Correlation & Correlation & RegressionRegression

CorrelationCorrelation

Finding the relationship between two quantitative variables without being able to infer causal relationships

Correlation is a statistical technique used to determine the degree to which two variables are related

• 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

Example

Scatter diagram of weight and systolic blood Scatter diagram of weight and systolic blood pressurepressure

8 0

1 0 0

1 2 0

1 4 0

1 6 0

1 8 0

2 0 0

2 2 0

6 0 7 0 8 0 9 0 1 0 0 1 1 0 1 2 0w t (k g )

8 0

1 0 0

1 2 0

1 4 0

1 6 0

1 8 0

2 0 0

2 2 0

6 0 7 0 8 0 9 0 1 0 0 1 1 0 1 2 0W t ( k g )

Scatter diagram of weight and systolic blood pressure

Scatter plots

The pattern of data is indicative of the type of relationship between your two variables:

positive relationship negative relationship no relationship

Positive relationshipPositive relationship

0

2

4

6

8

10

12

14

16

18

0 10 20 30 40 50 60 70 80 90

Age in Weeks

Heig

ht in

CM

Negative relationshipNegative relationship

Reliability

Age of Car

No relationNo relation

Correlation CoefficientCorrelation Coefficient

Statistic showing the degree of relation between two variables

Simple Correlation coefficient Simple Correlation coefficient (r)(r)

It is also called Pearson's correlation It is also called Pearson's correlation or product moment correlation or product moment correlationcoefficient. coefficient.

It measures the It measures the naturenature and and strengthstrength between two variables ofbetween two variables ofthe the quantitativequantitative type. type.

The The signsign of of rr denotes the nature of denotes the nature of association association

while the while the valuevalue of of rr denotes the denotes the strength of association.strength of association.

If the sign is If the sign is +ve+ve this means the relation this means the relation is is direct direct (an increase in one variable is (an increase in one variable is associated with an increase in theassociated with an increase in theother variable and a decrease in one other variable and a decrease in one variable is associated with avariable is associated with adecrease in the other variable).decrease in the other variable).

While if the sign is While if the sign is -ve-ve this means an this means an inverse or indirectinverse or indirect relationship (which relationship (which means an increase in one variable is means an increase in one variable is associated with a decrease in the other).associated with a decrease in the other).

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

association as illustratedassociation as illustratedby the following diagram.by the following diagram.

-1 10-0.25-0.75 0.750.25

strong strongintermediate intermediateweak weak

no relation

perfect correlation

perfect correlation

Directindirect

If If rr = Zero = Zero this means no association or this means no association or correlation between the two variables.correlation between the two variables.

If If 0 < 0 < rr < 0.25 < 0.25 = weak correlation. = weak correlation.

If If 0.25 ≤ 0.25 ≤ rr < 0.75 < 0.75 = intermediate correlation. = intermediate correlation.

If If 0.75 ≤ 0.75 ≤ rr < 1 < 1 = strong correlation. = strong correlation.

If If r r = l= l = perfect correlation. = perfect correlation.

ny)(

y.nx)(

x

nyx

xyr

22

22

How to compute the simple correlation coefficient (r)

ExampleExample:: A sample of 6 children was selected, data about their A sample of 6 children was selected, data about their

age in years and weight in kilograms was recorded as age in years and weight in kilograms was recorded as shown in the following table . It is required to find the shown in the following table . It is required to find the correlation between age and weight.correlation between age and weight.

serial No

Age (years)

Weight (Kg)

17122683812451056116913

These 2 variables are of the quantitative type, one These 2 variables are of the quantitative type, one variable (Age) is called the independent and variable (Age) is called the independent and denoted as (X) variable and the other (weight)denoted as (X) variable and the other (weight)is called the dependent and denoted as (Y) is called the dependent and denoted as (Y) variables to find the relation between age and variables to find the relation between age and weight compute the simple correlation coefficient weight compute the simple correlation coefficient using the following formula:using the following formula:

ny)(

y.nx)(

x

nyx

xyr

22

22

Serial n.

Age (years)

(x)

Weight (Kg)

(y)xyX2Y2

17128449144268483664381296641444510502510056116636121691311781169

Total∑x=41

∑y=66

∑xy= 461

∑x2=291

∑y2=742

r = 0.759r = 0.759strong direct correlation strong direct correlation

6(66)742.

6(41)291

66641461

r22

EXAMPLE: Relationship between Anxiety and EXAMPLE: Relationship between Anxiety and Test ScoresTest Scores

AnxietyAnxiety ))XX((

Test Test score (Y)score (Y)

XX22YY22XYXY

101022100100442020883364649924242299448181181811771149497755662525363630306655363625253030

∑∑X = 32X = 32∑∑Y = 32Y = 32∑∑XX22 = 230 = 230∑∑YY22 = 204 = 204∑∑XY=129XY=129

Calculating Correlation CoefficientCalculating Correlation Coefficient

94.)200)(356(

102477432)204(632)230(6

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

r

r = - 0.94

Indirect strong correlation

exerciseexercise

Regression AnalysesRegression Analyses

Regression: technique concerned with predicting some variables by knowing others

The process of predicting variable Y using variable X

RegressionRegression Uses a variable (x) to predict some outcome Uses a variable (x) to predict some outcome

variable (y)variable (y) Tells you how values in y change as a function Tells you how values in y change as a function

of changes in values of xof changes in values of x

Correlation and RegressionCorrelation and Regression

Correlation describes the strength of a Correlation describes the strength of a linear relationship between two variables

Linear means “straight line”

Regression tells us how to draw the straight line described by the correlation

Regression Calculates the “best-fit” line for a certain set of dataCalculates the “best-fit” line for a certain set of dataThe regression line makes the sum of the squares of The regression line makes the sum of the squares of

the residuals smaller than for any other linethe residuals smaller than for any other lineRegression minimizes residuals

8 0

1 0 0

1 2 0

1 4 0

1 6 0

1 8 0

2 0 0

2 2 0

6 0 7 0 8 0 9 0 1 0 0 1 1 0 1 2 0W t ( k g )

By using the least squares method (a procedure By using the least squares method (a procedure that minimizes the vertical deviations of plotted that minimizes the vertical deviations of plotted points surrounding a straight line) we arepoints surrounding a straight line) we areable to construct a best fitting straight line to the able to construct a best fitting straight line to the scatter diagram points and then formulate a scatter diagram points and then formulate a regression equation in the form of:regression equation in the form of:

nx)(

x

nyx

xyb 2

21)xb(xyy b

bXay

Regression Equation

Regression equation describes the regression line mathematically Intercept Slope 8 0

1 0 0

1 2 0

1 4 0

1 6 0

1 8 0

2 0 0

2 2 0

6 0 7 0 8 0 9 0 1 0 0 1 1 0 1 2 0W t ( k g )

Linear EquationsLinear EquationsY

Y = bX + a

a = Y-interceptX

Changein Y

Change in Xb = Slope

bXay

Hours studying and Hours studying and gradesgrades

Regressing grades on hours 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

Final grade in course = 59.95 + 3.17 * studyR-Square = 0.88

Predicted final grade in class =

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

Predict the final grade ofPredict 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)

ExerciseExercise

A sample of 6 persons was selected the A sample of 6 persons was selected the value of their age ( x variable) and their value of their age ( x variable) and their weight is demonstrated in the following weight is demonstrated in the following table. Find the regression equation and table. Find the regression equation and what is the predicted weight when age is what is the predicted weight when age is 8.5 years8.5 years..

Serial no.Age (x)Weight (y)123456

768569

128

12101113

AnswerAnswer

Serial no.Age (x)Weight (y)xyX2Y2

123456

768569

128

12101113

8448965066

117

493664253681

14464

144100121169

Total4166461291742

6.83641x 11

666

y

92.0

6)41(

291

666414612

b

Regression equation

6.83)0.9(x11y (x)

0.92x4.675y (x)

12.50Kg8.5*0.924.675y (8.5)

Kg58.117.5*0.924.675y (7.5)

11.411.611.8

1212.212.412.6

7 7.5 8 8.5 9

Age (in years)

Wei

ght (

in K

g)

we create a regression line by plotting two estimated values for y against their X component,

then extending the line right and left.

Exercise 2Exercise 2

The following are the The following are the age (in years) and age (in years) and systolic blood systolic blood pressure of 20 pressure of 20 apparently healthy apparently healthy adults.adults.

Age (x)

B.P (y)

Age (x)

B.P (y)

20436326533158465870

120128141126134128136132140144

46536020634326193123

128136146124143130124121126123

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

Serialxyxyx21201202400400243128550418493631418883396942612632766765531347102280963112839689617581367888336484613260722116958140812033641070144100804900

Serialxyxyx21146128588821161253136720828091360146876036001420124248040015631439009396916431305590184917261243224676181912122993611931126390696120231232829529

Total852263011448

641678

nx)(

x

nyx

xyb 2

21 4547.0

2085241678

2026308521144862

=

=112.13 + 0.4547 x

for age 25

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

y