차이

문서의 선택한 두 판 사이의 차이를 보여줍니다.

--- 삼원배치법_모수모형_반복없음 [2012/07/24 22:57]
moonrepeat [분산분석표]
+++ 삼원배치법_모수모형_반복없음 [2021/03/10 21:42]
@@ 줄 1: / 줄 1: @@
-====== 삼원배치법 (모수모형) (반복없음) ======
-===== 데이터 구조 =====
- [요인]&nbsp&nbsp $$A$$ 는 [모수인자]
- [요인]&nbsp&nbsp $$B$$ 는 [모수인자]
- [요인]&nbsp&nbsp $$C$$ 는 [모수인자]
-  $$ y_{ijk} = \mu + a_{i} + b_{j} + c_{k} + (ab)_{ij} + (ac)_{ik} + (bc)_{jk} + e_{ijk} $$
-   $$y_{ijk}$$ &nbsp&nbsp : &nbsp&nbsp $$A_{i}$$ 와&nbsp&nbsp $$B_{j}$$ , 그리고 $$C_{k}$$ 에서 얻은 [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
-   $$e_{ijk}$$ &nbsp&nbsp : &nbsp&nbsp $$A_{i}$$ 와&nbsp&nbsp $$B_{j}$$ , 그리고 $$C_{k}$$ 에서 얻은 [측정값]의 [오차]  ( $$e_{ijk} \sim N(0, \sigma_{E}^{ \ 2})$$ 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 $$( k = 1,2, \cdots ,n )$$
-===== 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&nbsp&nbsp $$y_{ijk}$$ 와 총편균&nbsp&nbsp $$\overline{\overline{y}}$$ 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&nbsp&nbsp $$i, \ j, \ k$$ 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&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, [오차변동]인&nbsp&nbsp $$S_{A}$$ , $$S_{B}$$ , $$S_{C}$$ , $$S_{A \times B}$$ , $$S_{A \times C}$$ , $$S_{B \times C}$$ , $$S_{E}$$ 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]]
- $$F_{0}=\frac{V_{_{B \times C}}}{V_{E}}$$
- [[기각역]] : $F_{0} > F_{1-\alpha}(\nu_{_{B \times C}},\nu_{_{E}})$
-===== 각 수준의 모평균의 추정 (주효과만이 유의한 경우) =====
- 주효과인 인자&nbsp $$A, B, C$$ 만이 유의한 경우 [교호작용]들이 모두 오차항에 [풀링]되어 버린다.
- (단,&nbsp&nbsp $$S_{E}\acute{}=S_{E}+S_{A \times B}+S_{A \times C}+S_{B \times C}, \ \nu_{E}\acute{}=\nu_{E}+\nu_{A \times B}+\nu_{A \times C}+\nu_{B \times C}, \ V_{E}\acute{}=S_{E}\acute{}/\nu_{E}\acute{}$$ 이다.)
- * '''[인자]&nbsp&nbsp $$A$$ 의 [모평균]에 관한 [추정]'''
-  $$i$$ [수준]에서의 [모평균]&nbsp&nbsp $$\mu(A_{i})$$ 의 [점추정]값
-   $$\hat{\mu}(A_{i})=\widehat{\mu + a_{i}} = \overline{y}_{i..}$$
-  $$i$$ [수준]에서의 [모평균]&nbsp&nbsp $$\mu(A_{i})$$ 의&nbsp&nbsp $$100(1-\alpha) \% $$ [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]&nbsp&nbsp $$B$$ 의 [모평균]에 관한 [추정]'''
-  $$j$$ [수준]에서의 [모평균]&nbsp&nbsp $$\mu(B_{j})$$ 의 [점추정]값
-   $$\hat{\mu}(B_{j})=\widehat{\mu + b_{j}} = \overline{y}_{.j.}$$
-  $$j$$ [수준]에서의 [모평균]&nbsp&nbsp $$\mu(B_{j})$$ 의&nbsp&nbsp $$100(1-\alpha) \% $$ [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]&nbsp&nbsp $$C$$ 의 [모평균]에 관한 [추정]'''
-  $$k$$ [수준]에서의 [모평균]&nbsp&nbsp $$\mu(C_{k})$$ 의 [점추정]값
-   $$\hat{\mu}(C_{k})=\widehat{\mu + c_{k}} = \overline{y}_{..k}$$
-  $$k$$ [수준]에서의 [모평균]&nbsp&nbsp $$\mu(C_{k})$$ 의&nbsp&nbsp $$100(1-\alpha) \% $$ [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]&nbsp&nbsp $$A$$ 와&nbsp&nbsp $$B$$ &nbsp&nbsp그리고&nbsp&nbsp $$C$$ 의 [모평균]에 관한 [추정]'''
-  $$A$$ [인자]의&nbsp&nbsp $$i$$ [수준]과&nbsp&nbsp $$B$$ [인자]의&nbsp&nbsp $$j$$ [수준],&nbsp&nbsp $$C$$ [인자]의&nbsp&nbsp $$k$$ [수준]에서의 [모평균]&nbsp&nbsp $$\mu(A_{i}B_{j}C_{k})$$ 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&nbsp&nbsp $$i$$ [수준]과&nbsp&nbsp $$B$$ [인자]의&nbsp&nbsp $$j$$ [수준],&nbsp&nbsp $$C$$ [인자]의&nbsp&nbsp $$k$$ [수준]에서의 [모평균]&nbsp&nbsp $$\mu(A_{i}B_{j}C_{k})$$ 의&nbsp&nbsp $$100(1-\alpha) \% $$ [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,&nbsp&nbsp $$n_{e}$$ 는 [유효반복수]이고&nbsp&nbsp $$n_{e} = \frac{lmn}{l+m+n-2}$$ 이다.
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-  * [[실험계획법]]
-  * [[삼원배치법]]

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