meta data for this page
차이
문서의 선택한 두 판 사이의 차이를 보여줍니다.
| 양쪽 이전 판 이전 판 다음 판 | 이전 판 | ||
|
삼원배치법_모수모형_반복있음 [2012/07/28 09:35] moonrepeat [데이터 구조] |
삼원배치법_모수모형_반복있음 [2021/03/10 21:42] (현재) |
||
|---|---|---|---|
| 줄 26: | 줄 26: | ||
| * $p$ : 실험의 [[반복]] 수 $( p = 1,2, \cdots ,r )$ | * $p$ : 실험의 [[반복]] 수 $( p = 1,2, \cdots ,r )$ | ||
| ===== 자료의 구조 ===== | ===== 자료의 구조 ===== | ||
| - | ||<|2> [인자] $$B$$ ||<|2> [인자] $$C$$ |||||||| [인자] $$A$$ || | + | ^ [[인자]]\\ $B$ ^ [[인자]]\\ $C$ ^ [[인자]] $A$ |||| |
| - | || $$A_{1}$$ || $$A_{2}$$ || $$\cdots$$ || $$A_{l}$$ || | + | ^:::^:::^ $$A_{1}$$ ^ $$A_{2}$$ ^ $$\cdots$$ ^ $$A_{l}$$ | |
| - | |||||||||||| || | + | ^ $$B_{1}$$ ^ $$C_{1}$$ | $$y_{1111}$$ | $$y_{2111}$$ | $$\cdots$$ | $$y_{l111}$$ | |
| - | ||<|10> $$B_{1}$$ ||<|3> $$C_{1}$$ || $$y_{1111}$$ || $$y_{2111}$$ || $$\cdots$$ || $$y_{l111}$$ || | + | ^:::^:::| $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | |
| - | || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || | + | ^:::^:::| $$y_{111r}$$ | $$y_{211r}$$ | $$\cdots$$ | $$y_{l11r}$$ | |
| - | || $$y_{111r}$$ || $$y_{211r}$$ || $$\cdots$$ || $$y_{l11r}$$ || | + | ^:::^ $$C_{2}$$ | $$y_{1121}$$ | $$y_{2121}$$ | $$\cdots$$ | $$y_{l121}$$ | |
| - | ||<|3> $$C_{2}$$ || $$y_{1121}$$ || $$y_{2121}$$ || $$\cdots$$ || $$y_{l121}$$ || | + | ^:::^:::| $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | |
| - | || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || | + | ^:::^:::| $$y_{112r}$$ | $$y_{212r}$$ | $$\cdots$$ | $$y_{l12r}$$ | |
| - | || $$y_{112r}$$ || $$y_{212r}$$ || $$\cdots$$ || $$y_{l12r}$$ || | + | ^:::^ $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | | $$\vdots$$ | |
| - | || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || || $$\vdots$$ || | + | ^:::^ $$C_{n}$$ | $$y_{11n1}$$ | $$y_{21n1}$$ | $$\cdots$$ | $$y_{l1n1}$$ | |
| - | ||<|3> $$C_{n}$$ || $$y_{11n1}$$ || $$y_{21n1}$$ || $$\cdots$$ || $$y_{l1n1}$$ || | + | ^:::^:::| $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | |
| - | || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || | + | ^:::^:::| $$y_{11nr}$$ | $$y_{21nr}$$ | $$\cdots$$ | $$y_{l1nr}$$ | |
| - | || $$y_{11nr}$$ || $$y_{21nr}$$ || $$\cdots$$ || $$y_{l1nr}$$ || | + | ^ $$B_{2}$$ ^ $$C_{1}$$ | $$y_{1211}$$ | $$y_{2211}$$ | $$\cdots$$ | $$y_{l211}$$ | |
| - | ||<|10> $$B_{2}$$ ||<|3> $$C_{1}$$ || $$y_{1211}$$ || $$y_{2211}$$ || $$\cdots$$ || $$y_{l211}$$ || | + | ^:::^:::| $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | |
| - | || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || | + | ^:::^:::| $$y_{121r}$$ | $$y_{221r}$$ | $$\cdots$$ | $$y_{l21r}$$ | |
| - | || $$y_{121r}$$ || $$y_{221r}$$ || $$\cdots$$ || $$y_{l21r}$$ || | + | ^:::^ $$C_{2}$$ | $$y_{1221}$$ | $$y_{2221}$$ | $$\cdots$$ | $$y_{l221}$$ | |
| - | ||<|3> $$C_{2}$$ || $$y_{1221}$$ || $$y_{2221}$$ || $$\cdots$$ || $$y_{l221}$$ || | + | ^:::^:::| $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | |
| - | || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || | + | ^:::^:::| $$y_{122r}$$ | $$y_{222r}$$ | $$\cdots$$ | $$y_{l22r}$$ | |
| - | || $$y_{122r}$$ || $$y_{222r}$$ || $$\cdots$$ || $$y_{l22r}$$ || | + | ^:::^ $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | | $$\vdots$$ | |
| - | || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || || $$\vdots$$ || | + | ^:::^ $$C_{n}$$ | $$y_{12n1}$$ | $$y_{22n1}$$ | $$\cdots$$ | $$y_{l2n1}$$ | |
| - | ||<|3> $$C_{n}$$ || $$y_{12n1}$$ || $$y_{22n1}$$ || $$\cdots$$ || $$y_{l2n1}$$ || | + | ^:::^:::| $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | |
| - | || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || | + | ^:::^:::| $$y_{12nr}$$ | $$y_{22nr}$$ | $$\cdots$$ | $$y_{l2nr}$$ | |
| - | || $$y_{12nr}$$ || $$y_{22nr}$$ || $$\cdots$$ || $$y_{l2nr}$$ || | + | ^ $$\vdots$$ || $$\vdots$$ |||| |
| - | |||| $$\vdots$$ |||||||| $$\vdots$$ || | + | ^ $$B_{m}$$ ^ $$C_{1}$$ | $$y_{1m11}$$ | $$y_{2m11}$$ | $$\cdots$$ | $$y_{lm11}$$ | |
| - | ||<|10> $$B_{m}$$ ||<|3> $$C_{1}$$ || $$y_{1m11}$$ || $$y_{2m11}$$ || $$\cdots$$ || $$y_{lm11}$$ || | + | ^:::^:::| $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | |
| - | || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || | + | ^:::^:::| $$y_{1m1r}$$ | $$y_{2m1r}$$ | $$\cdots$$ | $$y_{lm1r}$$ | |
| - | || $$y_{1m1r}$$ || $$y_{2m1r}$$ || $$\cdots$$ || $$y_{lm1r}$$ || | + | ^:::^ $$C_{2}$$ | $$y_{1m21}$$ | $$y_{2m21}$$ | $$\cdots$$ | $$y_{lm21}$$ | |
| - | ||<|3> $$C_{2}$$ || $$y_{1m21}$$ || $$y_{2m21}$$ || $$\cdots$$ || $$y_{lm21}$$ || | + | ^:::^:::| $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | |
| - | || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || | + | ^:::^:::| $$y_{1m2r}$$ | $$y_{2m2r}$$ | $$\cdots$$ | $$y_{lm2r}$$ | |
| - | || $$y_{1m2r}$$ || $$y_{2m2r}$$ || $$\cdots$$ || $$y_{lm2r}$$ || | + | ^:::^ $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | | $$\vdots$$ | |
| - | || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || || $$\vdots$$ || | + | ^:::^ $$C_{n}$$ | $$y_{1mn1}$$ | $$y_{2mn1}$$ | $$\cdots$$ | $$y_{lmn1}$$ | |
| - | ||<|3> $$C_{n}$$ || $$y_{1mn1}$$ || $$y_{2mn1}$$ || $$\cdots$$ || $$y_{lmn1}$$ || | + | ^:::^:::| $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | |
| - | || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || | + | ^:::^:::| $$y_{1mnr}$$ | $$y_{2mnr}$$ | $$\cdots$$ | $$y_{lmnr}$$ | |
| - | || $$y_{1mnr}$$ || $$y_{2mnr}$$ || $$\cdots$$ || $$y_{lmnr}$$ || | + | |
| - | $$AB$$ 2원표 | + | $AB$ 2원표 |
| - | ||<|2> [인자] $$B$$ |||||||| [인자] $$A$$ ||<|2> 합계 || | + | ^ [[인자]] $B$ ^ [[인자]] $A$ ^^^^ 합계 | |
| - | || $$A_{1}$$ || $$A_{2}$$ || $$\cdots$$ || $$A_{l}$$ || | + | ^:::^ $$A_{1}$$ ^ $$A_{2}$$ ^ $$\cdots$$ ^ $$A_{l}$$ ^:::| |
| - | |||||||||||| || | + | ^ $$B_{1}$$ | $$T_{11..}$$ | $$T_{21..}$$ | $$\cdots$$ | $$T_{l1..}$$ | $$T_{.1..}$$ | |
| - | || $$B_{1}$$ || $$T_{11..}$$ || $$T_{21..}$$ || $$\cdots$$ || $$T_{l1..}$$ || $$T_{.1..}$$ || | + | ^ $$B_{2}$$ | $$T_{12..}$$ | $$T_{22..}$$ | $$\cdots$$ | $$T_{l2..}$$ | $$T_{.2..}$$ | |
| - | || $$B_{2}$$ || $$T_{12..}$$ || $$T_{22..}$$ || $$\cdots$$ || $$T_{l2..}$$ || $$T_{.2..}$$ || | + | ^ $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | | $$\vdots$$ | $$\vdots$$ | |
| - | || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || || $$\vdots$$ || $$\vdots$$ || | + | ^ $$B_{m}$$ | $$T_{1m..}$$ | $$T_{2m..}$$ | $$\cdots$$ | $$T_{lm..}$$ | $$T_{.m..}$$ | |
| - | || $$B_{m}$$ || $$T_{1m..}$$ || $$T_{2m..}$$ || $$\cdots$$ || $$T_{lm..}$$ || $$T_{.m..}$$ || | + | ^ 합계 ^ $$T_{1...}$$ ^ $$T_{2...}$$ ^ $$\cdots$$ ^ $$T_{l...}$$ ^ $$T$$ | |
| - | |||||||||||| || | + | |
| - | || 합계 || $$T_{1...}$$ || $$T_{2...}$$ || $$\cdots$$ || $$T_{l...}$$ || $$T$$ || | + | |
| - | $$AC$$ 2원표 | + | $AC$ 2원표 |
| - | ||<|2> [인자] $$C$$ |||||||| [인자] $$A$$ ||<|2> 합계 || | + | ^ [[인자]] $C$ ^ [[인자]] $A$ ^^^^ 합계 | |
| - | || $$A_{1}$$ || $$A_{2}$$ || $$\cdots$$ || $$A_{l}$$ || | + | ^:::^ $$A_{1}$$ ^ $$A_{2}$$ ^ $$\cdots$$ ^ $$A_{l}$$ ^:::| |
| - | |||||||||||| || | + | ^ $$C_{1}$$ | $$T_{1.1.}$$ | $$T_{2.1.}$$ | $$\cdots$$ | $$T_{l.1.}$$ | $$T_{..1.}$$ | |
| - | || $$C_{1}$$ || $$T_{1.1.}$$ || $$T_{2.1.}$$ || $$\cdots$$ || $$T_{l.1.}$$ || $$T_{..1.}$$ || | + | ^ $$C_{2}$$ | $$T_{1.2.}$$ | $$T_{2.2.}$$ | $$\cdots$$ | $$T_{l.2.}$$ | $$T_{..2.}$$ | |
| - | || $$C_{2}$$ || $$T_{1.2.}$$ || $$T_{2.2.}$$ || $$\cdots$$ || $$T_{l.2.}$$ || $$T_{..2.}$$ || | + | ^ $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | | $$\vdots$$ | $$\vdots$$ | |
| - | || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || || $$\vdots$$ || $$\vdots$$ || | + | ^ $$C_{n}$$ | $$T_{1.n.}$$ | $$T_{2.n.}$$ | $$\cdots$$ | $$T_{l.n.}$$ | $$T_{..n.}$$ | |
| - | || $$C_{n}$$ || $$T_{1.n.}$$ || $$T_{2.n.}$$ || $$\cdots$$ || $$T_{l.n.}$$ || $$T_{..n.}$$ || | + | ^ 합계 ^ $$T_{1...}$$ ^ $$T_{2...}$$ ^ $$\cdots$$ ^ $$T_{l...}$$ ^ $$T$$ | |
| - | |||||||||||| || | + | |
| - | || 합계 || $$T_{1...}$$ || $$T_{2...}$$ || $$\cdots$$ || $$T_{l...}$$ || $$T$$ || | + | |
| - | $$BC$$ 2원표 | + | $BC$ 2원표 |
| - | ||<|2> [인자] $$C$$ |||||||| [인자] $$B$$ ||<|2> 합계 || | + | ^ [[인자]] $C$ ^ [[인자]] $B$ ^^^^ 합계 | |
| - | || $$B_{1}$$ || $$B_{2}$$ || $$\cdots$$ || $$B_{m}$$ || | + | ^:::^ $$B_{1}$$ ^ $$B_{2}$$ ^ $$\cdots$$ ^ $$B_{m}$$ ^:::| |
| - | |||||||||||| || | + | ^ $$C_{1}$$ | $$T_{.11.}$$ | $$T_{.21.}$$ | $$\cdots$$ | $$T_{.m1.}$$ | $$T_{..1.}$$ | |
| - | || $$C_{1}$$ || $$T_{.11.}$$ || $$T_{.21.}$$ || $$\cdots$$ || $$T_{.m1.}$$ || $$T_{..1.}$$ || | + | ^ $$C_{2}$$ | $$T_{.12.}$$ | $$T_{.22.}$$ | $$\cdots$$ | $$T_{.m2.}$$ | $$T_{..2.}$$ | |
| - | || $$C_{2}$$ || $$T_{.12.}$$ || $$T_{.22.}$$ || $$\cdots$$ || $$T_{.m2.}$$ || $$T_{..2.}$$ || | + | ^ $$\vdots$$ | $$\vdots$$ | $$\vdots$$ | | $$\vdots$$ | $$\vdots$$ | |
| - | || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || || $$\vdots$$ || $$\vdots$$ || | + | ^ $$C_{n}$$ | $$T_{.1n.}$$ | $$T_{.2n.}$$ | $$\cdots$$ | $$T_{.mn.}$$ | $$T_{..n.}$$ | |
| - | || $$C_{n}$$ || $$T_{.1n.}$$ || $$T_{.2n.}$$ || $$\cdots$$ || $$T_{.mn.}$$ || $$T_{..n.}$$ || | + | ^ 합계 ^ $$T_{.1..}$$ ^ $$T_{.2..}$$ ^ $$\cdots$$ ^ $$T_{.m..}$$ ^ $$T$$ | |
| - | |||||||||||| || | + | |
| - | || 합계 || $$T_{.1..}$$ || $$T_{.2..}$$ || $$\cdots$$ || $$T_{.m..}$$ || $$T$$ || | + | |
| - | || $$T_{i...} = \sum_{j=1}^{m} \sum_{k=1}^{n} \sum_{p=1}^{r} y_{ijkp}$$ || $$\overline{y}_{i...} = \frac{T_{i...}}{mnr}$$ || | + | | $$T_{i...} = \sum_{j=1}^{m} \sum_{k=1}^{n} \sum_{p=1}^{r} y_{ijkp}$$ | $$\overline{y}_{i...} = \frac{T_{i...}}{mnr}$$ | |
| - | || $$T_{.j..} = \sum_{i=1}^{l} \sum_{k=1}^{n} \sum_{p=1}^{r} y_{ijkp}$$ || $$\overline{y}_{.j..} = \frac{T_{.j..}}{lnr}$$ || | + | | $$T_{.j..} = \sum_{i=1}^{l} \sum_{k=1}^{n} \sum_{p=1}^{r} y_{ijkp}$$ | $$\overline{y}_{.j..} = \frac{T_{.j..}}{lnr}$$ | |
| - | || $$T_{..k.} = \sum_{i=1}^{l} \sum_{j=1}^{m} \sum_{p=1}^{r} y_{ijkp}$$ || $$\overline{y}_{..k.} = \frac{T_{..k.}}{lmr}$$ || | + | | $$T_{..k.} = \sum_{i=1}^{l} \sum_{j=1}^{m} \sum_{p=1}^{r} y_{ijkp}$$ | $$\overline{y}_{..k.} = \frac{T_{..k.}}{lmr}$$ | |
| - | || $$T_{ij..} = \sum_{k=1}^{n} \sum_{p=1}^{r} y_{ijkp}$$ || $$\overline{y}_{ij..} = \frac{T_{ij..}}{nr}$$ || | + | | $$T_{ij..} = \sum_{k=1}^{n} \sum_{p=1}^{r} y_{ijkp}$$ | $$\overline{y}_{ij..} = \frac{T_{ij..}}{nr}$$ | |
| - | || $$T_{i.k.} = \sum_{j=1}^{m} \sum_{p=1}^{r} y_{ijkp}$$ || $$\overline{y}_{i.k.} = \frac{T_{i.k.}}{mr}$$ || | + | | $$T_{i.k.} = \sum_{j=1}^{m} \sum_{p=1}^{r} y_{ijkp}$$ | $$\overline{y}_{i.k.} = \frac{T_{i.k.}}{mr}$$ | |
| - | || $$T_{.jk.} = \sum_{i=1}^{l} \sum_{p=1}^{r} y_{ijkp}$$ || $$\overline{y}_{.jk.} = \frac{T_{.jk.}}{lr}$$ || | + | | $$T_{.jk.} = \sum_{i=1}^{l} \sum_{p=1}^{r} y_{ijkp}$$ | $$\overline{y}_{.jk.} = \frac{T_{.jk.}}{lr}$$ | |
| - | || $$T_{ijk.} = \sum_{p=1}^{r} y_{ijkp}$$ || $$\overline{y}_{ijk.} = \frac{T_{ijk.}}{r}$$ || | + | | $$T_{ijk.} = \sum_{p=1}^{r} y_{ijkp}$$ | $$\overline{y}_{ijk.} = \frac{T_{ijk.}}{r}$$ | |
| - | || $$T = \sum_{i=1}^{l} \sum_{j=1}^{m} \sum_{k=1}^{n} \sum_{p=1}^{r} y_{ijkp}$$ || $$\overline{\overline{y}} = \frac{T}{lmnr} = \frac{T}{N}$$ || | + | | $$T = \sum_{i=1}^{l} \sum_{j=1}^{m} \sum_{k=1}^{n} \sum_{p=1}^{r} y_{ijkp}$$ | $$\overline{\overline{y}} = \frac{T}{lmnr} = \frac{T}{N}$$ | |
| - | || $$N = lmnr$$ || $$CT = \frac{T^{2}}{lmnr} = \frac{T^{2}}{N}$$ || | + | | $$N = lmnr$$ | $$CT = \frac{T^{2}}{lmnr} = \frac{T^{2}}{N}$$ | |
| - | ---- | + | ===== 제곱합 ===== |
| - | ===== [제곱합] ===== | + | 개개의 데이터 $y_{ijkp}$와 총평균 $\overline{\overline{y}}$의 차이는 다음과 같이 8부분으로 나뉘어진다. |
| - | 개개의 데이터   $$y_{ijkp}$$ 와 총편균   $$\overline{\overline{y}}$$ 의 차이는 다음과 같이 8부분으로 나뉘어진다. | + | |
| - | $$\begin{displaymath}\begin{split} (y_{ijkp}-\overline{\overline{y}}) &= (\overline{y}_{i...} - \overline{\overline{y}}) + (\overline{y}_{.j..} - \overline{\overline{y}}) + (\overline{y}_{..k.} - \overline{\overline{y}}) \\ &+ (\overline{y}_{ij..} - \overline{y}_{i...} - \overline{y}_{.j..} + \overline{\overline{y}}) + (\overline{y}_{i.k.} - \overline{y}_{i...} - \overline{y}_{..k.} + \overline{\overline{y}}) + (\overline{y}_{.jk.} - \overline{y}_{.j..} - \overline{y}_{..k.} + \overline{\overline{y}}) \\ &+ (y_{ijk.} - \overline{y}_{ij..} - \overline{y}_{i.k.} - \overline{y}_{.jk.} + \overline{y}_{i...} + \overline{y}_{.j..} + \overline{y}_{..k.} - \overline{\overline{y}}) \\ &+ (y_{ijkp}-\overline{y}_{ijk.}) \end{split}\end{displaymath}$$ | + | $$\begin{displaymath}\begin{split} (y_{ijkp}-\overline{\overline{y}}) &= (\overline{y}_{i...} - \overline{\overline{y}}) + (\overline{y}_{.j..} - \overline{\overline{y}}) + (\overline{y}_{..k.} - \overline{\overline{y}}) \\ &+ (\overline{y}_{ij..} - \overline{y}_{i...} - \overline{y}_{.j..} + \overline{\overline{y}}) + (\overline{y}_{i.k.} - \overline{y}_{i...} - \overline{y}_{..k.} + \overline{\overline{y}}) + (\overline{y}_{.jk.} - \overline{y}_{.j..} - \overline{y}_{..k.} + \overline{\overline{y}}) \\ &+ (y_{ijk.} - \overline{y}_{ij..} - \overline{y}_{i.k.} - \overline{y}_{.jk.} + \overline{y}_{i...} + \overline{y}_{.j..} + \overline{y}_{..k.} - \overline{\overline{y}}) \\ &+ (y_{ijkp}-\overline{y}_{ijk.}) \end{split}\end{displaymath}$$ |
| - | 양변을 제곱한 후에 모든   $$i, \ j, \ k, \ p$$ 에 대하여 합하면 아래의 등식을 얻을 수 있다. | + | 양변을 제곱한 후에 모든 $i, \ j, \ k, \ p$에 대하여 합하면 아래의 등식을 얻을 수 있다. |
| - | $$\begin{displaymath}\begin{split} \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijkp}-\overline{\overline{y}})^{2} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{i...} - \overline{\overline{y}})^{2} + \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{.j..} - \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{..k.} - \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{ij..} - \overline{y}_{i...} - \overline{y}_{.j..} + \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{i.k.} - \overline{y}_{i...} - \overline{y}_{..k.} + \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{.jk.} - \overline{y}_{.j..} - \overline{y}_{..k.} + \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijk.} - \overline{y}_{ij..} - \overline{y}_{i.k.} - \overline{y}_{.jk.} + \overline{y}_{i...} + \overline{y}_{.j..} + \overline{y}_{..k.} - \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijkp}-\overline{y}_{ijk.})^{2} \end{split}\end{displaymath}$$ | + | $$\begin{displaymath}\begin{split} \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijkp}-\overline{\overline{y}})^{2} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{i...} - \overline{\overline{y}})^{2} + \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{.j..} - \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{..k.} - \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{ij..} - \overline{y}_{i...} - \overline{y}_{.j..} + \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{i.k.} - \overline{y}_{i...} - \overline{y}_{..k.} + \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{.jk.} - \overline{y}_{.j..} - \overline{y}_{..k.} + \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijk.} - \overline{y}_{ij..} - \overline{y}_{i.k.} - \overline{y}_{.jk.} + \overline{y}_{i...} + \overline{y}_{.j..} + \overline{y}_{..k.} - \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijkp}-\overline{y}_{ijk.})^{2} \end{split}\end{displaymath}$$ |
| - | 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. | + | 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. |
| + | $$\begin{displaymath}\begin{split} S_{T} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijkp}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}y_{ijkp}^{ \ 2} - CT \end{split}\end{displaymath}$$ | ||
| - | $$\begin{displaymath}\begin{split} S_{T} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijkp}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}y_{ijkp}^{ \ 2} - CT \end{split}\end{displaymath}$$ | + | $$\begin{displaymath}\begin{split} S_{A} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{i...}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\frac{T_{i...}^{ \ 2}}{mnr}-CT \end{split}\end{displaymath}$$ |
| - | $$\begin{displaymath}\begin{split} S_{A} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{i...}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\frac{T_{i...}^{ \ 2}}{mnr}-CT \end{split}\end{displaymath}$$ | + | $$\begin{displaymath}\begin{split} S_{B} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{.j..}-\overline{\overline{y}})^{2} \\ &= \sum_{j=1}^{m}\frac{T_{.j..}^{ \ 2}}{lnr}-CT \end{split}\end{displaymath}$$ |
| - | $$\begin{displaymath}\begin{split} S_{B} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{.j..}-\overline{\overline{y}})^{2} \\ &= \sum_{j=1}^{m}\frac{T_{.j..}^{ \ 2}}{lnr}-CT \end{split}\end{displaymath}$$ | + | $$\begin{displaymath}\begin{split} S_{C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{..k.}-\overline{\overline{y}})^{2} \\ &= \sum_{k=1}^{n}\frac{T_{..k.}^{ \ 2}}{lmr}-CT \end{split}\end{displaymath}$$ |
| - | $$\begin{displaymath}\begin{split} S_{C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{..k.}-\overline{\overline{y}})^{2} \\ &= \sum_{k=1}^{n}\frac{T_{..k.}^{ \ 2}}{lmr}-CT \end{split}\end{displaymath}$$ | + | $$\begin{displaymath}\begin{split} S_{A \times B} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{ij..}-\overline{y}_{i...}-\overline{y}_{.j..}+\overline{\overline{y}})^{2} \\ &= S_{AB} - S_{A} - S_{B} \end{split}\end{displaymath}$$ |
| - | $$\begin{displaymath}\begin{split} S_{A \times B} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{ij..}-\overline{y}_{i...}-\overline{y}_{.j..}+\overline{\overline{y}})^{2} \\ &= S_{AB} - S_{A} - S_{B} \end{split}\end{displaymath}$$ | + | $$\begin{displaymath}\begin{split} S_{AB} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{ij..}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m} \frac{T_{ij..}^{ \ 2}}{nr} -CT \end{split}\end{displaymath}$$ |
| - | $$\begin{displaymath}\begin{split} S_{AB} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{ij..}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m} \frac{T_{ij..}^{ \ 2}}{nr} -CT \end{split}\end{displaymath}$$ | + | $$\begin{displaymath}\begin{split} S_{A \times C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{i.k.}-\overline{y}_{i...}-\overline{y}_{..k.}+\overline{\overline{y}})^{2} \\ &= S_{AC} - S_{A} - S_{C} \end{split}\end{displaymath}$$ |
| - | $$\begin{displaymath}\begin{split} S_{A \times C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{i.k.}-\overline{y}_{i...}-\overline{y}_{..k.}+\overline{\overline{y}})^{2} \\ &= S_{AC} - S_{A} - S_{C} \end{split}\end{displaymath}$$ | + | $$\begin{displaymath}\begin{split} S_{AC} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{i.k.}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{k=1}^{n} \frac{T_{i.k.}^{ \ 2}}{mr} -CT \end{split}\end{displaymath}$$ |
| - | $$\begin{displaymath}\begin{split} S_{AC} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{i.k.}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{k=1}^{n} \frac{T_{i.k.}^{ \ 2}}{mr} -CT \end{split}\end{displaymath}$$ | + | $$\begin{displaymath}\begin{split} S_{B \times C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{.jk.}-\overline{y}_{.j..}-\overline{y}_{..k.}+\overline{\overline{y}})^{2} \\ &= S_{BC} - S_{B} - S_{C} \end{split}\end{displaymath}$$ |
| - | $$\begin{displaymath}\begin{split} S_{B \times C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{.jk.}-\overline{y}_{.j..}-\overline{y}_{..k.}+\overline{\overline{y}})^{2} \\ &= S_{BC} - S_{B} - S_{C} \end{split}\end{displaymath}$$ | + | $$\begin{displaymath}\begin{split} S_{BC} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{.jk.}-\overline{\overline{y}})^{2} \\ &= \sum_{j=1}^{m}\sum_{k=1}^{n} \frac{T_{.jk.}^{ \ 2}}{lr} -CT \end{split}\end{displaymath}$$ |
| - | $$\begin{displaymath}\begin{split} S_{BC} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{.jk.}-\overline{\overline{y}})^{2} \\ &= \sum_{j=1}^{m}\sum_{k=1}^{n} \frac{T_{.jk.}^{ \ 2}}{lr} -CT \end{split}\end{displaymath}$$ | + | $$\begin{displaymath}\begin{split} S_{A \times B \times C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijk.}-\overline{y}_{ij..}-\overline{y}_{i.k.}-\overline{y}_{.jk.}+\overline{y}_{i...}+\overline{y}_{.j..}+\overline{y}_{..k.}-\overline{\overline{y}})^{2} \\ &= S_{ABC}-(S_{A}+S_{B}+S_{C}+S_{A \times B}+S_{A \times C}+S_{B \times C}) \end{split}\end{displaymath}$$ |
| - | $$\begin{displaymath}\begin{split} S_{A \times B \times C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijk.}-\overline{y}_{ij..}-\overline{y}_{i.k.}-\overline{y}_{.jk.}+\overline{y}_{i...}+\overline{y}_{.j..}+\overline{y}_{..k.}-\overline{\overline{y}})^{2} \\ &= S_{ABC}-(S_{A}+S_{B}+S_{C}+S_{A \times B}+S_{A \times C}+S_{B \times C}) \end{split}\end{displaymath}$$ | + | $$\begin{displaymath}\begin{split} S_{ABC} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijk.}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\frac{T_{ijk.}^{ \ 2}}{r} -CT \end{split}\end{displaymath}$$ |
| - | $$\begin{displaymath}\begin{split} S_{ABC} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijk.}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\frac{T_{ijk.}^{ \ 2}}{r} -CT \end{split}\end{displaymath}$$ | + | $$\begin{displaymath}\begin{split} S_{E} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijkp}-\overline{\overline{y}})^{2} \\ &= S_{T} - S_{ABC} \end{split}\end{displaymath}$$ |
| - | + | ||
| - | $$\begin{displaymath}\begin{split} S_{E} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijkp}-\overline{\overline{y}})^{2} \\ &= S_{T} - S_{ABC} \end{split}\end{displaymath}$$ | + | |
| - | ---- | + | |
| ===== 자유도 ===== | ===== 자유도 ===== | ||
| $$\nu_{A}=l-1$$ | $$\nu_{A}=l-1$$ | ||