regression
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regression [2023/05/17 08:09] – hkimscil | regression [2023/05/24 08:41] – [E.g., 3. Simple regression: Adjusted R squared & Slope test] hkimscil | ||
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- | **__r-square:__** | + | ===== r-square |
* $\displaystyle r^2=\frac{SS_{total}-SS_{res}}{SS_{total}} = \frac{\text{Explained sample variability}}{\text{Total sample variability}}$ | * $\displaystyle r^2=\frac{SS_{total}-SS_{res}}{SS_{total}} = \frac{\text{Explained sample variability}}{\text{Total sample variability}}$ | ||
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- | **__Adjusted | + | ===== Adjusted |
* $\displaystyle r^2=\frac{SS_{total}-SS_{res}}{SS_{total}} = 1 - \frac{SS_{res}}{SS_{total}} $ , | * $\displaystyle r^2=\frac{SS_{total}-SS_{res}}{SS_{total}} = 1 - \frac{SS_{res}}{SS_{total}} $ , | ||
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* Therefore, the Adjusted r< | * Therefore, the Adjusted r< | ||
- | **__Slope test__** | + | ===== Slope test ===== |
If we take a look at the ANOVA result: | If we take a look at the ANOVA result: | ||
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| b Dependent Variable: y ||||||| | | b Dependent Variable: y ||||||| | ||
<WRAP clear /> | <WRAP clear /> | ||
+ | F test recap. | ||
* ANOVA, F-test, $F=\frac{MS_{between}}{MS_{within}}$ | * ANOVA, F-test, $F=\frac{MS_{between}}{MS_{within}}$ | ||
- | | + | |
- | * MS_within? | + | * MS_within? |
- | * MS for residual | + | * regression에서 within 에 해당하는 것 == residual |
* $s = \sqrt{s^2} = \sqrt{\frac{SS_{res}}{n-2}} $ | * $s = \sqrt{s^2} = \sqrt{\frac{SS_{res}}{n-2}} $ | ||
- | * random difference (MS< | + | |
* MS for regression . . . Obtained difference | * MS for regression . . . Obtained difference | ||
* do the same procedure at the above in MS for < | * do the same procedure at the above in MS for < | ||
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* t-test | * t-test | ||
- | |||
* $\displaystyle t=\frac{b_{1} - \text{Hypothesized value of }\beta_{1}}{s_{b_{1}}}$ | * $\displaystyle t=\frac{b_{1} - \text{Hypothesized value of }\beta_{1}}{s_{b_{1}}}$ | ||
- | |||
* Hypothesized value of beta 값은 대개 0. 따라서 t 값은 | * Hypothesized value of beta 값은 대개 0. 따라서 t 값은 | ||
- | |||
* $\displaystyle t=\frac{b_{1}}{s_{b_{1}}}$ | * $\displaystyle t=\frac{b_{1}}{s_{b_{1}}}$ | ||
+ | * 기울기에 대한 표준오차는 (se) 아래와 같이 구한다 | ||
+ | |||
+ | \begin{eqnarray*} | ||
+ | \displaystyle s_{b_{1}} & = & \sqrt {\frac {MSE}{SS_{X}}} \\ | ||
+ | & = & \displaystyle \sqrt { \frac{1}{n-2} * \frac{SSE}{SS_{X}}} \\ | ||
+ | & = & \displaystyle \sqrt { \frac{1}{n-2} * \frac{ \Sigma{(Y-\hat{Y})^2} }{ \Sigma{ (X_{i} - \bar{X})^2 } } } \\ | ||
+ | \end{eqnarray*} | ||
- | * $\displaystyle s_{b_{1}} = \sqrt {\frac {MSE}{SS_{X}}} = \frac{\sqrt{\frac{SSE}{n-2}}}{\sqrt{SS_{X}}} = \displaystyle \frac{\sqrt{\frac{\Sigma{(Y-\hat{Y})^2}}{n-2}}}{\sqrt{\Sigma{(X_{i}-\bar{X})^2}}} $ | ||
^ X ^ Y ^ $X-\bar{X}$ | ^ X ^ Y ^ $X-\bar{X}$ | ||
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SSE = Sum of Square Error | SSE = Sum of Square Error | ||
기울기 beta(b)에 대한 표준오차값은 아래와 같이 구한다. | 기울기 beta(b)에 대한 표준오차값은 아래와 같이 구한다. | ||
+ | |||
\begin{eqnarray*} | \begin{eqnarray*} | ||
se_{\beta} & = & \frac {\sqrt{SSE/ | se_{\beta} & = & \frac {\sqrt{SSE/ |
regression.txt · Last modified: 2024/05/22 08:19 by hkimscil