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Correlation &Regression

Dr. Moataza Mahmoud Abdel Wahab

Lecturer of BiostatisticsHigh Institute of Public Health

University of Alexandria

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CorrelationFinding the relationship between two

quantitative variables without being able to

infer causal relationships

Correlationis a statistical technique used to

determine the degree to which two

variables are related

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Rectangular coordinate

Two quantitative variables

One variable is called independent (X) and

the second is called dependent (Y)

Points are not joined

No frequency table

Scatter diagram

Y

* *

*

X

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Wt.

(kg)

67 69 85 83 74 81 97 92 114 85

SBP

mHg)

120 125 140 160 130 180 150 140 200 130

Example

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80

100

120

140

160

180

200

220

60 70 80 90 100 110 120Wt (kg)

SBP(mmHg)

Scatter diagram of weight and systolic blood pressure

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Scatter plots

The pattern of data is indicative of the type of

relationship between your two variables:

positive relationship

negative relationship

no relationship

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Positive relationship

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0

2

4

6

8

10

12

14

16

18

0 10 20 30 40 50 60 70 80 90

Age in Weeks

Heightin

CM

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Negative relationship

Reliability

Age of Car

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No relation

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Correlation Coefficient

Statistic showing the degree of relation

between two variables

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Simple Correlation coefficient (r)

It is also called Pearson's correlation

or product moment correlation

coefficient.

It measures the natureand strength

between two variables of

the quantitativetype.

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The signof rdenotes the nature of

association

while the valueof rdenotes the

strength of association.

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If the sign is +ve this means the relationis direct (an increase in one variable isassociated with an increase in theother variable and a decrease in onevariable is associated with a

decrease in the other variable).

While if the sign is -ve this means an

inverse or indirect relationship (whichmeans an increase in one variable isassociated with a decrease in the other).

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The value of r ranges between ( -1) and ( +1)

The value of r denotes the strength of theassociation as illustrated

by the following diagram.

-110

-0.25-0.75 0.750.25

strong strongintermediate intermediateweak weak

no relation

perfect

correlation

perfect

correlation

Directindirect

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If r= Zero this means no association orcorrelation 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.

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n

y)(y.

n

x)(x

n

yxxy

r2

2

2

2

How to compute the simple correlation

coefficient (r)

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Example:

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.

Weight

(Kg)

Age

(years)

serial

No

1271

862

1283

1054

1165

1396

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These 2 variables are of the quantitative type, one

variable (Age) is called the independent and

denoted as (X) variable and the other (weight)is called the dependent and denoted as (Y)

variables to find the relation between age and

weight compute the simple correlation coefficient

using the following formula:

n

y)(y.

n

x)(x

n

yxxy

r2

22

2

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Y2X2xy

Weight

(Kg)

(y)

Age

(years)

(x)

Serial

n.

14449841271

643648862

14464961283

10025501054

12136661165

169811171396

y2=

742

x2=

291

xy=

461

y=

66

x=

41

Total

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r = 0.759

strong direct correlation

6

(66)742.

6

(41)291

6

6641461

r22

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EXAMPLE: Relationship between Anxiety and

Test Scores

Anxiety

(X)

Test

score (Y)X2 Y2 XY

10 2 100 4 20

8 3 64 9 242 9 4 81 18

1 7 1 49 7

5 6 25 36 306 5 36 25 30

X = 32 Y = 32 X2= 230 Y2= 204 XY=129

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Calculating Correlation Coefficient

94.

)200)(356(

1024774

32)204(632)230(6

)32)(32()129)(6(22

r

r = - 0.94

Indirect strong correlation

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Spearman Rank Correlat ion Coeff ic ient

(rs)

It is a non-parametric measure of correlation.

This procedure makes use of the two sets ofranks that may be assigned to the sample

values of x and Y.Spearman Rank correlation coefficient could becomputed in the following cases:

Both variables are quantitative.

Both variables are qualitative ordinal.

One variable is quantitative and the other isqualitative ordinal.

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Procedure:

1. Rank the values of X from 1 to n where nis the numbers of pairs of values of X and

Y in the sample.

2.

Rank the values of Y from 1 to n.3. Compute the value of di for each pair of

observation by subtracting the rank of Yi

from the rank of Xi

4. Square each di and compute di2 which

is the sum of the squared values.

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5. Apply the following formula

1)n(n

(di)61r2

2

s

The value of rsdenotes the magnitude

and nature of association giving the same

interpretation as simple r.

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Example

In a study of the relationship between leveleducation and income the following data was

obtained. Find the relationship between themand comment.

Income

(Y)

level education

(X)

sample

numbers

25Preparatory.A

10Primary.B

8University.C

10secondaryD15secondaryE

50illiterateF

60University.G

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Answer:

di2diRank

Y

Rank

X(Y)(X)423525PreparatoryA

0.250.55.5610Primary.B

30.25-5.571.58University.C

4-25.53.510secondaryD

0.25-0.543.515secondaryE

2552750illiterateF

0.250.511.560university.G

di2=64

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Comment:

There is an indirect weak correlation

between level of education and income.

1.0)48(7

646

1

sr

i

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exercise

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Regression Analyses

Regression: technique concerned with predictingsome variables by knowing others

The process of predicting variable Y usingvariable X

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Regression

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

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Correlation and Regression

Correlation describes the strength of a linear

relationship between two variables

Linearmeans straight line

Regressiontells us how to draw the straight line

described by the correlation

Regression

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Regression

Calculates the best-fit line for a certain set of data

The regression line makes the sum of the squares of

the residuals smaller than for any other lineRegression minimizes residuals

80

100

120

140

160

180

200

220

60 70 80 90 100 110 120Wt (kg)

SBP(mmHg)

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By using the least squares method (a procedure

that minimizes the vertical deviations of plotted

points surrounding a straight line) we 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

x)(x

n

yxxy

b2

2

1)xb(xyy b

bXay

R i E ti

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Regression Equation

Regression equation

describes the

regression line

mathematically

Intercept Slope

80

100

120

140

160

180

200

220

60 70 80 90 100 110 120Wt (kg)

SBP(mmHg)

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Linear Equations

Y

Y = bX + a

a = Y-intercept

X

Changein Y

Change in X

b = Slope

bXay

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Hours studying and grades

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Regressing grades on hours

Linear Regression

2.00 4.00 6.00 8.00 10.00

Number of hours spent studying

70.00

80.00

90.00

Finalgr

ade

in

course

Final grade in course = 59.95 + 3.17 * study

R-Square = 0.88

Predicted final grade in class =

59.95 + 3.17*(number of hours you study per week)

Predicted final grade in class = 59.95 + 3.17*(hours of study)

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Predict the final grade of

Someone who studies for 12 hours

Final grade = 59.95 + (3.17*12)

Final grade = 97.99

Someone who studies for 1 hour:

Final grade = 59.95 + (3.17*1)

Final grade = 63.12

Predicted final grade in class 59.95 3.17 (hours of study)

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Exercise

A sample of 6 persons was selected thevalue 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.

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Weight (y)Age (x)Serial no.

12

8

1210

11

13

7

6

85

6

9

1

2

34

5

6

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Answer

Y2X2xyWeight (y)Age (x)Serial no.

144

64

144100

121

169

49

36

6425

36

81

84

48

9650

66

117

12

8

1210

11

13

7

6

85

6

9

1

2

34

5

6

7422914616641Total

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6.836

41x 11

6

66y

92.0

6

)41(

291

6

6641461

2

b

Regression equation

6.83)0.9(x11y (x)

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0.92x4.675y (x)

12.50Kg8.5*0.924.675y(8.5)

Kg58.117.5*0.924.675y(7.5)

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11.4

11.6

11.8

12

12.2

12.4

12.6

7 7.5 8 8.5 9

Age (in years)

Weight(i

nKg)

we create a regression line by plotting two

estimated values for y against their X component,

then extending the line right and left.

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Exercise 2

The following are theage (in years) andsystolic bloodpressure of 20

apparently healthyadults.

B.P

(y)

Age

(x)

B.P

(y)

Age

(x)

128136

146

124

143

130

124

121126

123

4653

60

20

63

43

26

1931

23

120128

141

126

134

128

136

132140

144

2043

63

26

53

31

58

4658

70

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Find the correlation between ageand blood pressure using simple

and Spearman's correlationcoefficients, and comment.

Find the regression equation?

What is the predicted bloodpressure for a man aging 25 years?

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x2xyyxSerial

4002400120201

1849550412843239698883141633

6763276126264

28097102134535

9613968128316

33647888136587

21166072132468

336481201405894900100801447010

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x2xyyxSerial

211658881284611

280972081365312360087601466013

40024801242014

396990091436315184955901304316

67632241242617

3612299121191896139061263119

52928291232320

416781144862630852Total

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n

x)(x

n

yxxy

b2

21

4547.0

2085241678

20

2630852114486

2

=

=112.13 + 0.4547 x

for age 25

B.P = 112.13 + 0.4547 * 25=123.49 = 123.5 mm hg

y

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Multiple Regression

Multiple regression analysis is a

straightforward extension of simple

regression analysis which allows more

than one independent variable.

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