Effect
일원-결과
편차제곱합(변동)
$$SS_T=SS_{Reg} + SS_{Res}$$
$$\sum_{i=1}^{n}{Y_i}^2=n\bar{Y}^2 + \sum_{i=1}^{n}(Y_i-\bar{Y})^2$$
$$SS_Y=n\bar{Y}^2 +(n-1)S_Y^2$$
$$n\mathrm{E}[Y^2]=n\mathrm{E}[Y]^2 +(n-1)S_Y^2$$
$$\mathrm{E}[Y^2]=\mathrm{E}[Y]^2 +\dfrac{(n-1)}{n}S_Y^2$$
$$\mathrm{E}[Y^2]=\mathrm{E}[Y]^2 +\mathrm{E}[(Y-\bar{Y})^2]$$
$$\mathrm{E}[Y^2]-\mathrm{E}[Y]^2 =S_{Res}^2$$
자유도
$$n=1+ (n-1)$$
회귀편차제곱값은 표본평균제곱값, 1개만 가지므로
회귀편차제곱 자유도는 1
편차제곱평균(분산)
$$MS_T=\dfrac{SS_T}{n}=\dfrac{\sum\limits_{i=1}^{n}{Y_i}^2}{n}=\mathrm{E}[Y^2]$$
$$MS_{Reg}=\dfrac{SS_{Reg}}{1}=\dfrac{n{\bar Y}^2}{1}=n\mathrm{E}[Y]^2$$
$$MS_{Res}=\dfrac{SS_{Res}}{(n-1)}=\dfrac{\sum\limits_{i=1}^{n}(Y_i-\bar{Y})^2}{n-1}=S_Y^2$$
확률분포와 검정통계량
$$F_{1 , n-1}=\dfrac{MS_{Reg}}{MS_{Res}}=\dfrac{n\bar{Y}^2}{S_Y^2}$$
$$t_{n-1}=\sqrt{\dfrac{MS_{Reg}}{MS_{Res}}}=\sqrt{\dfrac{n\bar{Y}^2}{S_Y^2}}$$
$$t_{n-1}=\dfrac{\bar Y}{\mathrm{SE}(\bar Y)}=\dfrac{\bar Y}{\dfrac{S_Y}{\sqrt{n}}}$$
$$Z=\dfrac{\bar Y}{\mathrm{SE}(\bar Y)}=\dfrac{\bar Y}{\dfrac{\sigma_Y}{\sqrt{n}}}=\dfrac{\bar Y}{\sigma_{\overline Y}}$$
이원-결과
편차제곱합(변동)
$$SS_T=SS_{Reg} + SS_{Res}$$
$$\sum_{i=1}^{n}(Y_i-\mu_Y)^2=n(\bar{Y}-\mu_Y)^2 + \sum_{i=1}^{n}(Y_i-\bar{Y})^2$$
자유도
$$n=1+ (n-1)$$
편차제곱평균(분산)
$$MS_T=\frac{SS_{T}}{n}=\frac{\sum\limits_{i=1}^{n}(Y_i-\mu_Y)^2}{n}$$
$$MS_{Reg}=\frac{SS_{Reg}}{1}=\frac{n(\bar{Y}-\mu_Y)^2}{1}$$
$$ MS_{Res}=\frac{SS_{Res}}{n-1}=\frac{\sum\limits_{i=1}^{n}(Y_i-\bar{Y})^2}{n-1}=S_Y^2$$
확률분포와 검정통계량
$$F_{1 , n-1}=\dfrac{MS_{Reg}}{MS_{Res}}=\dfrac{n(\bar{Y}-\mu_Y)^2}{S_Y^2}$$
$$t_{n-1}=\sqrt{\dfrac{MS_{Reg}}{MS_{Res}}}=\sqrt{\dfrac{n(\bar{Y}-\mu_Y)^2}{S_Y^2}}$$
$$t_{n-1}=\dfrac{\bar Y-\mu_Y}{\mathrm{SE}(\bar Y)}=\dfrac{\bar Y-\mu_Y}{\dfrac{S_Y}{\sqrt{n}}}$$
$$Z=\dfrac{\bar Y-\mu_Y}{\mathrm{SE}(\bar Y)}=\dfrac{\bar Y-\mu_Y}{\dfrac{\sigma_Y}{\sqrt{n}}}=\dfrac{\bar Y-\mu_Y}{\sigma_{\overline Y}}$$
중재-효과
편차제곱합(변동)
$$SS_T=SS_{Reg} + SS_{Res}$$
자유도
$$n= 1 + (n-1)$$
편차제곱평균(분산)
$$MS_T=\dfrac{SS_{T}}{n}=\dfrac{\sum\limits_{i=1}^{n}{D_i}^2}{n}$$
$$MS_{Reg}=\dfrac{SS_{Reg}}{1}=\dfrac{n{\bar D}^2}{1}$$
$$MS_{Res}=\dfrac{SS_{Res}}{(n-1)}=\dfrac{\sum\limits_{i=1}^{n}(D_i-\bar{D})^2}{n-1}=S_D^2$$
확률분포와 검정통계량
$$F_{1,n-1}=\dfrac{MS_{Reg}}{MS_{Res}}={\dfrac{n\bar{D}^2}{S_D^2}}$$
$$t_{n-1}=\sqrt{\dfrac{MS_{Reg}}{MS_{Res}}}=\dfrac{\bar D}{\dfrac{S_D}{\sqrt{n}}}$$
$$t_{n-1}=\dfrac{\bar D}{\mathrm{SE}(\bar D)}=\dfrac{\bar D}{\dfrac{S_D}{\sqrt{n}}}$$
$$Z=\dfrac{\bar D}{\mathrm{SE}(\bar D)}=\dfrac{\bar D}{\dfrac{\sigma_D}{\sqrt{n}}}=\dfrac{\bar D}{\sigma_{\overline D}}$$
원인 - 결과 : 선형
편차제곱합(변동)
$$SS_T=SS_{Reg} + SS_{Res}$$
자유도
$$n-1= 1 + (n-2)$$
편차제곱평균(분산)
$$MS_T=\dfrac{SS_{T}}{n-1}$$
$$MS_{Reg}=\dfrac{SS_{Reg}}{1}$$
$$MS_{Res}=\dfrac{SS_{Res}}{n-2}=S_D^2$$
확률분포와 검정통계량
$$F_{1,n-2}=\dfrac{MS_{Reg}}{MS_{Res}}={\dfrac{n\bar{D}^2}{S_D^2}}$$
$$t_{n-2}=\sqrt{\dfrac{MS_{Reg}}{MS_{Res}}}=\dfrac{\bar D}{\dfrac{S_D}{\sqrt{n}}}$$
$$t_{n-2}=\dfrac{\bar D}{\mathrm{SE}(\bar D)}=\dfrac{\bar D}{\dfrac{S_p}{\sqrt{n}}}$$
$$Z=\dfrac{\bar D}{\mathrm{SE}(\bar D)}=\dfrac{\bar D}{\dfrac{\sigma_D}{\sqrt{n}}}=\dfrac{\bar D}{\sigma_{\overline D}}$$
원인 - 결과 : 비선형
편차제곱합(변동)
$$SS_T=SS_{Tr} + SS_E$$
$$SS_T=SS_{Reg} + SS_{Res}$$
자유도
$$n-1=(k-1) + (n-k)$$
$k$는 일원(범주형 원인변수)의 변수값의 수
편차제곱평균(분산)
$$MS_T=\frac{SS_{T}}{n-1}$$
$$MS_{Tr}=\frac{SS_{Tr}}{k-1}$$
$$ MS_E=\frac{SS_{E}}{n-k}$$
확률분포와 검정통계량
$$F_{k-1, n-k}=\dfrac{MS_{Tr}}{MS_E}$$
$$F_{k-1, n-k}=\dfrac{MS_{Reg}}{MS_{Res}}$$
요인 - 반응
편차곱합 - 펀차제곱합
$$SM_{XY}$$
$$SS_{Y}$$
$$SS_{X}$$
자유도
$$n-1$$
$$n-1$$
$$n-1$$
편차곱평균(공분산) - 편차제곱평균(분산)
$$MM_{XY}=\dfrac{SM_{XY}}{n-1}=S_{XY}$$
$$MS_{Y}=\dfrac{SS_{Y}}{n-1}=S_{Y}^2$$
$$MS_{X}=\dfrac{SS_{X}}{n-1}=S_{X}^2$$
표본상관계수, 표본결정계수, 표본회귀계수
표본상관계수 추정량 :
$$R_{XY}=\dfrac{MM_{XY}}{\sqrt{MS_X}\sqrt{MS_Y}}=\dfrac{S_{XY}}{S_X S_Y}=\hat{\beta}_1\dfrac{S_X}{S_Y}$$
표본상관계수제곱 추정량 :
$$R_{XY}^2=\dfrac{(MM_{XY})^2}{MS_X MS_Y}=\dfrac{S_{XY}^2}{S_X^2 S_Y^2}=\hat{\beta}_1^2\dfrac{S_X^2}{S_Y^2}$$
표본회귀계수(기울기) 추정량 :
$$\hat{\beta}_1=\dfrac{MM_{XY}}{MS_X}=\dfrac{S_{XY}}{S_X^2}=R_{XY}\dfrac{S_Y}{S_X}$$
표본회귀계수(절편) 추정량 :
$$\hat{\beta}_0=\bar{Y}-\dfrac{MM_{XY}}{MS_X}\bar{X}=\bar{Y}-\dfrac{S_{XY}}{S_X^2}\bar{X}$$
두 변수 상관
편차제곱합(변동)
$$SS_{T}=SS_{Y}=SS_{Reg}+SS_{Res}$$
$$\sum_{i=1}^{n}{Y_i}^2=n\hat{Y}^2 + \sum_{i=1}^{n}(Y_i-\hat{Y})^2$$
$$SS_Y=n\hat{Y}^2 +(n-1)S_\hat{Y}^2$$
$$n\mathrm{E}[Y^2]=n\mathrm{E}[Y]^2 +(n-1)S_\hat{Y}^2$$
$$\mathrm{E}[Y^2]=\mathrm{E}[Y]^2 +\dfrac{(n-1)}{n}S_\hat{Y}^2$$
$$\mathrm{E}[Y^2]=\mathrm{E}[Y]^2 +\mathrm{E}[(Y-\hat{Y})^2]$$
$$\mathrm{E}[Y^2]-\mathrm{E}[Y]^2 =S_{Res}^2$$
자유도
$$n-1=1+(n-2)$$
편차제곱평균(분산)
$$MS_{T}=MS_{Y}=\dfrac{SS_Y}{n-1}=S_{Y}^2$$
$MS_{Reg}=\dfrac{SS_{Reg}}{1}=S_{Reg}^2$
$$MS_{Res}=\dfrac{SS_{Res}}{n-2}=S_{Res}^2$$
확률분포와 검정통계량
$$F_{1,n-2}=\dfrac{MS_{Reg}}{MS_{Res}}=(n-2)\dfrac{R^2}{1-R^2}$$
$$t_{n-2}=\sqrt{\dfrac{MS_{Reg}}{MS_{Res}}}=\sqrt{(n-2)\dfrac{R^2}{1-R^2}}$$
$$t_{n-2}=\dfrac{R}{\mathrm{SE}(R)}=\dfrac{R}{\sqrt{\dfrac{1-R^2}{n-2}}}$$
표본결정계수(회귀적합성) :
$$R^2≡\dfrac{SS_{Reg}}{SS_T}=\dfrac{MS_{Reg}}{MS_{Reg}+(n-2)MS_{Res}}=\dfrac{F_{1,n-2}}{F_{1,n-2}+(n-2)}$$
$$R^2≡\dfrac{SS_{Reg}}{SS_T}=\dfrac{S_{Reg}^2}{(n-1)S_Y^2}$$
한 원인변수와 한 결과변수 : 회귀(결정계수)
편차제곱합(변동)
$$SS_{T}=SS_{Reg}+SS_{Res}$$
자유도
$$n-1=1+(n-2)$$
편차제곱평균(분산)
$$MS_{T}=MS_{Y}=\dfrac{SS_Y}{n-1}$$
$MS_{Reg}=\dfrac{SS_{Reg}}{1}=S_{Reg}^2$
$$MS_{Res}=\dfrac{SS_{Res}}{n-2}=S_{Res}^2$$
확률분포와 검정통계량
$$F_{1,n-2}=\dfrac{MS_{Reg}}{MS_{Res}}=(n-2)\dfrac{R^2}{1-R^2}$$
$$t_{n-2}=\sqrt{\dfrac{MS_{Reg}}{MS_{Res}}}=\sqrt{(n-2)\dfrac{R^2}{1-R^2}}$$
$$t_{n-2}=\dfrac{R}{\mathrm{SE}(R)}=\dfrac{R}{\sqrt{\dfrac{1-R^2}{n-2}}}$$
표본결정계수(회귀적합성) :
$$R^2≡\dfrac{SS_{Reg}}{SS_T}=\dfrac{MS_{Reg}}{MS_{Reg}+(n-2)MS_{Res}}=\dfrac{F_{1,n-2}}{F_{1,n-2}+(n-2)}$$
$$R^2≡\dfrac{SS_{Reg}}{SS_T}=\dfrac{S_{Reg}^2}{(n-1)S_Y^2}=\dfrac{F_{1,n-2}}{F_{1,n-2}+(n-2)}$$
여러 원인변수와 한 결과변수 : 회귀(결정계수)
편차제곱합(변동)
$$SS_{T}=SS_{Reg}+SS_{Res}$$
자유도
$$n-1=1+(n-2)$$
편차제곱평균(분산)
$$MS_{T}=MS_{Y}=\dfrac{SS_Y}{n-1}$$
$MS_{Reg}=\dfrac{SS_{Reg}}{1}=S_{Reg}^2$
$$MS_{Res}=\dfrac{SS_{Res}}{n-2}=S_{Res}^2$$
확률분포와 검정통계량
$$F_{1,n-2}=\dfrac{MS_{Reg}}{MS_{Res}}=(n-2)\dfrac{R^2}{1-R^2}$$
$$t_{n-2}=\sqrt{\dfrac{MS_{Reg}}{MS_{Res}}}=\sqrt{(n-2)\dfrac{R^2}{1-R^2}}$$
$$t_{n-2}=\dfrac{R}{\mathrm{SE}(R)}=\dfrac{R}{\sqrt{\dfrac{1-R^2}{n-2}}}$$
표본결정계수(회귀적합성) :
$$R^2≡\dfrac{SS_{Reg}}{SS_T}=\dfrac{MS_{Reg}}{MS_{Reg}+(n-2)MS_{Res}}=\dfrac{F_{1,n-2}}{F_{1,n-2}+(n-2)}$$
$$R^2≡\dfrac{SS_{Reg}}{SS_T}=\dfrac{S_{Reg}^2}{(n-1)S_Y^2}=\dfrac{F_{1,n-2}}{F_{1,n-2}+(n-2)}$$