Correlation & Regression

Correlation
•
Introduction to Correlation & Regression
– We often see things that are related to one another.
• height/weight
• IQ/Performance in School
• Age/Income
– We call this relationship
Correlation
•
Pearson r is the most common method of measuring relationship.

Correlation
• Formula for calculating Pearson’s r
– Let x and y be two sets of paired observations with standard deviations = s x
•
How might we and s y measure relationship between two sets of scores?

Correlation
How might we measure relationship between two sets of scores?

Correlation
•
Is this a good measure of relationship?
– It does give different values for different degrees of relationship.
– It does not provide consistency which allows it to be interpreted.
– Every set of scores will yield a different score
• The result will vary with the size of the scores.
•
How can we equalize these scores so they will give consistent and meaningful results every time?

Correlation
•
How can we equalize these scores so they will give consistent and meaningful results every time?
•
We can change the scores to standard scores and take the average product of the standard scores for the X and Y variables.
r r
Z Z
Y
Y

X
5
10
5
11
12
4
3
2
7
1
Y
5
10
5
11
12
4
3
2
7
1
Correlation x
-5
-6
2
3
1
-4
1
4
-1
5 y
-5
-6
2
3
1
-4
1
4
-1
5
Mean X Mean Y SD X
6 6 3.66
SD Y
3.66
Zx Zy ZxZy
0.273
0.273
0.0746
-1.093
-1.093
1.1940
0.273
0.273
0.0746
-1.366
-1.366
1.8657
-1.639
-1.639
2.6866
0.546
0.820
0.546
0.820
0.2985
0.6716
1.093
1.093
1.1940
-0.273
-0.273
0.0746
1.366
1.366
1.8657
Sum
10
R= 1 r r
 
Z Z
Y
Y

Correlation r r
 
• This is called the standard score formula .
•
It is a defining formula
•
It is not a formula that you would use to actually calculate the correlation coefficient .
Z Z
Y
Y
•
We call this the
Pearson Product
Moment r

Pearson Product Moment
Correlation Coefficient
• The most widely used method of measuring correlation is the Pearson Product Moment
Correlation .
•
We will also consider a Rank Order Correlation
Coefficient
– It is an Ordinal Level Correlation Method
– Spearman Rank Order Correlation
• Limits for Correlation are -1 0 +1

Pearson Product Moment
Correlation Coefficient
Calculating Pearson’s Product Moment r

Pearson Product Moment
Correlation Coefficient
Example Illustrating Computation of Pearson’s r

Pearson Product Moment
Correlation Coefficient
Calculating Pearson’s Product Moment r

Pearson Product Moment r r r

     
 .
.
774

r

Pearson Product Moment r
H
» Computation of r from raw scores
W
7 1 4 0
7 1 2 0
6 . 5 1 3 0


N
N

X
1
2
N
N

XY
1


N
 
N 
X

1 1
Y



N
1
X 
 2




N
N

Y
1
2


N
1
Y 
 2



6 1 3 0
6 1 2 0
6 1 1 0
N


1
N
1
X
X
2


70 5
N 
1
N 
1
Y
Y
2

1380

161000
N 
1
X Y

6 1 1 0
5 . 5 1 2 0
5 . 5 1 0 0
5 1 1 0
5 1 0 0
5 9 0
8 1 9 5

r

Pearson Product Moment r r
 r



Computation of r from raw scores
N
N

XY
1


N
 
N 
X

1 1
Y

N
N

X
1
2 

N
1
X 
 2




N
N

Y
1
2 

N
1
Y 
 2




    
     
 
2



    

1380
2
 

1050
  

1050
1842300

.
774

Spearman Rank Difference
Correlation (Rho)(
D
)
• Rho
– We sometimes have data we want to correlate which doesn’t meet the r.
requirements for a Pearson   
• Not at Interval Level
– Rho is a correlation technique that requires only ordinal level of measurement.
n
6
 n
2
D
2

H W
5
5
7
7
6
6
140
120
6.5
130
6 130
120
110
6 110
5.5
120
5.5
100
5 110
100
90
Spearman Rank Difference
Correlation (Rho)(
D
) r1 r2 D D2
11.5
11.5
10 10.5
-0.5
7.5
10.5
-3
7.5
7.5
12 -0.5
0.25
8 3.5
12.25
8 -0.5
5 2.5
0.25
9
0.25
6.25
7.5
4.5
4.5
2
2
2
5
8
2.5
5
2.5
1
2.5
-3.5
2
-3
-0.5
1
6.25
12.25
4
9
0.25
1
61
  
 
 
 
366
  
1716

Spearman Rank Difference
Correlation (Rho)
•
Advantages and Disadvantages of Rho
– Advantages
:
• Ease of Computation
• Skewness influences r but not Rho
– Disadvantages :
• It is somewhat less consistent from sample to sample.

Spearman Rank Difference
Correlation (Rho)
Next We will focus on interpreting a correlation coefficient and regression.
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