regression
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| Both sides previous revisionPrevious revisionNext revision | Previous revisionNext revisionBoth sides next revision | ||
| regression [2023/05/17 08:02] – 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 |
| * $\displaystyle r^2=\frac{SS_{total}-SS_{res}}{SS_{total}} = \frac{\text{Explained sample variability}}{\text{Total sample variability}}$ | * $\displaystyle r^2=\frac{SS_{total}-SS_{res}}{SS_{total}} = \frac{\text{Explained sample variability}}{\text{Total sample variability}}$ | ||
| Line 672: | Line 672: | ||
| - | **__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}} $ , | ||
| Line 693: | Line 693: | ||
| * R2 value goes down -- which means | * R2 value goes down -- which means | ||
| * more (many) IVs is not always good | * more (many) IVs is not always good | ||
| - | * 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: | ||
| Line 706: | Line 706: | ||
| | 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 ^ $X-\bar{X}$ | ||
| Line 756: | Line 759: | ||
| 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