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--- partial_and_semipartial_correlation [2019/10/12 23:51] – [regression gpa against sat] hkimscil
+++ partial_and_semipartial_correlation [2019/10/13 21:25] – [regression gpa against clep] hkimscil
@@ Line 84: / Line 84: @@
 linear model
-y hat = 0.0024 X + 1.7848
+''y hat = 0.0024 X + 1.7848''
+''gpa hat = 0.0024 sat + 1.7848''
@@ Line 93: / Line 94: @@
 [1] 0.7180529</code>
+Collect
-<code>> cor.gpa.sat <- as.data.frame(cbind(sat, gpa, lm.gpa.sat$fitted.values, lm.gpa.sat$residuals))
+  - sat,
+  - gpa,
+  - predicted value (y hat),
+  - residuals (error)
+And see correlation among themselves.
+<code>
+> cor.gpa.sat <- as.data.frame(cbind(sat, gpa, lm.gpa.sat$fitted.values, lm.gpa.sat$residuals))
 > colnames(cor.gpa.sat) <- c("sat", "gpa", "pred", "resid")
 > round(cor.gpa.sat,5)
@@ Line 109: / Line 116: @@
 550 2.9 3.13544 -0.23544
 >
-> round(cor(cor.gpa.sat),3)
+round(cor(cor.gpa.sat),4)
         sat   gpa  pred resid
 sat   1.000 0.718 1.000 0.000
@@ Line 116: / Line 123: @@
 resid 0.000 0.696 0.000 1.000
 >
-> </code>
+</code>
+Note that
+  * r (sat and gpa) = .718 (sqrt(r<sup>2</sup>=0.5156)
+  * r (sat and pred) = 1. In other words, predicted values (y hats) are the linear function of x (sat) values (''y hat = 0.0024 X + 1.7848'').
+  * r (sat and resid) = 0. residuals are orthogonal to the independent (sat) values.
 ===== regression gpa against clep =====
 <code># import test score data "tests_cor.csv"
@@ Line 145: / Line 156: @@
 Residual standard error: 0.1637 on 8 degrees of freedom
 Multiple R-squared:  0.7679,	Adjusted R-squared:  0.7388
-F-statistic: 26.46 on 1 and 8 DF,  p-value: 0.0008808</code>
+F-statistic: 26.46 on 1 and 8 DF,  p-value: 0.0008808
+</code>
+''y hat = 0.06054 * clep + 1.17438''
 <code>
@@ Line 151: / Line 167: @@
 res.lm.gpa.clep <- lm.gpa.clep$residuals
 </code>
 {{lm.gpa.clep.png?500}}
 <code>
 # get cor between gpa, sat, pred, and resid from. lm.gpa.clep
-cor.gpa.clep <- as.data.frame(cbind(gpa, clep, lm.gpa.clep$fitted.values, lm.gpa.clep$residuals))
+cor.gpa.clep <- as.data.frame(cbind(clep, gpa, lm.gpa.clep$fitted.values, lm.gpa.clep$residuals))
-colnames(cor.gpa.clep) <- c("gpa", "clep", "pred", "resid")
+colnames(cor.gpa.clep) <- c("clep", "gpa", "pred", "resid")
 cor(cor.gpa.clep)
 </code>
-<code>         gpa   clep   pred  resid
+<code>
-gpa   1.0000 0.8763 0.8763 0.4818
+> round(cor(cor.gpa.clep),4)
-clep  0.8763 1.0000 1.0000 0.0000
+        clep    gpa   pred  resid
-pred  0.8763 1.0000 1.0000 0.0000
+clep  1.0000 0.8763 1.0000 0.0000
-resid 0.4818 0.0000 0.0000 1.0000
+gpa   0.8763 1.0000 0.8763 0.4818
-> </code>
+pred  1.0000 0.8763 1.0000 0.0000
+resid 0.0000 0.4818 0.0000 1.0000
+>
+        sat   gpa  pred resid
+sat   1.0000 0.7180 1.0000 0.0000
+gpa   0.7180 1.0000 0.7180 0.6960
+pred  1.0000 0.7180 1.0000 0.0000
+resid 0.0000 0.6960 0.0000 1.0000
+>
+</code>