Regression and correlation

1. Correlation and
Regression
1
BY UNSA SHAKIR

2. Correlation and Regression
2
Correlation describes the strength of a
linear relationship between two variables
Regression tells us how to draw the straight
line described by the correlation

3. Correlation and Regression
• For example:
A sociologist may be interested in the relationship
between education and self-esteem or Income and
Number of Children in a family.
Independent Variables
Education
Family Income
Dependent Variables
Self-Esteem
Number of Children
3

4. Correlation and Regression
• For example:
• May expect: As education increases, self-esteem
increases (positive relationship).
• May expect: As family income increases, the number
of children in families declines (negative relationship).
Family Income
Dependent Variables
Self-Esteem
Number of Children
Independent Variables
Education
+
4
-

5. Correlation
5

6. Correlation
6
• Correlation is a statistical technique used to
determine the degree to which two variables
are related
• A correlation is a relationship between two
variables. The data can be represented by the
ordered pairs (x, y) where x is the independent
(or explanatory) variable, and y is the
dependent (or response) variable.

7. Correlation
x 1 2 3 4 5
y – 4 – 2 – 1 0 2
A scatter plot can be used to determine
whether a linear (straight line) correlation
exists between two variables.
x
2 4
–2
– 4
y
7
2
6
Example:

8. Linear Correlation
y
x
Negative Linear Correlation
y
x
No Correlation
y
x
Positive Linear Correlation
y
x
Nonlinear Correlation
As x increases,
y tends to
decrease.
8
As x increases,
y tends to
increase.

9. Correlation Coefficient
• It is also called Pearson's correlation or
product moment correlation coefficient
• The correlation coefficient is a measure of
the strength and the direction of a linear
relationship between two variables. The
symbol r represents the sample correlation
coefficient. The formula for r is
.
9
r 
nxy xy
nx2
x2
ny2
y2

10. of r denotes the nature of
The sign
association
while the value of r denotes the strength of
association.
10

11. If the sign is +ve this means the relation is
direct (an increase in one variable is
11
other variable and a decrease in
variable is associated with
associated with an increase in the
one
a
decrease in the other variable).
While if the sign is -ve this means an
inverse or indirect relationship (which
means an increase in one variable is
associated with a decrease in the other).

12. -1 -0.75 -0.25 0 0.25 0.75 1
The value of r ranges between ( -1) and ( +1)
The value of r denotes the strength of the
association as illustrated
by the following diagram.
strong intermediate weak weak intermediate strong
no
relation
perfect
correlation
perfect
correlation
Direct
12
indirect

13. If r = Zero this means no association or
correlation between the two variables.
If 0 < r < 0.25 = weak correlation.
If 0.25 ≤ r < 0.75 = intermediate correlation.
If 0.75 ≤ r < 1 = strong correlation.
If r = l = perfect correlation.
13

14. Linear Correlation
x
Weak positive correlation
y
x
Nonlinear Correlation
y
r = 0.91
x
Strong negative correlation
y
r = 0.42
14
r = 0.88
x
Strong positive correlation
y
r = 0.07

15. Calculatinga CorrelationCoefficient
.
15
r 
nxy  xy
nx2
 x2
n y2
 y2
Calculating a Correlation Coefficient
In Words
1. Find the sum of the x-values.
2. Find the sum of the y-values.
In Symbols
x
 y
xy
3. Multiply each x-value by its
corresponding y-value and find
the sum.

16. Calculatinga CorrelationCoefficient
16
Calculating a Correlation Coefficient
x2
 y 2
In Words In Symbols
4. Square each x-value and
find the sum.
5. Square each y-value and
find the sum.
6. Use these five sums to
calculate the correlation
coefficient.

17. Correlation Coefficient
17
Example:
Calculate the correlation coefficient r for the following
data.
x y xy x2 y2
1 – 3 – 3 1 9
2 – 1 – 2 4 1
3 0 0 9 0
4 1 4 16 1
5 2 10 25 4
x 15  y  1 xy  9 x2
 55  y2
 15

18. Correlation Coefficient
nxy  xy
r 
Example:
Calculate the correlation coefficient r for the following
data.
nx2
 x2
n y2
 y2
5(9)  151

5(55) 152
5(15)  12
 6 0
18
0.986
5 0 7 4
There is a strong positive linear correlation
between x and y.

19. Correlation Coefficient
19
Hours,
x
0 1 2 3 3 5 5 5 6 7 7 10
Test score,
y
96 85 82 74 95 68 76 84 58 65 75 50
Example:
The following data represents the number of hours, 12
different students watched television during the
weekend and the scores of each student who took a test
the following Monday.
a) Display the scatter plot.
b) Calculate the correlation coefficient r.

20. Correlation Coefficient
Test
score
y
100
80
60
40
20
x
2 4 6 8 10
Hours watching TV
Hours,
x
0 1 2 3 3 5 5 5 6 7 7 10
Test score,
y
96 85 82 74 95 68 76 84 58 65 75 50
20

21. Correlation Coefficient
21
Hours, x 0 1 2 3 3 5 5 5 6 7 7 10
Test
score, y
96 85 82 74 95 68 76 84 58 65 75 50
xy 0 85
16
4
222
28
5
34
0
38
0
420 348
45
5
52
5
50
0
x2 0 1 4 9 9 25 25 25 36 49 49
10
0
y2
921
6
722
5
67
24
547
6
90
25
46
24
57
76
705
6
336
4
42
25
56
25
25
00
Example continued:
x  54  y  908 xy  3724 x2
 332  y2
 70836

22. Correlation Coefficient
Example continued:
r 
nxy xy
nx2
x2
ny2
y2
12(3724)  54908
22

12(332)  542
12(70836)  9082  0.831
• There is a strong negative linear correlation.
• As the number of hours spent watching TV increases,
the test scores tend to decrease.

23. Example:
23
A sample of 6 children was selected, data about their
age in years and weight in kilograms was recorded
as shown in the following table . It is required to find
the correlation between age and weight.
serial
No
Age
(years)
Weight
(Kg)
1 7 12
2 6 8
3 8 12
4 5 10
5 6 11
6 9 13

24. Serial
n.
Age
(year)
(x)
Weight
(Kg)
(y)
xy X2 Y2
1 7 12 84 49 144
2 6 8 48 36 64
3 8 12 96 64 144
4 5 10 50 25 100
5 6 11 66 36 121
6 9 13 117 81 169
Total ∑x=
41
∑y=
66
∑xy=
461
∑x2=
291
∑y2=
742
24

25. r = 0.759
strong direct correlation

25

 


6
.742 
6
291
461
41 66
r  6
(41)2
  (66)2


26. EXAMPLE: Relationship betweenAnxiety and Test
Scores
26
Anxiety
(X)
Test
score (Y)
X2 Y2 XY
10 2 100 4 20
8 3 64 9 24
2 9 4 81 18
1 7 1 49 7
5 6 25 36 30
6 5 36 25 30
∑X = 32 ∑Y = 32 ∑X2 = 230 ∑Y2 = 204 ∑XY=129

27. Calculating Correlation Coefficient
(356)(200)
27
 .94
774 1024

(6)(129)  (32)(32)
6(230)  322
6(204) 322

r 
r = - 0.94
Indirect strong correlation

28. Example
Tree
Height
y
Trunk
Diameter
x xy y2 x2
35 8 280 1225 64
49 9 441 2401 81
27 7 189 729 49
33 6 198 1089 36
60 13 780 3600 169
21 7 147 441 49
45 11 495 2025 121
51 12 612 2601 144
Σ =321 Σ =73 Σ =3142 Σ =14111 Σ =713
28

29. 13
10
0
20
30
40
50
60
70
0 2 12 14
2
29
4 6 8 10
Trunk Diameter, x
Tree
Height,
y
Example
• r = 0.886 → relatively
strong positive linear
association between x
and y

30. 30

31. Regression
31

32. Regression Analyses
32
• Regression technique is concerned with
predicting some variables by knowing others
• The process of predicting variable Y using
variable X

33. 20
Types of Regression Models
Positive Linear Relationship
Negative Linear Relationship
Relationship NOT Linear
No Relationship
33

34. Regression
34
Uses a variable (x) to predict some outcome
variable (y)
Tells you how values in y change as a
function of changes in values of x

35. The regression line makes the sum of the squares of the
residuals smaller than for any other line
Regression minimizes residuals
220
200
180
160
140
120
100
80
60 70 80 90 100 110
Wt (kg)
120
SBP(mmHg)
35

36. By using the least squares method (a procedure that
minimizes the vertical deviations of
surrounding a straight line)
plotted
we
points
are
able to construct a best fitting straight line to the scatter
diagram points and then formulate a regression equation
in the form of:

n
 n
  x  y
 xy
b
(  x) 2
 x 2
1
ŷ  y  b(x  x)
ŷ  a  bX
Regression equation describes the regression line
mathematically by showing Intercept and Slope
36

37. Correlation and Regression
37
• The statistics equation for a line:
Y = a + bx
Where^
: Y = the line’s position on
the v
^ertical axis at any point
(estimated value of dependent
variable)
X = the line’s position on the
horizontal axis at any point (value of
the independent variable for which you
want an estimate of Y)
b = the slope of the line
(called the coefficient)
a = the intercept with the Y axis,
where X equals zero

38. Linear Equations
Y
Y=bX+a
Change
inY
ChangeinX
a=Y-intercept
X
b=Slope
38

39. Exercise
A sample of 6 persons was selected the value of
their age ( x variable) and their weight is
demonstrated in the following table. Find the
regression equation and what is the predicted
weight when age is 8.5 years.
39
Serial no. Age (x) Weight (y)
1 7 12
2 6 8
3 8 12
4 5 10
5 6 11
6 9 13

40. Answer
40
Serial no. Age (x) Weight (y) xy X2 Y2
1 7 12 84 49 144
2 6 8 48 36 64
3 8 12 96 64 144
4 5 10 50 25 100
5 6 11 66 36 121
6 9 13 117 81 169
Total 41 66 461 291 742

41. 6
x 
41
 6.83
6
y 
66
11
6
41
(41)2
291 
461
41 66
b  6  0.92
Regression equation
ŷ(x) 11 0.9(x  6.83)

42. ŷ(x)
42
 4.675  0.92x
ŷ(8.5)  4.675  0.92*8.5 12.50Kg
ŷ(7.5) 4.6750.92*7.511.58Kg

43. we create a regression line by plotting two estimated
values for y against their X component, then extending
the line right and left.
43

44. Regression Line
44
Example:
a) Find the equation of the regression line.
b) Use the equation to find the expected value when
value of x is 2.3
x y xy x2 y2
1 – 3 – 3 1 9
2 – 1 – 2 4 1
3 0 0 9 0
4 1 4 16 1
5 2 10 25 4
x 15  y  1 xy  9 x2
 55  y2
 15

45. Regression Line
x
2
1
1
2
3
1 2 3 4 5
m 
nxy  xy
nx2
 x2
y

5(9) 151
45
5(55) 152
50
 60
1.2

46. Regression Line
46
Example:
The following data represents the number of hours 12
different students watched television during the
weekend and the scores of each student who took a
test the following Monday.
a) Find the equation of the regression line.
b) Use the equation to find the expected test score
for a student who watches 9 hours of TV.

47. Regression Line
47
Hours, x 0 1 2 3 3 5 5 5 6 7 7 10
Test score,
y
96 85 82 74 95 68 76 84 58 65 75 50
xy 0 85 164 222 285 340 380 420 348 455 525 500
x2 0 1 4 9 9 25 25 25 36 49 49 100
y2 9216
722
5
672
4
547
6
902
5
462
4
577
6
705
6
336
4
422
5
562
5
250
0
x  54  y  908 xy  3724 x2
 332  y2
 70836

48. Exercise
• Find the correlation between age and blood
pressure using simple and Spearman's
correlation coefficients, and comment.
• Find the regression equation?
• What is the predicted blood pressure for a
man aging 25 years?
48

49. Serial x y xy x2
1 20 120 2400 400
2 43 128 5504 1849
3 63 141 8883 3969
4 26 126 3276 676
5 53 134 7102 2809
6 31 128 3968 961
7 58 136 7888 3364
8 46 132 6072 2116
9 58 140 8120 3364
10 70 144 10080 4900
49

50. Serial x y xy x2
11 46 128 5888 2116
12 53 136 7208 2809
13 60 146 8760 3600
14 20 124 2480 400
15 63 143 9009 3969
16 43 130 5590 1849
17 26 124 3224 676
18 19 121 2299 361
19 31 126 3906 961
20 23 123 2829 529
Total 852 2630 114486 41678
50

51. 
n
xy  x y
b  n
(x)2
x2
1
20
51
8522
20  0.4547
41678 
114486 
852  2630
=
ŷ =112.13 + 0.4547 x
for age 25
B.P = 112.13 + 0.4547 * 25=123.49
= 123.5 mm hg