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--- r:linear_regression [2018/06/15 08:53] – [e.g. 3] hkimscil
+++ r:linear_regression [2019/06/13 10:15] (current) – hkimscil
@@ Line 51: / Line 51: @@
 Does the model fit the data well?
   * **Plot the residuals** and check the regression diagnostics.
+    * see [[https://drsimonj.svbtle.com/visualising-residuals|visualizing residuals]]
 Does the data satisfy the assumptions behind linear regression?
   * Check whether the diagnostics confirm that a linear model is reasonable for your data.
@@ Line 146: / Line 147: @@
 </WRAP>
+<WRAP info>
+What about beta coefficient?
+<blockquote>In R we demonstrate the use of the lm.beta() function in the QuantPsyc package (due to Thomas D. Fletcher of State Farm). The function is short and sweet, and takes a linear model object as argument:</blockquote>
+<code>> lm.beta(mod)
+EngineSize
+-0.7100032
+> cor(MPG.city,EngineSize)
+[1] -0.7100032
+>
+</code></WRAP>
 ====== Multiple Regression ======
+regression output table
 | anova(m)  | ANOVA table  |
 | coefficients(m) = coef(m)  | Model coefficients  |
@@ Line 184: / Line 198: @@
 F-statistic: 54.83 on 2 and 90 DF,  p-value: 2.674e-16
 </code>
-Questions:
+<WRAP box help>Questions:
   * What is R<sup>2</sup>?
   * How many cars are involved in this test? (cf. df = 90)
+    * df + # of variables involved (3) = 93
+    * check 'str(Cars93)'
   * If I eliminate the R<sup>2</sup> from the above output, can you still identify what it is?
+</WRAP>
+<WRAP box info>The last question:
+  * If I eliminate the R<sup>2</sup> from the above output, can you still identify what it is?
+  * to answer the question, use the regression output table:
+R<sup>2</sup> = SS<sub>reg</sub>/SS<sub>total</sub> = 1 - SS<sub>res</sub>/SS<sub>total</sub>
+=
+<code>> anova(lm.model)
+Analysis of Variance Table
+Response: Cars93$MPG.city
+                  Df Sum Sq Mean Sq F value  Pr(>F)
+Cars93$EngineSize  1   1465    1465  100.65 2.4e-16 ***
+Cars93$Price       1    131     131    9.01  0.0035 **
+Residuals         90   1310      15
+---
+Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
+> sstotal = 1465+131+1310
+> ssreg <- 1465+131
+> ssreg/sstotal
+[1] 0.54921
+>
+> # or
+> 1-(deviance(lm.model)/sstotal)
+[1] 0.54932
+</code>
+</WRAP>
 Regression formula:
   * $\hat{Y} = -2.99 * EngineSize + -0.15 * Price +  33.35$
   * $\hat{Y} = \widehat{\text{MPG.city}}$
+<WRAP box info>in the meantime,
+<code>> lm.beta(lm.model)
+Cars93$EngineSize      Cars93$Price
+       -0.5517121        -0.2649553
+> cor(MPG.city,EngineSize)
+[1] -0.7100032
+> cor(EngineSize,Price)
+[1] 0.5974254
+> cor(MPG.city,Price)
+[1] -0.5945622
+>
+</code>
+Or . . . .
+<code>> temp <- subset(Cars93, select=c(MPG.city,EngineSize,Price))
+> temp
+. . . .
+> cor(temp)
+             MPG.city EngineSize      Price
+MPG.city    1.0000000 -0.7100032 -0.5945622
+EngineSize -0.7100032  1.0000000  0.5974254
+Price      -0.5945622  0.5974254  1.0000000
+>
+</code>
+Beta coefficients are not equal to correlations among variables.
+</WRAP>
 <code>plot(lm.model$residuals)