This document discusses correlation and regression analysis. It defines concepts like covariance, correlation, and regression. Examples are provided to illustrate covariance, correlation coefficients from scatter plots, and linear regression analysis in R using the mtcars dataset. The regression case study finds a negative correlation between weight and miles per gallon for cars, with a slope of -5.3 and intercept of 37 for the best fit line. In conclusion, the document covers key concepts for understanding the relationship between variables using statistical techniques like correlation and regression.

1. Correlation and
Regression in R
Hamid Reza Bolhasani
PhD, Data Scientist
Jan 2020
1

2. Table of contents
- Covariance
- Correlation
- Examples
- Regression
- Case Study in R
- Conclusion
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3. Covariance
1
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Variance
Gives information of a single variable
Covariance
Gives information on the degree to which two variables
vary together.
 When X and Y cov (x,y) = pos.
 When X and Y cov (x,y) = neg.
 When no constant relationship: cov (x,y) = 0
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4. Covariance Example
High variance data Low variance data
Subject x y x error * y
error
x y X error * y
error
1 101 100 2500 54 53 9
2 81 80 900 53 52 4
3 61 60 100 52 51 1
4 51 50 0 51 50 0
5 41 40 100 50 49 1
6 21 20 900 49 48 4
7 1 0 2500 48 47 9
Mean 51 50 51 50
Sum of x error * y error : 7000 Sum of x error * y error : 28
Covariance: 1166.67 Covariance: 4.67
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5. Correlation & Regression
Correlation
- Is there any relationship between 2 variables (x,y)?
- X is independent (Explanatory) and Y is dependent (Response)
- Correlation ≠ Causation
Regression
How well a certain independent variable predict dependent variable?
     2 2
2 2
N XY X Y
N X X N Y Y
xyr
 
  

   
xy
xy
x y
s
r
s s
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6. Correlation in Scatter Diagrams
x
y
Strong negative correlation
x
y
Weak positive correlation
x
y
Strong positive correlation
x
y
Nonlinear Correlation
r = 0.91 r = 0.88
r = 0.42
r = 0.07
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7. Regression Example
Smoking vs Lung Capacity
N Cigarettes (X ) Lung Capacity (Y )
1 0 45
2 5 42
3 10 33
4 15 31
5 20 29
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8. Example Analysis
Smoking vs Lung Capacity
20
25
30
35
40
45
50
-5 0 5 10 15 20 25
LungCapacity
Smoking (yrs)
Lung Capacity (Y )
8
1
( 215) 53.75
4
xyS    
0.96xyr  
When smoking is above its group
means, lung capacity tends to be
below its group mean.
Greater smoking exposure implies
greater likelihood of lung damage.

9. Regression
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- The process of predicting variable Y using variable X.
- Tells us how values in Y changes as a function of changes in value X.
- Calculates the “best-fit” line for a certain of data.
x
y
d1
d2
d3
Observed y-value
Predicted y-value
intercept
ε
ŷ = ax + b
ε = residual error
= y i , true value
slope
= ŷ, predicted value

10. Regression: Case Study in R
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library(ggplot2)
ggplot(data=mtcars, aes(x=wt, y=mpg))+geom_point()
- Data = mtcars
- wt: weight
- Mpg: miles per gallon
- S(x,y) = -5.11
- r(x,y) = -0.86

11. Regression: Case Study in R
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- Intercept = 37
- Slope = -5.3
ggplot(data=mtcars, aes(x=wt, y=mpg))+geom_point()+geom_smooth(method="lm",se=FALSE)
lm(data=mtcars, mpg ~ wt)

12. Thanks!Hamid Reza Bolhasani
bolhasani@gmail.com
Jan 2020
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