Correlation
CORRELATION &
LINEAR REGRESSION
Prof. Jemabel Gonzaga-Sidayen
Spearman rank order correlation coefficient rho (rs)
• Spearman rho is really a linear correlation coefficient applied to data that meet the requirements of ordinal scaling
• Formula:
rs = 1 - 6 Σ D i 2
N3 - N
– Di = difference between the ith pair of ranks
– R(Xi) = rank of the ith X score
– R(Yi) = rank of the ith Y score
– N = number of pairs of ranks
Try this
Subject
Proportion of
Similar
Attitudes (X)
Attraction
(Y)
Rank of Xi
Rank of Yi
Di
Di 2
1
0.30
8.9
5
7
-2
4
2
0.44
9.3
7
8
-1
1
3
0.67
9.6
11
10
1
1
4
0.00
6.2
1
1
0
0
5
0.50
8.8
8
6
2
4
6
0.15
8.1
3
5
-2
4
7
0.58
9.5
9
9
0
0
8
0.32
7.1
6
2
4
16
9
0.72
11.0
12
13
-1
1
10
1.00
11.7
15
15
0
0
11
0.87
11.5
14
14
0
0
12
0.09
7.3
2
3
-1
1
13
0.82
10.0
13
11.5
1.5
2.25
14
0.64
10.0
10
11.5
-1.5
2.25
15
0.24
7.5
4
4
0
0
Solution
ΣDi 2 = 36.5 rs = 1 - 6 (36.5)
= 1 = 0.93
153 - 15
219_
3360
Pearson r correlation
r
=
(ΣX)( ΣY)
Σ XY - ------------N
---------------------------------------------------(ΣX) 2
(ΣY)2
ΣX 2 - ------ΣY2 - ------N
N
Subject
Proportion of
Similar
Attitudes (X)
X2
Attraction
(Y)
Y2
XY
1
0.30
0.09
8.9
79.21
2.67
2
0.44
0.1936
9.3
86.49
4.092
3
0.67
0.4489
9.6
92.16
6.432
4
0.00
0.00
6.2
38.44
0.00
5
0.50
0.25
8.8
77.44
4.4
Σ
1.91
0.9825
42.8
373.74
17.594
r
=
r
=
(ΣX)( ΣY)
Σ XY - ------------N
-------------------------------------------------------(ΣX)2
(ΣY)2
ΣX2 - ------ΣY2 - ------N
N
17.594 - (1.91)(42.8) / 5
-----------------------------------------------------------0.9825 – (1.91) 2 /5
373.74 – (42.8) 2 /5
Regression
• Regression and correlation are closely related. At the most basic level, they both involve relationship between two variables, and they both utilize the same set of basic data: paired scores taken from the same or matched subjects.
•
LINEAR REGRESSION
Prof. Jemabel Gonzaga-Sidayen
Spearman rank order correlation coefficient rho (rs)
• Spearman rho is really a linear correlation coefficient applied to data that meet the requirements of ordinal scaling
• Formula:
rs = 1 - 6 Σ D i 2
N3 - N
– Di = difference between the ith pair of ranks
– R(Xi) = rank of the ith X score
– R(Yi) = rank of the ith Y score
– N = number of pairs of ranks
Try this
Subject
Proportion of
Similar
Attitudes (X)
Attraction
(Y)
Rank of Xi
Rank of Yi
Di
Di 2
1
0.30
8.9
5
7
-2
4
2
0.44
9.3
7
8
-1
1
3
0.67
9.6
11
10
1
1
4
0.00
6.2
1
1
0
0
5
0.50
8.8
8
6
2
4
6
0.15
8.1
3
5
-2
4
7
0.58
9.5
9
9
0
0
8
0.32
7.1
6
2
4
16
9
0.72
11.0
12
13
-1
1
10
1.00
11.7
15
15
0
0
11
0.87
11.5
14
14
0
0
12
0.09
7.3
2
3
-1
1
13
0.82
10.0
13
11.5
1.5
2.25
14
0.64
10.0
10
11.5
-1.5
2.25
15
0.24
7.5
4
4
0
0
Solution
ΣDi 2 = 36.5 rs = 1 - 6 (36.5)
= 1 = 0.93
153 - 15
219_
3360
Pearson r correlation
r
=
(ΣX)( ΣY)
Σ XY - ------------N
---------------------------------------------------(ΣX) 2
(ΣY)2
ΣX 2 - ------ΣY2 - ------N
N
Subject
Proportion of
Similar
Attitudes (X)
X2
Attraction
(Y)
Y2
XY
1
0.30
0.09
8.9
79.21
2.67
2
0.44
0.1936
9.3
86.49
4.092
3
0.67
0.4489
9.6
92.16
6.432
4
0.00
0.00
6.2
38.44
0.00
5
0.50
0.25
8.8
77.44
4.4
Σ
1.91
0.9825
42.8
373.74
17.594
r
=
r
=
(ΣX)( ΣY)
Σ XY - ------------N
-------------------------------------------------------(ΣX)2
(ΣY)2
ΣX2 - ------ΣY2 - ------N
N
17.594 - (1.91)(42.8) / 5
-----------------------------------------------------------0.9825 – (1.91) 2 /5
373.74 – (42.8) 2 /5
Regression
• Regression and correlation are closely related. At the most basic level, they both involve relationship between two variables, and they both utilize the same set of basic data: paired scores taken from the same or matched subjects.
•