Differences

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--- regression [2023/05/17 08:40] – hkimscil
+++ regression [2023/05/24 08:53] (current) – [Slope test] hkimscil
@@ Line 663: / Line 663: @@
-**__r-square:__**
+===== r-square =====
   * $\displaystyle r^2=\frac{SS_{total}-SS_{res}}{SS_{total}} = \frac{\text{Explained sample variability}}{\text{Total sample variability}}$
@@ Line 672: / Line 672: @@
-**__Adjusted r-square:__**
+===== Adjusted r-square =====
   * $\displaystyle r^2=\frac{SS_{total}-SS_{res}}{SS_{total}} = 1 - \frac{SS_{res}}{SS_{total}} $ ,
@@ Line 695: / Line 695: @@
   * Therefore, the Adjusted r<sup>2</sup> = 1- (.367 / 1.5) = 0.756 (green color cell)
-**__Slope test__**
+===== Slope test =====
 If we take a look at the ANOVA result:
@@ Line 706: / Line 706: @@
 | b Dependent Variable: y    |||||||
 <WRAP clear />
+F test recap.
   * ANOVA, F-test, $F=\frac{MS_{between}}{MS_{within}}$
-  * MS_between?
+    * MS_between?
-  * MS_within?
+    * MS_within?
-  * MS for residual
+  * regression에서 within 에 해당하는 것 == residual
    * $s = \sqrt{s^2} = \sqrt{\frac{SS_{res}}{n-2}} $
-   * random difference (MS<sub>within</sub> ): $s^2 = \frac{SS_{res}}{n-2} $
+   * 왜냐하면 이 ss residual이 random difference 를 말하는 것이므로 (MS<sub>within</sub> ): $s^2 = \frac{SS_{res}}{n-2} $
   * MS for regression . . . Obtained difference
    * do the same procedure at the above in MS for <del>residual</del> regression.
@@ Line 729: / Line 729: @@
   * Why do we do t-test for the slope of X variable? The below is a mathematical explanation for this.
-  * Sampling distribution of b:
+  * Sampling distribution of error around the slope line b:
    * $\displaystyle \sigma_{b_{1}} = \frac{\sigma}{\sqrt{SS_{x}}}$
+     * We remember that $\displaystyle \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$ ?
    * estimation of $\sigma_{b_{1}}$ : substitute sigma with s
+만약에 error들이 (residual들) slope b를 중심으로 포진해 있고, 이것을 따로 떼어내서 distribution curve를 그려보면 평균이 0이고 standard deviation이 위의 standard error값을 갖는 normal distribution을 이루게 될 것이다.
   * t-test
    * $\displaystyle t=\frac{b_{1} - \text{Hypothesized value of }\beta_{1}}{s_{b_{1}}}$
+   * Hypothesized value of b 값은 (혹은 beta) 0. 따라서 t 값은
-   * Hypothesized value of beta 값은 대개 0. 따라서 t 값은
    * $\displaystyle t=\frac{b_{1}}{s_{b_{1}}}$
+   * 기울기에 대한 표준오차는 (se) 아래와 같이 구한다
 \begin{eqnarray*}
 \displaystyle s_{b_{1}} & = & \sqrt {\frac {MSE}{SS_{X}}} \\
- & = & \sqrt { \frac{1}{n-2}* \frac{SSE}{SS_{X}}} \\
+ & = & \displaystyle \sqrt { \frac{1}{n-2} * \frac{SSE}{SS_{X}}} \\
- & = & \frac{\sqrt{\frac{SSE}{n-2}}}{\sqrt{SS_{X}}} \\
+ & = & \displaystyle \sqrt { \frac{1}{n-2} * \frac{ \Sigma{(Y-\hat{Y})^2} }{ \Sigma{ (X_{i} - \bar{X})^2 } } } \\
- & = & \displaystyle \frac{\sqrt{\frac{\Sigma{(Y-\hat{Y})^2}}{n-2}}}{\sqrt{\Sigma{(X_{i}-\bar{X})^2}}} \\
 \end{eqnarray*}
@@ Line 760: / Line 757: @@
 Regression formula: y<sub>predicted</sub> = -0.1 + 0.7 X
-SSE = Sum of Square Error
+SSE = Sum of Square Error = SS_residual
 기울기 beta(b)에 대한 표준오차값은 아래와 같이 구한다.
 \begin{eqnarray*}
 se_{\beta} & = & \frac {\sqrt{SSE/n-2}}{\sqrt{SSX}} \\
@@ Line 770: / Line 768: @@
 따라서 t = b / se = 3.655631
-<code>x <- c(1, 2, 3, 4, 5)
-y <- c(1, 1, 2, 2, 4)
-mody <- lm(y ~ x)
-</code>
-<code>
-> x <- c(1, 2, 3, 4, 5)
-> y <- c(1, 1, 2, 2, 4)
-> mody <- lm(y ~ x)
-> summary(mody)
-Call:
-lm(formula = y ~ x)
-Residuals:
-          2          3          4          5
-.000e-01 -3.000e-01 -3.886e-16 -7.000e-01  6.000e-01
-Coefficients:
-            Estimate Std. Error t value Pr(>|t|)
-(Intercept)  -0.1000     0.6351  -0.157   0.8849
-x             0.7000     0.1915   3.656   0.0354 *
----
-Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
-Residual standard error: 0.6055 on 3 degrees of freedom
-Multiple R-squared:  0.8167,	Adjusted R-squared:  0.7556
-F-statistic: 13.36 on 1 and 3 DF,  p-value: 0.03535
->
-</code>
 ====== E.g., 4. Simple regression ======
 Another example of simple regression: from {{:elemapi.sav}} \\