Chapter – II

1

Correlation

and

Regression Analysis - I

CORRELATION

4

• Explanatory Variable

or

Influencing variable

or

Predictor variable

• Response Variable

5

Independent Variable

Dependent Variable

6

X Y

2 55

3 58

5 49

7 48

9 35

15 33

Types of correlation

7

Positive Linear Relationship Negative Linear Relationship

Non-Linear Relationship No Relationship

8

Pearson’s Correlation Coefficient

Pearson’s Correlation Coefficient

10

YYXX

XY

SS

Sr

=

( )( )YYXXS XY −−=

( )2

−= XXS XX

( ) −=2

YYSYY

Pearson’s Correlation Coefficient

11

−−

−=

2222 )()( yynxxn

yxxynr

Pearson’s Correlation Coefficient

12

−

−

−

=

n

yy

n

xx

n

yxxy

r2

2

2

2)()(

Example• Find the Pearson’s correlation coefficient of the given

variables

13

Price Proft

2 6

3 7

5 10

7 11

10 13.5

14

y x xy x2

y2

2 6

3 7

5 10

7 11

10 13.5

27 47.5 295 488.25 187

Example

Nine students held their breath, once afterbreathing normally and relaxing for oneminute, and once after hyperventilatingfor one minute.

The table indicates how long (in sec) theywere able to hold their breath.

Is there an association between the twovariables?

15

16

Subject Normal Hypervent

A 56 87

B 56 91

C 65 85

D 65 91

E 50 75

F 25 28

G 87 122

H 44 66

I 35 58

Example • Identify whether there is an association between the given

variables

17

Price Demand

2 55

5 49

7 48

3 58

15 33

9 35

18

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0.20 0.40 0.60 0.80 1.00-1.00 -0.80 -0.60 -0.40 -0.20 0.00

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19

- 1 perfect positive linear relationship

0.80 - 0.99 very strong positive linear relationship

0.60 - 0.79 strong positive linear relationship

0.40 - 0.59 moderate positive linear relationship

0.20 - 0.39 weak positive linear relationship

0.01 - 0.19 very weak positive linear relationship

- 0 no linear relationship

-0.19 - - 0.01 very weak negative linear relationship

- 0.39 - - 0.20 weak negative linear relationship

- 0.59 - - 0.40 moderate negative linear relationship

- 0.79 - - 0.60 strong negative linear relationship

- 0.99 - - 0.80 very strong negative linear relationship

- - 1 perfect negative linear relationship

100 r2

known as the

coefficient of determination

which gives us the percentagedependency of the response variable inthe explanatory variable

20

Spearman’s Correlation coefficient

• Sometimes it is called Rank Correlation

•Denoted by “ R “

21

Spearman Correlation Coefficient

Spearman’s Rank Correlation Coefficient

23

)1(

61

2

2

−−=

nn

dR

Example

• An expert was asked to rank, according totaste, seven wines costing below £ 4. Herrankings ( with 1 being the worst taste and7 the best ) and the prices per bottle wereas follows.

24

Sample Rank of taste Price (£)

A 1 2.49

B 2 2.99

C 3 3.49

D 4 2.99

E 5 3.59

F 6 3.99

G 7 3.99

25

Sample Rank of

taste

Rank of

price

d d2

A 1

B 2

C 3

D 4

E 5

F 6

G 726

Which correlation to use

• If you intend to use regression then thechoice should be Pearson’s correlationcoefficient

27

REGRESSION Analysis

• Here we try to find a linear relation between theexplanatory variable ( x ) and the responsevariable ( y ) in the form

y = 0 + 1 x +

where ‘0’ and ‘1’ are constant to be determined

29

30

Regression

Graphical Method

Method of least Squares

Formulae for ‘0’ and ‘1’

31

( )( )( )22

1

−

−=

xxn

yxxyn

−

=

n

x

n

y10

Example

• A company has the following data on thelast year sales in each of its regions and thecorresponding number of sales personsemployed.

32

Region Sales Salespersons

A 236 11

B 234 12

C 298 18

D 250 15

E 246 13

F 202 10

33

x y x2 x y

11 236

12 234

18 298

15 250

13 246

10 202

79 1,466 1,083 19,736

34

Interpreting ‘0’ and ‘1’

35

Forecasting

•Once the equation of the regression linehas been computed, it is straight forwardprocess to obtain forecasts.

36

Example

• Considering the previous example forecastthe number of sales that would beexpected next year in regions thatemployed

• 14 salespersons

• 25 salespersons

37

Judging the Validity of Forecasts

• Validity of forecasts is given by thecoefficient of determination 100 r 2

38

Profits

£ m

Advertising expenditure

£ m

11.3 0.52

12.1 0.61

14.1 0.63

14.6 0.70

15.1 0.70

15.2 0.7539

A company has the following data on its profits

and advertising expenditure over the last six

years:

40

i. Identify the predictor and the response

variable

ii. Find the correlation coefficient and

interpret

iii. Find the best fit line

iv. Interpret your best fit line

v. Forecast the profits for the next year if an

advertising budget of £ 800,000 is

allocated,

vi. What is the validity of your forecast

Hours Studied Test Score

4 31

9 58

10 65

14 73

4 37

7 44

12 60

22 91

1 21

17 84 41

Consider the following data on the number of hours which 10

persons studied for a math test and their test scores :

42

i. Identify the predictor and the response

variable

ii. Find the correlation coefficient and interpret

iii. Find the least square regression line

iv. Interpret your regression line

v. If a student studies for 20 hours what would

be his expected test score (round to the

nearest integer)

vi. What is the validity of your forecast