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차이
문서의 선택한 두 판 사이의 차이를 보여줍니다.
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이원배치법_혼합모형_반복있음 [2012/07/26 09:08] moonrepeat [[자유도]] |
이원배치법_혼합모형_반복있음 [2021/03/10 21:42] (현재) |
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| * $k$ : 실험의 [[반복]] 수 $( j = 1,2, \cdots ,r )$ | * $k$ : 실험의 [[반복]] 수 $( j = 1,2, \cdots ,r )$ | ||
| ===== 자료의 구조 ===== | ===== 자료의 구조 ===== | ||
| - | ||<|2> [인자] $$B$$ |||||||||||||| [인자] $$A$$ ||<|2> 합계 ||<|2> [평균] || | + | ^ [[인자]]\\ $B$ ^ [[인자]] $A$ ^^^^^^^ 합계 ^ [[평균]] | |
| - | |||| $$A_{1}$$ |||| $$A_{2}$$ || $$\cdots$$ |||| $$A_{l}$$ || | + | ^:::^ $$A_{1}$$ ^^ $$A_{2}$$ ^^ $$\cdots$$ ^ $$A_{l}$$ ^^:::^:::| |
| - | |||||||||||||||||||| || | + | ^ $$B_{1}$$ | $$y_{111}$$ | $$T_{11.}$$ | $$y_{211}$$ | $$T_{21.}$$ | $$\cdots$$ | $$y_{l11}$$ | $$T_{l1.}$$ | $$T_{.1.}$$ | $$\overline{y}_{.1.}$$ | |
| - | ||<|4> $$B_{1}$$ || $$y_{111}$$ ||<|2> $$T_{11.}$$ || $$y_{211}$$ ||<|2> $$T_{21.}$$ ||<|4> $$\cdots$$ || $$y_{l11}$$ ||<|2> $$T_{l1.}$$ ||<|4> $$T_{.1.}$$ ||<|4> $$\overline{y}_{.1.}$$ || | + | |:::| $$y_{112}$$ |:::| $$y_{212}$$ |:::|:::| $$y_{l12}$$ |:::|:::|:::| |
| - | || $$y_{112}$$ || $$y_{212}$$ || $$y_{l12}$$ || | + | |:::| $$\vdots$$ | $$\overline{y}_{11.}$$ | $$\vdots$$ | $$\overline{y}_{21.}$$ |:::| $$\vdots$$ | $$\overline{y}_{l1.}$$ |:::|:::| |
| - | || $$\vdots$$ ||<|2> $$\overline{y}_{11.}$$ || $$\vdots$$ ||<|2> $$\overline{y}_{21.}$$ || $$\vdots$$ ||<|2> $$\overline{y}_{l1.}$$ || | + | |:::| $$y_{11r}$$ |:::| $$y_{21r}$$ |:::|:::| $$y_{l1r}$$ |:::|:::|:::| |
| - | || $$y_{11r}$$ || $$y_{21r}$$ || $$y_{l1r}$$ || | + | ^ $$B_{2}$$ | $$y_{121}$$ | $$T_{12.}$$ | $$y_{221}$$ | $$T_{22.}$$ | $$\cdots$$ | $$y_{l21}$$ | $$T_{l2.}$$ | $$T_{.2.}$$ | $$\overline{y}_{.2.}$$ | |
| - | ||<|4> $$B_{2}$$ || $$y_{121}$$ ||<|2> $$T_{12.}$$ || $$y_{221}$$ ||<|2> $$T_{22.}$$ ||<|4> $$\cdots$$ || $$y_{l21}$$ ||<|2> $$T_{l2.}$$ ||<|4> $$T_{.2.}$$ ||<|4> $$\overline{y}_{.2.}$$ || | + | |:::| $$y_{122}$$ |:::| $$y_{222}$$ |:::|:::| $$y_{l22}$$ |:::|:::|:::| |
| - | || $$y_{122}$$ || $$y_{222}$$ || $$y_{l22}$$ || | + | |:::| $$\vdots$$ | $$\overline{y}_{12.}$$ | $$\vdots$$ | $$\overline{y}_{22.}$$ |:::| $$\vdots$$ | $$\overline{y}_{l2.}$$ | :::|:::| |
| - | || $$\vdots$$ ||<|2> $$\overline{y}_{12.}$$ || $$\vdots$$ ||<|2> $$\overline{y}_{22.}$$ || $$\vdots$$ ||<|2> $$\overline{y}_{l2.}$$ || | + | |:::| $$y_{12r}$$ |:::| $$y_{22r}$$ |:::|:::| $$y_{l2r}$$ |:::|:::|:::| |
| - | || $$y_{12r}$$ || $$y_{22r}$$ || $$y_{l2r}$$ || | + | ^ $$\vdots$$ | $$\vdots$$ || $$\vdots$$ || $$\vdots$$ | $$\vdots$$ || $$\vdots$$ | $$\vdots$$ | |
| - | || $$\vdots$$ |||| $$\vdots$$ |||| $$\vdots$$ || $$\vdots$$ |||| $$\vdots$$ || $$\vdots$$ || $$\vdots$$ || | + | ^ $$B_{m}$$ | $$y_{1m1}$$ | $$T_{1m.}$$ | $$y_{2m1}$$ | $$T_{2m.}$$ | $$\cdots$$ | $$y_{lm1}$$ | $$T_{lm.}$$ | $$T_{.m.}$$ | $$\overline{y}_{.m.}$$ | |
| - | ||<|4> $$B_{m}$$ || $$y_{1m1}$$ ||<|2> $$T_{1m.}$$ || $$y_{2m1}$$ ||<|2> $$T_{2m.}$$ ||<|4> $$\cdots$$ || $$y_{lm1}$$ ||<|2> $$T_{lm.}$$ ||<|4> $$T_{.m.}$$ ||<|4> $$\overline{y}_{.m.}$$ || | + | |:::| $$y_{1m2}$$ |:::| $$y_{2m2}$$ |:::|:::| $$y_{lm2}$$ |:::|:::|:::| |
| - | || $$y_{1m2}$$ || $$y_{2m2}$$ || $$y_{lm2}$$ || | + | |:::| $$\vdots$$ | $$\overline{y}_{1m.}$$ | $$\vdots$$ | $$\overline{y}_{2m.}$$ |:::| $$\vdots$$ | $$\overline{y}_{lm.}$$ |:::|:::| |
| - | || $$\vdots$$ ||<|2> $$\overline{y}_{1m.}$$ || $$\vdots$$ ||<|2> $$\overline{y}_{2m.}$$ || $$\vdots$$ ||<|2> $$\overline{y}_{lm.}$$ || | + | |:::| $$y_{1mr}$$ |:::| $$y_{2mr}$$ |:::|:::| $$y_{lmr}$$ |:::|:::|:::| |
| - | || $$y_{1mr}$$ || $$y_{2mr}$$ || $$y_{lmr}$$ || | + | ^ 합계 ^ $$T_{1..}$$ ^^ $$T_{2..}$$ ^^ $$\cdots$$ ^ $$T_{l..}$$ ^^ $$T$$ ^ ^ |
| - | |||||||||||||||||||| || | + | ^ [[평균]] ^ $$\overline{y}_{1..}$$ ^^ $$\overline{y}_{2..}$$ ^^ $$\cdots$$ ^ $$\overline{y}_{l..}$$ ^^ ^ $$\overline{\overline{y}}$$ ^ |
| - | || 합계 |||| $$T_{1..}$$ |||| $$T_{2..}$$ || $$\cdots$$ |||| $$T_{l..}$$ || $$T$$ || || | + | |
| - | || [평균] |||| $$\overline{y}_{1..}$$ |||| $$\overline{y}_{2..}$$ || $$\cdots$$ |||| $$\overline{y}_{l..}$$ || || $$\overline{\overline{y}}$$ || | + | |
| - | || $$T_{i..} = \sum_{j=1}^{m} \sum_{k=1}^{r} y_{ijk}$$ || $$\overline{y}_{i..} = \frac{T_{i..}}{mr}$$ || | + | | $$T_{i..} = \sum_{j=1}^{m} \sum_{k=1}^{r} y_{ijk}$$ | $$\overline{y}_{i..} = \frac{T_{i..}}{mr}$$ | |
| - | || $$T_{.j.} = \sum_{i=1}^{l} \sum_{k=1}^{r} y_{ijk}$$ || $$\overline{y}_{.j.} = \frac{T_{.j.}}{lr}$$ || | + | | $$T_{.j.} = \sum_{i=1}^{l} \sum_{k=1}^{r} y_{ijk}$$ | $$\overline{y}_{.j.} = \frac{T_{.j.}}{lr}$$ | |
| - | || $$T_{ij.} = \sum_{k=1}^{r} y_{ijk}$$ || $$\overline{y}_{ij.} = \frac{T_{ij.}}{r}$$ || | + | | $$T_{ij.} = \sum_{k=1}^{r} y_{ijk}$$ | $$\overline{y}_{ij.} = \frac{T_{ij.}}{r}$$ | |
| - | || $$T = \sum_{i=1}^{l} \sum_{j=1}^{m} \sum_{k=1}^{r} y_{ijk}$$ || $$\overline{\overline{y}} = \frac{T}{lmr} = \frac{T}{N}$$ || | + | | $$T = \sum_{i=1}^{l} \sum_{j=1}^{m} \sum_{k=1}^{r} y_{ijk}$$ | $$\overline{\overline{y}} = \frac{T}{lmr} = \frac{T}{N}$$ | |
| - | || $$N = lmr$$ || $$CT = \frac{T^{2}}{lmr} = \frac{T^{2}}{N}$$ || | + | | $$N = lmr$$ | $$CT = \frac{T^{2}}{lmr} = \frac{T^{2}}{N}$$ | |
| - | ---- | + | ===== 제곱합 ===== |
| - | ===== [제곱합] ===== | + | 개개의 데이터 $y_{ijk}$와 총평균 $\overline{\overline{y}}$의 차이는 다음과 같이 네 부분으로 나뉘어진다. |
| - | 개개의 데이터 $$y_{ijk}$$ 와 총편균 $$\overline{\overline{y}}$$ 의 차이는 다음과 같이 네 부분으로 나뉘어진다. | + | |
| - | $$(y_{ijk}-\overline{\overline{y}})=(y_{i..}-\overline{\overline{y}})+(y_{.j.}-\overline{\overline{y}})+(\overline{y}_{ij.}-\overline{y}_{i..}-\overline{y}_{.j.}+\overline{\overline{y}})+(y_{ijk}-\overline{y}_{ij.})$$ | + | $$(y_{ijk}-\overline{\overline{y}})=(y_{i..}-\overline{\overline{y}})+(y_{.j.}-\overline{\overline{y}})+(\overline{y}_{ij.}-\overline{y}_{i..}-\overline{y}_{.j.}+\overline{\overline{y}})+(y_{ijk}-\overline{y}_{ij.})$$ |
| - | 양변을 제곱한 후에 모든 $$i, \ j, \ k$$ 에 대하여 합하면 아래의 등식을 얻을 수 있다. | + | 양변을 제곱한 후에 모든 $i, \ j, \ k$에 대하여 합하면 아래의 등식을 얻을 수 있다. |
| - | $$\begin{displaymath}\begin{split} \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(y_{ijk}-\overline{\overline{y}})^{2} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(y_{i..}-\overline{\overline{y}})^{2}+\sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(y_{.j.}-\overline{\overline{y}})^{2}+\sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(\overline{y}_{ij.}-\overline{y}_{i..}-\overline{y}_{.j.}+\overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(y_{ijk}-\overline{y}_{ij.})^{2} \end{split}\end{displaymath}$$ | + | $$\begin{displaymath}\begin{split} \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(y_{ijk}-\overline{\overline{y}})^{2} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(y_{i..}-\overline{\overline{y}})^{2}+\sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(y_{.j.}-\overline{\overline{y}})^{2}+\sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(\overline{y}_{ij.}-\overline{y}_{i..}-\overline{y}_{.j.}+\overline{\overline{y}})^{2} \\ &+ \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(y_{ijk}-\overline{y}_{ij.})^{2} \end{split}\end{displaymath}$$ |
| - | 위 식에서 왼쪽 항은 총변동 $$S_{T}$$ 이고, 오른쪽 항은 차례대로 $$A$$ 의 [변동], $$B$$ 의 [변동], $$A, \ B$$ 의 [교호작용]의 변동 [오차변동]인 $$S_{A}$$ , $$S_{B}$$ , $$S_{A \times B}$$ , $$S_{E}$$ 가 된다. | + | 위 식에서 왼쪽 항은 총변동 $S_{T}$이고, 오른쪽 항은 차례대로 $A$의 [[변동]], $B$의 [[변동]], $A, \ B$의 [[교호작용]]의 변동 [[오차변동]]인 $S_{A}$, $S_{B}$, $S_{A \times B}$, $S_{E}$가 된다. |
| + | $$\begin{displaymath}\begin{split} S_{T} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(y_{ijk}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}y_{ijk}^{ \ 2} - CT \end{split}\end{displaymath}$$ | ||
| - | $$\begin{displaymath}\begin{split} S_{T} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(y_{ijk}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}y_{ijk}^{ \ 2} - CT \end{split}\end{displaymath}$$ | + | $$\begin{displaymath}\begin{split} S_{A} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(y_{i..}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\frac{T_{i..}^{ \ 2}}{mr}-CT \end{split}\end{displaymath}$$ |
| - | $$\begin{displaymath}\begin{split} S_{A} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(y_{i..}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\frac{T_{i..}^{ \ 2}}{mr}-CT \end{split}\end{displaymath}$$ | + | $$\begin{displaymath}\begin{split} S_{B} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(y_{.j.}-\overline{\overline{y}})^{2} \\ &= \sum_{j=1}^{m}\frac{T_{.j.}^{ \ 2}}{lr}-CT \end{split}\end{displaymath}$$ |
| - | $$\begin{displaymath}\begin{split} S_{B} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(y_{.j.}-\overline{\overline{y}})^{2} \\ &= \sum_{j=1}^{m}\frac{T_{.j.}^{ \ 2}}{lr}-CT \end{split}\end{displaymath}$$ | + | $$\begin{displaymath}\begin{split} S_{A \times B} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(\overline{y}_{ij.}-\overline{y}_{i..}-\overline{y}_{.j.}+\overline{\overline{y}})^{2} \\ &= S_{AB} - S_{A} - S_{B} \end{split}\end{displaymath}$$ |
| - | $$\begin{displaymath}\begin{split} S_{A \times B} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(\overline{y}_{ij.}-\overline{y}_{i..}-\overline{y}_{.j.}+\overline{\overline{y}})^{2} \\ &= S_{AB} - S_{A} - S_{B} \end{split}\end{displaymath}$$ | + | $$\begin{displaymath}\begin{split} S_{AB} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(\overline{y}_{ij.}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m} \frac{T_{ij.}^{ \ 2}}{r} -CT \end{split}\end{displaymath}$$ |
| - | $$\begin{displaymath}\begin{split} S_{AB} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(\overline{y}_{ij.}-\overline{\overline{y}})^{2} \\ &= \sum_{i=1}^{l}\sum_{j=1}^{m} \frac{T_{ij.}^{ \ 2}}{r} -CT \end{split}\end{displaymath}$$ | + | $$\begin{displaymath}\begin{split} S_{E} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(y_{ijk}-\overline{y}_{ij.})^{2} \\ &= S_{T}-S_{AB} \end{split}\end{displaymath}$$ |
| - | + | ||
| - | $$\begin{displaymath}\begin{split} S_{E} &= \sum_{i=1}^{l}\sum_{j=1}^{m}\sum_{k=1}^{r}(y_{ijk}-\overline{y}_{ij.})^{2} \\ &= S_{T}-S_{AB} \end{split}\end{displaymath}$$ | + | |
| - | ---- | + | |
| ===== 자유도 ===== | ===== 자유도 ===== | ||
| $$\nu_{_{A}} = l-1$$ | $$\nu_{_{A}} = l-1$$ | ||