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partial_and_semipartial_correlation [2019/05/30 10:01]
hkimscil [Semipartial cor]
partial_and_semipartial_correlation [2019/11/27 15:10] (current)
hkimscil [Partial and semi-partial correlation]
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 references references
 {{https://​web.stanford.edu/​~hastie/​Papers/​ESLII.pdf|The Elements of Statistical Learning}} or local copy  {{https://​web.stanford.edu/​~hastie/​Papers/​ESLII.pdf|The Elements of Statistical Learning}} or local copy 
- +[{{  :​pasted:​20191127-150222.png?​250}}]
 Simple explanation of the below procedures is like this: Simple explanation of the below procedures is like this:
   * Separately regress Y and X1 against X2, that is,    * Separately regress Y and X1 against X2, that is, 
     * regress Y against X2 AND      * regress Y against X2 AND 
-    * regression ​X1 against X2.+    * regress ​X1 against X2.
   * Regress the Y residuals against the X1 residuals.   * Regress the Y residuals against the X1 residuals.
 In the below example, In the below example,
-  * regress gpa against sat +  * regress gpa against sat (and get residuals of gpa = a + b) 
-  * regress clep against sat +  * regress clep against sat (and get residuals of clep = b + c) 
-  * regress the gpa residuals against clep residuals. +  * regress the gpa residuals against clep residuals. ​(''​%%lm(a+b~b+c)%%''​) 
-Take a close look at the graphs, especially, the grey areas.+  * In this case, $r^{2} = \displaystyle \frac{b}{(a+b)}$ and $b$ is very small. 
 + 
 +Take a close look at the right graph, especially, the ''​%%b%%'' ​areas although clep's is significantly explains gpa (before controlling sat). 
 + 
  
 For more, see https://​stats.stackexchange.com/​questions/​28474/​how-can-adding-a-2nd-iv-make-the-1st-iv-significant For more, see https://​stats.stackexchange.com/​questions/​28474/​how-can-adding-a-2nd-iv-make-the-1st-iv-significant
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 > </​code>​ > </​code>​
 +
 +linear model 
 +''​y hat = 0.0024 X + 1.7848''​
 +''​gpa hat = 0.0024 sat + 1.7848''​
 +
 +
  
 {{lm.gpa.sat.png?​400}} {{lm.gpa.sat.png?​400}}
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 [1] 0.7180529</​code>​ [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"​) > colnames(cor.gpa.sat) <- c("​sat",​ "​gpa",​ "​pred",​ "​resid"​)
 > round(cor.gpa.sat,​5) > round(cor.gpa.sat,​5)
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 10 550 2.9 3.13544 -0.23544 10 550 2.9 3.13544 -0.23544
 > >
-round(cor(cor.gpa.sat),​3)+round(cor(cor.gpa.sat),​4)
         sat   ​gpa ​ pred resid         sat   ​gpa ​ pred resid
 sat   1.000 0.718 1.000 0.000 sat   1.000 0.718 1.000 0.000
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 resid 0.000 0.696 0.000 1.000 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 ===== ===== regression gpa against clep =====
 <​code>#​ import test score data "​tests_cor.csv"​ <​code>#​ import test score data "​tests_cor.csv"​
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 Residual standard error: 0.1637 on 8 degrees of freedom Residual standard error: 0.1637 on 8 degrees of freedom
 Multiple R-squared: ​ 0.7679,​ Adjusted R-squared: ​ 0.7388 ​ 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>​ <​code>​
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 res.lm.gpa.clep <- lm.gpa.clep$residuals res.lm.gpa.clep <- lm.gpa.clep$residuals
 </​code>​ </​code>​
 +
 {{lm.gpa.clep.png?​500}} {{lm.gpa.clep.png?​500}}
 +
 <​code>​ <​code>​
 # get cor between gpa, sat, pred, and resid from. lm.gpa.clep # 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) cor(cor.gpa.clep)
 </​code>​ </​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>​
  
  
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 </​code>​ </​code>​
 +
 +''​Multiple R-squared: ​ 0.7778''​
 +''​F (2, 7) = 12.25, p = 0.005157 ''​
 +
 +''​intercept 1.1607560 p = 0.0249 ''​
 +''​clep 0.0729294 ​ p = 0.0239''​
 +''​sat 0.0007015 ​ p = 0.5940 '' ​
 +
 +One other thing that we could do help determine a pragmatic argument is to regress GPA on both SAT and CLEP at the same time to see what happens. If we do that, we find that R-square for the model is .78, F = 12.25, p < .01. The intercept and b weight for CLEP are both significant,​ but the b weight for SAT is not significant. The values are
 +
 +  * ''​Intercept = 1.16, t=2.844, p < .05''​
 +  * ''​CLEP = 0.07, t=2.874, p < .05''​
 +  * ''​SATQ = -.0007, t=-0.558, n.s.''​
 +
 +In this case, we would conclude that the significant unique predictor is CLEP. Although SAT is highly correlated with GPA, it adds nothing to the prediction equation once the CLEP score is entered. (These data are fictional and the sample size is much too small to run this analysis. It's there for illustration only.)
 +
 +Now suppose we wanted to argue something a little different. Suppose we had a theory that said that all measures of math achievement share a common explanation,​ which is math ability. In other words, the reason that various (all) math achievement tests are correlated is that they share the math ability factor. In other words, math ability explains the correlation between achievement tests. In path diagram form, we might represent this something like this:
 +
 + 
 ===== checking partial cor 1 ===== ===== checking partial cor 1 =====
 <​code>​ <​code>​
partial_and_semipartial_correlation.1559179873.txt.gz · Last modified: 2019/05/30 10:01 by hkimscil