는 [모수인자]

[요인]&nbsp&nbsp $$B$$ 는 [모수인자]

[요인]&nbsp&nbsp $$C$$ 는 [모수인자]

$$ y_{ijkp} = \mu + a_{i} + b_{j} + c_{k} + (ab)_{ij} + (ac)_{ik} + (bc)_{jk} + (abc)_{ijk} + e_{ijkp} $$

 $$y_{ijkp}$$ &nbsp&nbsp : &nbsp&nbsp $$A_{i}$$ 와&nbsp&nbsp $$B_{j}$$ &nbsp&nbsp그리고&nbsp&nbsp $$C_{k}$$ 에서 얻은&nbsp&nbsp $$p$$ 번째 [측정값]

 $$\mu$$ &nbsp&nbsp : 실험전체의 [모평균]

 $$a_{i}$$ &nbsp&nbsp : &nbsp&nbsp $$A_{i}$$ 가 주는 효과

 $$b_{j}$$ &nbsp&nbsp : &nbsp&nbsp $$B_{j}$$ 가 주는 효과

 $$c_{k}$$ &nbsp&nbsp : &nbsp&nbsp $$C_{k}$$ 가 주는 효과

 $$(ab)_{ij}$$ &nbsp&nbsp : &nbsp&nbsp $$A_{i}$$ 와&nbsp&nbsp $$B_{j}$$ 의 [교호작용] 효과

 $$(ac)_{ik}$$ &nbsp&nbsp : &nbsp&nbsp $$A_{i}$$ 와&nbsp&nbsp $$C_{k}$$ 의 [교호작용] 효과

 $$(bc)_{jk}$$ &nbsp&nbsp : &nbsp&nbsp $$B_{j}$$ 와&nbsp&nbsp $$C_{k}$$ 의 [교호작용] 효과

 $$(abc)_{ijk}$$ &nbsp&nbsp : &nbsp&nbsp $$A_{i}$$ 와&nbsp&nbsp $$B_{J}$$ &nbsp&nbsp그리고&nbsp&nbsp $$C_{k}$$ 의 [교호작용] 효과

 $$e_{ijkp}$$ &nbsp&nbsp : &nbsp&nbsp $$A_{i}$$ 와&nbsp&nbsp $$B_{j}$$ &nbsp&nbsp그리고&nbsp&nbsp $$C_{k}$$ 에서 얻은&nbsp&nbsp $$p$$ 번째 [측정값]의 [오차]  ( $$e_{ijkp} \sim N(0, \sigma_{E}^{ \ 2})$$ 이고 서로 [독립])

  $$i$$ &nbsp&nbsp : 인자&nbsp&nbsp $$A$$ 의 [수준] 수&nbsp&nbsp $$( i = 1,2, \cdots ,l )$$

  $$j$$ &nbsp&nbsp : 인자&nbsp&nbsp $$B$$ 의 [수준] 수&nbsp&nbsp $$( j = 1,2, \cdots ,m )$$

  $$k$$ &nbsp&nbsp : 인자&nbsp&nbsp $$C$$ 의 [수준] 수&nbsp&nbsp $$( k = 1,2, \cdots ,n )$$

  $$p$$ &nbsp&nbsp : 실험의 [반복] 수&nbsp&nbsp $$( p = 1,2, \cdots ,r )$$

—-

||<|2> [인자] $$B$$ ||<|2> [인자] $$C$$ |||||||| [인자] $$A$$ || || $$A_{1}$$ || $$A_{2}$$ || $$\cdots$$ || $$A_{l}$$ || |||||||||||| || ||<|10> $$B_{1}$$ ||<|3> $$C_{1}$$ || $$y_{1111}$$ || $$y_{2111}$$ || $$\cdots$$ || $$y_{l111}$$ || || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || || $$y_{111r}$$ || $$y_{211r}$$ || $$\cdots$$ || $$y_{l11r}$$ || ||<|3> $$C_{2}$$ || $$y_{1121}$$ || $$y_{2121}$$ || $$\cdots$$ || $$y_{l121}$$ || || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || || $$y_{112r}$$ || $$y_{212r}$$ || $$\cdots$$ || $$y_{l12r}$$ || || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || || $$\vdots$$ || ||<|3> $$C_{n}$$ || $$y_{11n1}$$ || $$y_{21n1}$$ || $$\cdots$$ || $$y_{l1n1}$$ || || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || || $$y_{11nr}$$ || $$y_{21nr}$$ || $$\cdots$$ || $$y_{l1nr}$$ || ||<|10> $$B_{2}$$ ||<|3> $$C_{1}$$ || $$y_{1211}$$ || $$y_{2211}$$ || $$\cdots$$ || $$y_{l211}$$ || || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || || $$y_{121r}$$ || $$y_{221r}$$ || $$\cdots$$ || $$y_{l21r}$$ || ||<|3> $$C_{2}$$ || $$y_{1221}$$ || $$y_{2221}$$ || $$\cdots$$ || $$y_{l221}$$ || || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || || $$y_{122r}$$ || $$y_{222r}$$ || $$\cdots$$ || $$y_{l22r}$$ || || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || || $$\vdots$$ || ||<|3> $$C_{n}$$ || $$y_{12n1}$$ || $$y_{22n1}$$ || $$\cdots$$ || $$y_{l2n1}$$ || || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || || $$y_{12nr}$$ || $$y_{22nr}$$ || $$\cdots$$ || $$y_{l2nr}$$ || |||| $$\vdots$$ |||||||| $$\vdots$$ || ||<|10> $$B_{m}$$ ||<|3> $$C_{1}$$ || $$y_{1m11}$$ || $$y_{2m11}$$ || $$\cdots$$ || $$y_{lm11}$$ || || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || || $$y_{1m1r}$$ || $$y_{2m1r}$$ || $$\cdots$$ || $$y_{lm1r}$$ || ||<|3> $$C_{2}$$ || $$y_{1m21}$$ || $$y_{2m21}$$ || $$\cdots$$ || $$y_{lm21}$$ || || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || || $$y_{1m2r}$$ || $$y_{2m2r}$$ || $$\cdots$$ || $$y_{lm2r}$$ || || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || || $$\vdots$$ || ||<|3> $$C_{n}$$ || $$y_{1mn1}$$ || $$y_{2mn1}$$ || $$\cdots$$ || $$y_{lmn1}$$ || || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || || $$y_{1mnr}$$ || $$y_{2mnr}$$ || $$\cdots$$ || $$y_{lmnr}$$ ||

$$AB$$ 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 || $$T_{.1..}$$ || $$T_{.2..}$$ || $$\cdots$$ || $$T_{.m..}$$ || $$T$$ ||

 || $$T_{i...} = \sum_{j=1}^{m} \sum_{k=1}^{n} \sum_{p=1}^{r} y_{ijkp}$$ || $$\overline{y}_{i...} = \frac{T_{i...}}{mnr}$$ ||
 || $$T_{.j..} = \sum_{i=1}^{l} \sum_{k=1}^{n} \sum_{p=1}^{r} y_{ijkp}$$ || $$\overline{y}_{.j..} = \frac{T_{.j..}}{lnr}$$ ||
 || $$T_{..k.} = \sum_{i=1}^{l} \sum_{j=1}^{m} \sum_{p=1}^{r} y_{ijkp}$$ || $$\overline{y}_{..k.} = \frac{T_{..k.}}{lmr}$$ ||
 || $$T_{ij..} = \sum_{k=1}^{n} \sum_{p=1}^{r} y_{ijkp}$$ || $$\overline{y}_{ij..} = \frac{T_{ij..}}{nr}$$ ||
 || $$T_{i.k.} = \sum_{j=1}^{m} \sum_{p=1}^{r} y_{ijkp}$$ || $$\overline{y}_{i.k.} = \frac{T_{i.k.}}{mr}$$ ||
 || $$T_{.jk.} = \sum_{i=1}^{l} \sum_{p=1}^{r} y_{ijkp}$$ || $$\overline{y}_{.jk.} = \frac{T_{.jk.}}{lr}$$ ||
 || $$T_{ijk.} = \sum_{p=1}^{r} y_{ijkp}$$ || $$\overline{y}_{ijk.} = \frac{T_{ijk.}}{r}$$ ||
 || $$T = \sum_{i=1}^{l} \sum_{j=1}^{m} \sum_{k=1}^{n} \sum_{p=1}^{r} y_{ijkp}$$ || $$\overline{\overline{y}} = \frac{T}{lmnr} = \frac{T}{N}$$ ||
 || $$N = lmnr$$ || $$CT = \frac{T^{2}}{lmnr} = \frac{T^{2}}{N}$$ ||

—-

개개의 데이터&nbsp&nbsp $$y_{ijkp}$$ 와 총편균&nbsp&nbsp $$\overline{\overline{y}}$$ 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&nbsp&nbsp $$i, \ j, \ k, \ p$$ 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&nbsp&nbsp $$A$$ 의 [변동],&nbsp&nbsp $$B$$ 의 [변동],&nbsp&nbsp $$C$$ 의 [변동],&nbsp&nbsp $$A, \ B$$ 의 [교호작용]의 변동,&nbsp&nbsp $$A, \ C$$ 의 [교호작용]의 변동,&nbsp&nbsp $$B, \ C$$ 의 [교호작용]의 변동,&nbsp&nbsp $$A, \ B, \ C$$ 의 [교호작용]의 변동, [오차변동]인&nbsp&nbsp $$S_{A}$$ , $$S_{B}$$ , $$S_{C}$$ , $$S_{A \times B}$$ , $$S_{A \times C}$$ , $$S_{B \times C}$$ , $$S_{A \times B \times C}$$ , $$S_{E}$$ 가 된다.

$$\begin{displaymath}\begin{split} S_{T} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijkp}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}y_{ijkp}^{ \ 2} - CT \end{split}\end{displaymath}$$

$$\begin{displaymath}\begin{split} S_{A} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{i...}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\frac{T_{i...}^{ \ 2}}{mnr}-CT \end{split}\end{displaymath}$$

$$\begin{displaymath}\begin{split} S_{B} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{.j..}-\overline{\overline{y}})^{2} \\ &= \sum_{j=1}^{m}\frac{T_{.j..}^{ \ 2}}{lnr}-CT \end{split}\end{displaymath}$$

$$\begin{displaymath}\begin{split} S_{C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{..k.}-\overline{\overline{y}})^{2} \\ &= \sum_{k=1}^{n}\frac{T_{..k.}^{ \ 2}}{lmr}-CT \end{split}\end{displaymath}$$

$$\begin{displaymath}\begin{split} S_{A \times B} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{ij..}-\overline{y}_{i...}-\overline{y}_{.j..}+\overline{\overline{y}})^{2} \\ &= S_{AB} - S_{A} - S_{B} \end{split}\end{displaymath}$$

$$\begin{displaymath}\begin{split} S_{AB} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{ij..}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m} \frac{T_{ij..}^{ \ 2}}{nr} -CT \end{split}\end{displaymath}$$

$$\begin{displaymath}\begin{split} S_{A \times C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{i.k.}-\overline{y}_{i...}-\overline{y}_{..k.}+\overline{\overline{y}})^{2} \\ &= S_{AC} - S_{A} - S_{C} \end{split}\end{displaymath}$$

$$\begin{displaymath}\begin{split} S_{AC} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{i.k.}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{k=1}^{n} \frac{T_{i.k.}^{ \ 2}}{mr} -CT \end{split}\end{displaymath}$$

$$\begin{displaymath}\begin{split} S_{B \times C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{.jk.}-\overline{y}_{.j..}-\overline{y}_{..k.}+\overline{\overline{y}})^{2} \\ &= S_{BC} - S_{B} - S_{C} \end{split}\end{displaymath}$$

$$\begin{displaymath}\begin{split} S_{BC} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{.jk.}-\overline{\overline{y}})^{2} \\ &= \sum_{j=1}^{m}\sum_{k=1}^{n} \frac{T_{.jk.}^{ \ 2}}{lr} -CT \end{split}\end{displaymath}$$

$$\begin{displaymath}\begin{split} S_{A \times B \times C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijk.}-\overline{y}_{ij..}-\overline{y}_{i.k.}-\overline{y}_{.jk.}+\overline{y}_{i...}+\overline{y}_{.j..}+\overline{y}_{..k.}-\overline{\overline{y}})^{2} \\ &= S_{ABC}-(S_{A}+S_{B}+S_{C}+S_{A \times B}+S_{A \times C}+S_{B \times C}) \end{split}\end{displaymath}$$

$$\begin{displaymath}\begin{split} S_{ABC} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijk.}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\frac{T_{ijk.}^{ \ 2}}{r} -CT \end{split}\end{displaymath}$$

$$\begin{displaymath}\begin{split} S_{E} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijkp}-\overline{\overline{y}})^{2} \\ &= S_{T} - S_{ABC} \end{split}\end{displaymath}$$

—-

$$\nu_{A}=l-1$$

$$\nu_{B}=m-1$$

$$\nu_{C}=n-1$$

$$\nu_{A \times B}=\nu_{A} \times \nu_{B}=(l-1)(m-1)$$

$$\nu_{A \times C}=\nu_{A} \times \nu_{C}=(l-1)(n-1)$$

$$\nu_{B \times C}=\nu_{B} \times \nu_{C}=(m-1)(n-1)$$

$$\nu_{A \times B \times C}=\nu_{A} \times \nu_{B} \times \nu_{C} =(l-1)(m-1)(n-1)$$

$$\nu_{E}=\nu_{T}-(\nu_{A}+\nu_{B}+\nu_{C}+\nu_{A \times B}+\nu_{A \times C}+\nu_{B \times C}+\nu_{A \times B \times C})=lmn(r-1)$$

$$\nu_{T}=lmnr-1=N-1$$

$$V_{A}=\frac{S_{A}}{\nu_{A}}$$

$$V_{B}=\frac{S_{B}}{\nu_{B}}$$

$$V_{C}=\frac{S_{C}}{\nu_{C}}$$

$$V_{A \times B}=\frac{S_{A \times B}}{\nu_{A \times B}}$$

$$V_{A \times C}=\frac{S_{A \times C}}{\nu_{A \times C}}$$

$$V_{B \times C}=\frac{S_{B \times C}}{\nu_{B \times C}}$$

$$V_{A \times B \times C}=\frac{S_{A \times B \times C}}{\nu_{A \times B \times C}}$$

$$V_{E}=\frac{S_{E}}{\nu_{E}}$$

$$E(V_{A})=\sigma_{E}^{ \ 2} +mnr \sigma_{A}^{ \ 2}$$

$$E(V_{B})=\sigma_{E}^{ \ 2} +lnr \sigma_{B}^{ \ 2}$$

$$E(V_{C})=\sigma_{E}^{ \ 2} +lmr \sigma_{C}^{ \ 2}$$

$$E(V_{A \times B})=\sigma_{E}^{ \ 2} +nr \sigma_{A \times B}^{ \ 2}$$

$$E(V_{A \times C})=\sigma_{E}^{ \ 2} +mr \sigma_{A \times C}^{ \ 2}$$

$$E(V_{B \times C})=\sigma_{E}^{ \ 2} +lr \sigma_{A \times B}^{ \ 2}$$

$$E(V_{A \times B \times C})=\sigma_{E}^{ \ 2} +r \sigma_{A \times B \times C}^{ \ 2}$$

$$E(V_{E})=\sigma_{E}^{ \ 2}$$

|| '[요인]' || '[제곱합]' $$SS$$ || '[자유도]' $$DF$$ || '[평균제곱]' $$MS$$ || $$E(MS)$$ || $$F_{0}$$ || '기각치' || '[순변동]' $$ S\acute{} $$ || '[기여율]' $$\rho$$ || |||||||||||||||||| || || $$A$$ || $$S_{_{A}}$$ || $$\nu_{_{A}}=l-1$$ || $$V_{_{A}}=S_{_{A}}/\nu_{_{A}}$$ || $$\sigma_{_{E}}^{ \ 2}+mnr \ \sigma_{_{A}}^{2}$$ || $$V_{_{A}}/V_{_{E}}$$ || $$F_{1-\alpha}(\nu_{_{A}} \ , \ \nu_{_{E}})$$ || $$S_{_{A}}\acute{}$$ || $$S_{_{A}}\acute{}/S_{_{T}}$$ || || $$B$$ || $$S_{_{B}}$$ || $$\nu_{_{B}}=m-1$$ || $$V_{_{B}}=S_{_{B}}/\nu_{_{B}}$$ || $$\sigma_{_{E}}^{ \ 2}+lnr \ \sigma_{_{B}}^{2}$$ || $$V_{_{B}}/V_{_{E}}$$ || $$F_{1-\alpha}(\nu_{_{B}} \ , \ \nu_{_{E}})$$ || $$S_{_{B}}\acute{}$$ || $$S_{_{B}}\acute{}/S_{_{T}}$$ || || $$C$$ || $$S_{_{C}}$$ || $$\nu_{_{C}}=n-1$$ || $$V_{_{C}}=S_{_{C}}/\nu_{_{C}}$$ || $$\sigma_{_{E}}^{ \ 2}+lmr \ \sigma_{_{C}}^{2}$$ || $$V_{_{C}}/V_{_{E}}$$ || $$F_{1-\alpha}(\nu_{_{C}} \ , \ \nu_{_{E}})$$ || $$S_{_{C}}\acute{}$$ || $$S_{_{C}}\acute{}/S_{_{T}}$$ || || $$A \times B$$ || $$S_{_{A \times B}}$$ || $$\nu_{_{A \times B}}=(l-1)(m-1)$$ || $$V_{_{A \times B}}=S_{_{A \times B}}/\nu_{_{A \times B}}$$ || $$\sigma_{_{E}}^{ \ 2}+nr \ \sigma_{_{A \times B}}^{2}$$ || $$V_{_{A \times B}}/V_{_{E}}$$ || $$F_{1-\alpha}(\nu_{_{A \times B}} \ , \ \nu_{_{E}})$$ || $$S_{_{A \times B}}\acute{}$$ || $$S_{_{A \times B}}\acute{}/S_{_{T}}$$ || || $$A \times C$$ || $$S_{_{A \times C}}$$ || $$\nu_{_{A \times C}}=(l-1)(n-1)$$ || $$V_{_{A \times C}}=S_{_{A \times C}}/\nu_{_{A \times C}}$$ || $$\sigma_{_{E}}^{ \ 2}+mr \ \sigma_{_{A \times C}}^{2}$$ || $$V_{_{A \times C}}/V_{_{E}}$$ || $$F_{1-\alpha}(\nu_{_{A \times C}} \ , \ \nu_{_{E}})$$ || $$S_{_{A \times C}}\acute{}$$ || $$S_{_{A \times C}}\acute{}/S_{_{T}}$$ || || $$B \times C$$ || $$S_{_{B \times C}}$$ || $$\nu_{_{B \times C}}=(m-1)(n-1)$$ || $$V_{_{B \times C}}=S_{_{B \times C}}/\nu_{_{B \times C}}$$ || $$\sigma_{_{E}}^{ \ 2}+lr \ \sigma_{_{B \times C}}^{2}$$ || $$V_{_{B \times C}}/V_{_{E}}$$ || $$F_{1-\alpha}(\nu_{_{B \times C}} \ , \ \nu_{_{E}})$$ || $$S_{_{B \times C}}\acute{}$$ || $$S_{_{B \times C}}\acute{}/S_{_{T}}$$ || || $$A \times B \times C$$ || $$S_{_{A \times B \times C}}$$ || $$\nu_{_{A \times B \times C}}=(l-1)(m-1)(n-1)$$ || $$V_{_{A \times B \times C}}=S_{_{A \times B \times C}}/\nu_{_{A \times B \times C}}$$ || $$\sigma_{_{E}}^{ \ 2}+r \ \sigma_{_{A \times B \times C}}^{ \ 2}$$ || $$V_{_{A \times B \times C}}/V_{_{E}}$$ || $$F_{1-\alpha}(\nu_{_{A \times B \times C}} \ , \ \nu_{_{E}})$$ || $$S_{_{A \times B \times C}}\acute{}$$ || $$S_{_{A \times B \times C}}\acute{}/S_{_{T}}$$ || || $$E$$ || $$S_{_{E}}$$ || $$\nu_{_{E}}=lmn(r-1)$$ || $$V_{_{E}}=S_{_{E}}/\nu_{_{E}}$$ || $$\sigma_{_{E}}^{ \ 2}$$ || || || $$S_{_{E}}\acute{}$$ || $$S_{_{E}}\acute{}/S_{_{T}}$$ || |||||||||||||||||| || || $$T$$ || $$S_{_{T}}$$ || $$\nu_{_{T}}=lmnr-1$$ || || || || || $$S_{_{T}}$$ || $$1$$ ||

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 교호작용에 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 교호작용에 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 교호작용에 대한 분산분석

• $$F_{0}=\frac{V_{B \times C}}{V_{E}}$$

[[기각역]] : $F_{0} > F_{1-\alpha}(\nu_{_{B \times C}},\nu_{_{E}})$

인자 $A , \ B , \ C$의 교호작용에 대한 분산분석

• $$F_{0}=\frac{V_{A \times B \times C}}{V_{E}}}$$

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.)

인자 $A$의 모평균에 관한 추정

$i$ 수준에서의 모평균 $\mu(A_{i})$의 점추정값

• $$\hat{\mu}(A_{i})=\widehat{\mu + a_{i}} = \overline{y}_{i...}$$

$i$ 수준에서의 모평균 $\mu(A_{i})$의 $100(1-\alpha) \% $ 신뢰구간은 아래와 같다.

• $$\hat{\mu}(A_{i})= \left( \overline{y}_{i...} - t_{\alpha/2}(\nu_{E}\acute{} \ ) \sqrt{\frac{V_{E}\acute{}}{mnr}} \ , \ \overline{y}_{i...} + t_{\alpha/2}(\nu_{E}\acute{} \ ) \sqrt{\frac{V_{E}\acute{}}{mnr}} \right)$$

인자 $B$의 모평균에 관한 추정

$j$ 수준에서의 모평균 $\mu(B_{j})$의 점추정값

• $$\hat{\mu}(B_{j})=\widehat{\mu + b_{j}} = \overline{y}_{.j..}$$

$j$ 수준에서의 모평균 $\mu(B_{j})$의 $100(1-\alpha) \%$ 신뢰구간은 아래와 같다.

• $$\hat{\mu}(B_{j})= \left( \overline{y}_{.j..} - t_{\alpha/2}(\nu_{E}\acute{} \ ) \sqrt{\frac{V_{E}\acute{}}{lnr}} \ , \ \overline{y}_{.j..} + t_{\alpha/2}(\nu_{E}\acute{} \ ) \sqrt{\frac{V_{E}\acute{}}{lnr}} \right)$$

인자 $C$의 모평균에 관한 추정

$k$ 수준에서의 모평균 $\mu(C_{k})$의 점추정값

• $$\hat{\mu}(C_{k})=\widehat{\mu + c_{k}} = \overline{y}_{..k.}$$

$k$ 수준에서의 모평균 $\mu(C_{k})$의 $100(1-\alpha) \% $ 신뢰구간은 아래와 같다.

• $$\hat{\mu}(C_{k})= \left( \overline{y}_{..k.} - t_{\alpha/2}(\nu_{E}\acute{} \ ) \sqrt{\frac{V_{E}\acute{}}{lmr}} \ , \ \overline{y}_{..k.} + t_{\alpha/2}(\nu_{E}\acute{} \ ) \sqrt{\frac{V_{E}\acute{}}{lmr}} \right)$$

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

• $$\hat{\mu}(A_{i}B_{j}C_{k})=\widehat{\mu+a_{i}+b_{j}+c_{k}}=\overline{y}_{i...} + \overline{y}_{.j..} + \overline{y}_{..k.} - 2 \overline{\overline{y}}$$

$A$ 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.

• $$\hat{\mu}(A_{i}B_{j}C_{k})= \left( (\overline{y}_{i...} + \overline{y}_{.j..} + \overline{y}_{..k.} - 2\overline{\overline{y}}) - t_{\alpha/2}(\nu_{E}\acute{} \ )\sqrt{\frac{V_{E}\acute{}}{n_{e}}} \ , \ (\overline{y}_{i...} + \overline{y}_{.j..} + \overline{y}_{..k.} - 2\overline{\overline{y}}) - t_{\alpha/2}(\nu_{E}\acute{} \ )\sqrt{\frac{V_{E}\acute{}}{n_{e}}} \right)$$

단, $n_{e}$는 유효반복수이고 $n_{e} = \frac{lmnr}{l+m+n-2}$이다.

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