차이

문서의 선택한 두 판 사이의 차이를 보여줍니다.

--- 삼원배치법_모수모형_반복있음 [2012/07/27 23:46]
moonrepeat [[자유도]]
+++ 삼원배치법_모수모형_반복있음 [2021/03/10 21:42] (현재)
@@ 줄 1: / 줄 1: @@
 ====== 삼원배치법 (모수모형) (반복있음) ======
 ===== 데이터 구조 =====
- [요인]&nbsp&nbsp $$A$$ 는 [모수인자]
+ [[요인]] $A$는 [[모수인자]]
- [요인]&nbsp&nbsp $$B$$ 는 [모수인자]
+ [[요인]] $B$는 [[모수인자]]
- [요인]&nbsp&nbsp $$C$$ 는 [모수인자]
+ [[요인]] $C$는 [[모수인자]]
+ $$ y_{ijkp} = \mu + a_{i} + b_{j} + c_{k} + (ab)_{ij} + (ac)_{ik} + (bc)_{jk} + (abc)_{ijk} + e_{ijkp} $$
-  $$ y_{ijkp} = \mu + a_{i} + b_{j} + c_{k} + (ab)_{ij} + (ac)_{ik} + (bc)_{jk} + (abc)_{ijk} + e_{ijkp} $$
+  * $y_{ijkp}$  :  $A_{i}$ 와 $B_{j}$ 그리고 $C_{k}$ 에서 얻은 $p$ 번째 [[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
+  * $(abc)_{ijk}$ : $A_{i}$와 $B_{J}$ 그리고 $C_{k}$의 [[교호작용]] 효과
+  * $e_{ijkp}$ : $A_{i}$와 $B_{j}$ 그리고 $C_{k}$에서 얻은 $p$번째 [[측정값]]의 [[오차]] ($e_{ijkp} \sim N(0, \sigma_{E}^{ \ 2})$이고 서로 [[독립]])
-   $$y_{ijkp}$$ &nbsp&nbsp : &nbsp&nbsp $$A_{i}$$ 와&nbsp&nbsp $$B_{j}$$ &nbsp&nbsp그리고&nbsp&nbsp $$C_{k}$$ 에서 얻은&nbsp&nbsp $$p$$ 번째 [측정값]
+  * $i$ : [[인자]] $A$의 [[수준]] 수 $( i = 1,2, \cdots ,l )$
+  * $j$ : [[인자]] $B$의 [[수준]] 수 $( j = 1,2, \cdots ,m )$
-   $$\mu$$ &nbsp&nbsp : 실험전체의 [모평균]
+  * $k$ : [[인자]] $C$의 [[수준]] 수 $( k = 1,2, \cdots ,n )$
+  * $p$ : 실험의 [[반복]] 수 $( p = 1,2, \cdots ,r )$
-   $$a_{i}$$ &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 $$( p = 1,2, \cdots ,r )$$
-----
 ===== 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
-  ||<|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
+ $AB$ 2원표
-  ||<|2> [인자] $$C$$ |||||||| [인자] $$A$$ ||<|2> 합계 ||
+^  [[인자]] $B$  ^  [[인자]] $A$  ^^^^  합계  |
-  || $$A_{1}$$ || $$A_{2}$$ || $$\cdots$$ || $$A_{l}$$ ||
+^:::^  $$A_{1}$$  ^  $$A_{2}$$  ^  $$\cdots$$  ^  $$A_{l}$$  ^:::|
-  |||||||||||| ||
+^  $$B_{1}$$  |  $$T_{11..}$$  |  $$T_{21..}$$  |  $$\cdots$$  |  $$T_{l1..}$$  |  $$T_{.1..}$$  |
-  || $$C_{1}$$ || $$T_{1.1.}$$ || $$T_{2.1.}$$ || $$\cdots$$ || $$T_{l.1.}$$ || $$T_{..1.}$$ ||
+^  $$B_{2}$$  |  $$T_{12..}$$  |  $$T_{22..}$$  |  $$\cdots$$  |  $$T_{l2..}$$  |  $$T_{.2..}$$  |
-  || $$C_{2}$$ || $$T_{1.2.}$$ || $$T_{2.2.}$$ || $$\cdots$$ || $$T_{l.2.}$$ || $$T_{..2.}$$ ||
+^  $$\vdots$$  |  $$\vdots$$  |  $$\vdots$$  |   |  $$\vdots$$  |  $$\vdots$$  |
-  || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || || $$\vdots$$ || $$\vdots$$ ||
+^  $$B_{m}$$  |  $$T_{1m..}$$  |  $$T_{2m..}$$  |  $$\cdots$$  |  $$T_{lm..}$$  |  $$T_{.m..}$$  |
-  || $$C_{n}$$ || $$T_{1.n.}$$ || $$T_{2.n.}$$ || $$\cdots$$ || $$T_{l.n.}$$ || $$T_{..n.}$$ ||
+^  합계  ^  $$T_{1...}$$  ^  $$T_{2...}$$  ^  $$\cdots$$  ^  $$T_{l...}$$  ^  $$T$$  |
-  |||||||||||| ||
-  || 합계 || $$T_{1...}$$ || $$T_{2...}$$ || $$\cdots$$ || $$T_{l...}$$ || $$T$$ ||
-  $$BC$$ 2원표
+ $AC$ 2원표
-  ||<|2> [인자] $$C$$ |||||||| [인자] $$B$$ ||<|2> 합계 ||
+^  [[인자]] $C$  ^  [[인자]] $A$  ^^^^  합계  |
-  || $$B_{1}$$ || $$B_{2}$$ || $$\cdots$$ || $$B_{m}$$ ||
+^:::^  $$A_{1}$$  ^  $$A_{2}$$  ^  $$\cdots$$  ^  $$A_{l}$$  ^:::|
-  |||||||||||| ||
+^  $$C_{1}$$  |  $$T_{1.1.}$$  |  $$T_{2.1.}$$  |  $$\cdots$$  |  $$T_{l.1.}$$  |  $$T_{..1.}$$  |
-  || $$C_{1}$$ || $$T_{.11.}$$ || $$T_{.21.}$$ || $$\cdots$$ || $$T_{.m1.}$$ || $$T_{..1.}$$ ||
+^  $$C_{2}$$  |  $$T_{1.2.}$$  |  $$T_{2.2.}$$  |  $$\cdots$$  |  $$T_{l.2.}$$  |  $$T_{..2.}$$  |
-  || $$C_{2}$$ || $$T_{.12.}$$ || $$T_{.22.}$$ || $$\cdots$$ || $$T_{.m2.}$$ || $$T_{..2.}$$ ||
+^  $$\vdots$$  |  $$\vdots$$  |  $$\vdots$$  |   |  $$\vdots$$  |  $$\vdots$$  |
-  || $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || || $$\vdots$$ || $$\vdots$$ ||
+^  $$C_{n}$$  |  $$T_{1.n.}$$  |  $$T_{2.n.}$$  |  $$\cdots$$  |  $$T_{l.n.}$$  |  $$T_{..n.}$$  |
-  || $$C_{n}$$ || $$T_{.1n.}$$ || $$T_{.2n.}$$ || $$\cdots$$ || $$T_{.mn.}$$ || $$T_{..n.}$$ ||
+^  합계  ^  $$T_{1...}$$  ^  $$T_{2...}$$  ^  $$\cdots$$  ^  $$T_{l...}$$  ^  $$T$$  |
-  |||||||||||| ||
-  || 합계 || $$T_{.1..}$$ || $$T_{.2..}$$ || $$\cdots$$ || $$T_{.m..}$$ || $$T$$ ||
-   || $$T_{i...} = \sum_{j=1}^{m} \sum_{k=1}^{n} \sum_{p=1}^{r} y_{ijkp}$$ || $$\overline{y}_{i...} = \frac{T_{i...}}{mnr}$$ ||
+ $BC$ 2원표
-   || $$T_{.j..} = \sum_{i=1}^{l} \sum_{k=1}^{n} \sum_{p=1}^{r} y_{ijkp}$$ || $$\overline{y}_{.j..} = \frac{T_{.j..}}{lnr}$$ ||
+^  [[인자]] $C$  ^  [[인자]] $B$  ^^^^  합계  |
-   || $$T_{..k.} = \sum_{i=1}^{l} \sum_{j=1}^{m} \sum_{p=1}^{r} y_{ijkp}$$ || $$\overline{y}_{..k.} = \frac{T_{..k.}}{lmr}$$ ||
+^:::^  $$B_{1}$$  ^  $$B_{2}$$  ^  $$\cdots$$  ^  $$B_{m}$$  ^:::|
-   || $$T_{ij..} = \sum_{k=1}^{n} \sum_{p=1}^{r} y_{ijkp}$$ || $$\overline{y}_{ij..} = \frac{T_{ij..}}{nr}$$ ||
+^  $$C_{1}$$  |  $$T_{.11.}$$  |  $$T_{.21.}$$  |  $$\cdots$$  |  $$T_{.m1.}$$  |  $$T_{..1.}$$  |
-   || $$T_{i.k.} = \sum_{j=1}^{m} \sum_{p=1}^{r} y_{ijkp}$$ || $$\overline{y}_{i.k.} = \frac{T_{i.k.}}{mr}$$ ||
+^  $$C_{2}$$  |  $$T_{.12.}$$  |  $$T_{.22.}$$  |  $$\cdots$$  |  $$T_{.m2.}$$  |  $$T_{..2.}$$  |
-   || $$T_{.jk.} = \sum_{i=1}^{l} \sum_{p=1}^{r} y_{ijkp}$$ || $$\overline{y}_{.jk.} = \frac{T_{.jk.}}{lr}$$ ||
+^  $$\vdots$$  |  $$\vdots$$  |  $$\vdots$$  |   |  $$\vdots$$  |  $$\vdots$$  |
-   || $$T_{ijk.} = \sum_{p=1}^{r} y_{ijkp}$$ || $$\overline{y}_{ijk.} = \frac{T_{ijk.}}{r}$$ ||
+^  $$C_{n}$$  |  $$T_{.1n.}$$  |  $$T_{.2n.}$$  |  $$\cdots$$  |  $$T_{.mn.}$$  |  $$T_{..n.}$$  |
-   || $$T = \sum_{i=1}^{l} \sum_{j=1}^{m} \sum_{k=1}^{n} \sum_{p=1}^{r} y_{ijkp}$$ || $$\overline{\overline{y}} = \frac{T}{lmnr} = \frac{T}{N}$$ ||
+^  합계  ^  $$T_{.1..}$$  ^  $$T_{.2..}$$  ^  $$\cdots$$  ^  $$T_{.m..}$$  ^  $$T$$  |
-   || $$N = lmnr$$ || $$CT = \frac{T^{2}}{lmnr} = \frac{T^{2}}{N}$$ ||
-----
-===== [제곱합] =====
- 개개의 데이터&nbsp&nbsp $$y_{ijkp}$$ 와 총편균&nbsp&nbsp $$\overline{\overline{y}}$$ 의 차이는 다음과 같이 8부분으로 나뉘어진다.
-  $$\begin{displaymath}\begin{split} (y_{ijkp}-\overline{\overline{y}}) &= (\overline{y}_{i...} - \overline{\overline{y}}) + (\overline{y}_{.j..} - \overline{\overline{y}}) + (\overline{y}_{..k.} - \overline{\overline{y}}) \\ &+ (\overline{y}_{ij..} - \overline{y}_{i...} - \overline{y}_{.j..} + \overline{\overline{y}}) + (\overline{y}_{i.k.} - \overline{y}_{i...} - \overline{y}_{..k.} + \overline{\overline{y}}) + (\overline{y}_{.jk.} - \overline{y}_{.j..} - \overline{y}_{..k.} + \overline{\overline{y}}) \\ &+ (y_{ijk.} - \overline{y}_{ij..} - \overline{y}_{i.k.} - \overline{y}_{.jk.} + \overline{y}_{i...} + \overline{y}_{.j..} + \overline{y}_{..k.} - \overline{\overline{y}}) \\ &+ (y_{ijkp}-\overline{y}_{ijk.}) \end{split}\end{displaymath}$$
+| $$T_{i...} = \sum_{j=1}^{m} \sum_{k=1}^{n} \sum_{p=1}^{r} y_{ijkp}$$ | $$\overline{y}_{i...} = \frac{T_{i...}}{mnr}$$ |
+| $$T_{.j..} = \sum_{i=1}^{l} \sum_{k=1}^{n} \sum_{p=1}^{r} y_{ijkp}$$ | $$\overline{y}_{.j..} = \frac{T_{.j..}}{lnr}$$ |
+| $$T_{..k.} = \sum_{i=1}^{l} \sum_{j=1}^{m} \sum_{p=1}^{r} y_{ijkp}$$ | $$\overline{y}_{..k.} = \frac{T_{..k.}}{lmr}$$ |
+| $$T_{ij..} = \sum_{k=1}^{n} \sum_{p=1}^{r} y_{ijkp}$$ | $$\overline{y}_{ij..} = \frac{T_{ij..}}{nr}$$ |
+| $$T_{i.k.} = \sum_{j=1}^{m} \sum_{p=1}^{r} y_{ijkp}$$ | $$\overline{y}_{i.k.} = \frac{T_{i.k.}}{mr}$$ |
+| $$T_{.jk.} = \sum_{i=1}^{l} \sum_{p=1}^{r} y_{ijkp}$$ | $$\overline{y}_{.jk.} = \frac{T_{.jk.}}{lr}$$ |
+| $$T_{ijk.} = \sum_{p=1}^{r} y_{ijkp}$$ | $$\overline{y}_{ijk.} = \frac{T_{ijk.}}{r}$$ |
+| $$T = \sum_{i=1}^{l} \sum_{j=1}^{m} \sum_{k=1}^{n} \sum_{p=1}^{r} y_{ijkp}$$ | $$\overline{\overline{y}} = \frac{T}{lmnr} = \frac{T}{N}$$ |
+| $$N = lmnr$$ | $$CT = \frac{T^{2}}{lmnr} = \frac{T^{2}}{N}$$ |
+===== 제곱합 =====
+ 개개의 데이터 $y_{ijkp}$와 총평균 $\overline{\overline{y}}$의 차이는 다음과 같이 8부분으로 나뉘어진다.
- 양변을 제곱한 후에 모든&nbsp&nbsp $$i, \ j, \ k, \ p$$ 에 대하여 합하면 아래의 등식을 얻을 수 있다.
+ $$\begin{displaymath}\begin{split} (y_{ijkp}-\overline{\overline{y}}) &= (\overline{y}_{i...} - \overline{\overline{y}}) + (\overline{y}_{.j..} - \overline{\overline{y}}) + (\overline{y}_{..k.} - \overline{\overline{y}}) \\ &+ (\overline{y}_{ij..} - \overline{y}_{i...} - \overline{y}_{.j..} + \overline{\overline{y}}) + (\overline{y}_{i.k.} - \overline{y}_{i...} - \overline{y}_{..k.} + \overline{\overline{y}}) + (\overline{y}_{.jk.} - \overline{y}_{.j..} - \overline{y}_{..k.} + \overline{\overline{y}}) \\ &+ (y_{ijk.} - \overline{y}_{ij..} - \overline{y}_{i.k.} - \overline{y}_{.jk.} + \overline{y}_{i...} + \overline{y}_{.j..} + \overline{y}_{..k.} - \overline{\overline{y}}) \\ &+ (y_{ijkp}-\overline{y}_{ijk.}) \end{split}\end{displaymath}$$
-  $$\begin{displaymath}\begin{split} \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijkp}-\overline{\overline{y}})^{2} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{i...} - \overline{\overline{y}})^{2} + \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{.j..} - \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{..k.} - \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{ij..} - \overline{y}_{i...} - \overline{y}_{.j..} + \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{i.k.} - \overline{y}_{i...} - \overline{y}_{..k.} + \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{.jk.} - \overline{y}_{.j..} - \overline{y}_{..k.} + \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijk.} - \overline{y}_{ij..} - \overline{y}_{i.k.} - \overline{y}_{.jk.} + \overline{y}_{i...} + \overline{y}_{.j..} + \overline{y}_{..k.} - \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijkp}-\overline{y}_{ijk.})^{2} \end{split}\end{displaymath}$$
+ 양변을 제곱한 후에 모든 $i, \ j, \ k, \ p$에 대하여 합하면 아래의 등식을 얻을 수 있다.
- 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.
+ $$\begin{displaymath}\begin{split} \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijkp}-\overline{\overline{y}})^{2} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{i...} - \overline{\overline{y}})^{2} + \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{.j..} - \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{..k.} - \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{ij..} - \overline{y}_{i...} - \overline{y}_{.j..} + \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{i.k.} - \overline{y}_{i...} - \overline{y}_{..k.} + \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{.jk.} - \overline{y}_{.j..} - \overline{y}_{..k.} + \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijk.} - \overline{y}_{ij..} - \overline{y}_{i.k.} - \overline{y}_{.jk.} + \overline{y}_{i...} + \overline{y}_{.j..} + \overline{y}_{..k.} - \overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijkp}-\overline{y}_{ijk.})^{2} \end{split}\end{displaymath}$$
+ 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.
-  $$\begin{displaymath}\begin{split} S_{T} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijkp}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}y_{ijkp}^{ \ 2} - CT \end{split}\end{displaymath}$$
+ $$\begin{displaymath}\begin{split} S_{T} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijkp}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}y_{ijkp}^{ \ 2} - CT \end{split}\end{displaymath}$$
-  $$\begin{displaymath}\begin{split} S_{A} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{i...}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\frac{T_{i...}^{ \ 2}}{mnr}-CT \end{split}\end{displaymath}$$
+ $$\begin{displaymath}\begin{split} S_{A} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{i...}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\frac{T_{i...}^{ \ 2}}{mnr}-CT \end{split}\end{displaymath}$$
-  $$\begin{displaymath}\begin{split} S_{B} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{.j..}-\overline{\overline{y}})^{2} \\ &= \sum_{j=1}^{m}\frac{T_{.j..}^{ \ 2}}{lnr}-CT \end{split}\end{displaymath}$$
+ $$\begin{displaymath}\begin{split} S_{B} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{.j..}-\overline{\overline{y}})^{2} \\ &= \sum_{j=1}^{m}\frac{T_{.j..}^{ \ 2}}{lnr}-CT \end{split}\end{displaymath}$$
-  $$\begin{displaymath}\begin{split} S_{C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{..k.}-\overline{\overline{y}})^{2} \\ &= \sum_{k=1}^{n}\frac{T_{..k.}^{ \ 2}}{lmr}-CT \end{split}\end{displaymath}$$
+ $$\begin{displaymath}\begin{split} S_{C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{..k.}-\overline{\overline{y}})^{2} \\ &= \sum_{k=1}^{n}\frac{T_{..k.}^{ \ 2}}{lmr}-CT \end{split}\end{displaymath}$$
-  $$\begin{displaymath}\begin{split} S_{A \times B} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{ij..}-\overline{y}_{i...}-\overline{y}_{.j..}+\overline{\overline{y}})^{2} \\ &= S_{AB} - S_{A} - S_{B} \end{split}\end{displaymath}$$
+ $$\begin{displaymath}\begin{split} S_{A \times B} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{ij..}-\overline{y}_{i...}-\overline{y}_{.j..}+\overline{\overline{y}})^{2} \\ &= S_{AB} - S_{A} - S_{B} \end{split}\end{displaymath}$$
-  $$\begin{displaymath}\begin{split} S_{AB} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{ij..}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m} \frac{T_{ij..}^{ \ 2}}{nr} -CT \end{split}\end{displaymath}$$
+ $$\begin{displaymath}\begin{split} S_{AB} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{ij..}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m} \frac{T_{ij..}^{ \ 2}}{nr} -CT \end{split}\end{displaymath}$$
-  $$\begin{displaymath}\begin{split} S_{A \times C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{i.k.}-\overline{y}_{i...}-\overline{y}_{..k.}+\overline{\overline{y}})^{2} \\ &= S_{AC} - S_{A} - S_{C} \end{split}\end{displaymath}$$
+ $$\begin{displaymath}\begin{split} S_{A \times C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{i.k.}-\overline{y}_{i...}-\overline{y}_{..k.}+\overline{\overline{y}})^{2} \\ &= S_{AC} - S_{A} - S_{C} \end{split}\end{displaymath}$$
-  $$\begin{displaymath}\begin{split} S_{AC} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{i.k.}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{k=1}^{n} \frac{T_{i.k.}^{ \ 2}}{mr} -CT \end{split}\end{displaymath}$$
+ $$\begin{displaymath}\begin{split} S_{AC} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{i.k.}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{k=1}^{n} \frac{T_{i.k.}^{ \ 2}}{mr} -CT \end{split}\end{displaymath}$$
-  $$\begin{displaymath}\begin{split} S_{B \times C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{.jk.}-\overline{y}_{.j..}-\overline{y}_{..k.}+\overline{\overline{y}})^{2} \\ &= S_{BC} - S_{B} - S_{C} \end{split}\end{displaymath}$$
+ $$\begin{displaymath}\begin{split} S_{B \times C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{.jk.}-\overline{y}_{.j..}-\overline{y}_{..k.}+\overline{\overline{y}})^{2} \\ &= S_{BC} - S_{B} - S_{C} \end{split}\end{displaymath}$$
-  $$\begin{displaymath}\begin{split} S_{BC} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{.jk.}-\overline{\overline{y}})^{2} \\ &= \sum_{j=1}^{m}\sum_{k=1}^{n} \frac{T_{.jk.}^{ \ 2}}{lr} -CT \end{split}\end{displaymath}$$
+ $$\begin{displaymath}\begin{split} S_{BC} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(\overline{y}_{.jk.}-\overline{\overline{y}})^{2} \\ &= \sum_{j=1}^{m}\sum_{k=1}^{n} \frac{T_{.jk.}^{ \ 2}}{lr} -CT \end{split}\end{displaymath}$$
-  $$\begin{displaymath}\begin{split} S_{A \times B \times C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijk.}-\overline{y}_{ij..}-\overline{y}_{i.k.}-\overline{y}_{.jk.}+\overline{y}_{i...}+\overline{y}_{.j..}+\overline{y}_{..k.}-\overline{\overline{y}})^{2} \\ &= S_{ABC}-(S_{A}+S_{B}+S_{C}+S_{A \times B}+S_{A \times C}+S_{B \times C}) \end{split}\end{displaymath}$$
+ $$\begin{displaymath}\begin{split} S_{A \times B \times C} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijk.}-\overline{y}_{ij..}-\overline{y}_{i.k.}-\overline{y}_{.jk.}+\overline{y}_{i...}+\overline{y}_{.j..}+\overline{y}_{..k.}-\overline{\overline{y}})^{2} \\ &= S_{ABC}-(S_{A}+S_{B}+S_{C}+S_{A \times B}+S_{A \times C}+S_{B \times C}) \end{split}\end{displaymath}$$
-  $$\begin{displaymath}\begin{split} S_{ABC} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijk.}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\frac{T_{ijk.}^{ \ 2}}{r} -CT \end{split}\end{displaymath}$$
+ $$\begin{displaymath}\begin{split} S_{ABC} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijk.}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\frac{T_{ijk.}^{ \ 2}}{r} -CT \end{split}\end{displaymath}$$
-  $$\begin{displaymath}\begin{split} S_{E} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijkp}-\overline{\overline{y}})^{2} \\ &= S_{T} - S_{ABC} \end{split}\end{displaymath}$$
+ $$\begin{displaymath}\begin{split} S_{E} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{n}\sum_{p=1}^{r}(y_{ijkp}-\overline{\overline{y}})^{2} \\ &= S_{T} - S_{ABC} \end{split}\end{displaymath}$$
-----
 ===== 자유도 =====
  $$\nu_{A}=l-1$$

Tools

menus and quick search

quick search

site status

문서 도구

meta data for this page

차이