chapter 5_ handouts.pdf

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QMT412 Pn. Sanizah's Notes 02/05/2013

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CHAPTER 5

CORRELATION AND REGRESSION

Introduction

Correlation and Regression

Scatter Plot/Diagram

Coefficient of Correlation

Simple Linear Regression

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Learning objectives

Explain the concept of correlation Calculate Pearsons correlation coefficientand

interpret the results

Calculate Spearmans rank correlation forqualitative and quantitative data and interpret theresults

Determine the regression equation for a set of data andinterpret the equation

Use the regression equation to forecast

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Introduction

Correlation: Do you have a relationship?

(Between two quantitative variables, x&y)

If you have a relationship:

1) What is the direction? (+ or -)

2) What is the strength (r: -1 to +1)

#Correlationmeasures LINEARrelationship.

If you have a significant correlation:

How well can you predicta subjects y-score ifyou know their x-score?

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Correlation & Regression Regressionand correlationare two concepts used to

describe the relationshipbetween variables.

Correlationis a statistical method used todetermineif a relationship between variablesexists.

Regressionis the statistical method used todescribethe nature of the relationship betweenvariables - that is, positive or negative, linear ornonlinear.

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Independent and Dependent Variable

In this chapter, we want to study the relationshipbetween 2 variables only.

Independent variable x

Dependent variable - y

For example:

Expenditure (x) andRevenue (y)

Price (x) andsales (y)

Number of days absent (x) andCGPA (y)

Age of a person (x) andhis/her blood pressure (y)

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Independent and Dependent Variable

Also called predictoror

explanatoryor manipulatedvariable the variable in regression that can

be controlled or manipulated

Independentvariable (x)

Also called the responsevariable the variable that cannot be

controlled or manipulated

Dependentvariable (y)

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Dependent(x) Vs. Independent(y)

Intentionally manipulated

Controlled

Vary at known rate

Cause

Intentionally left alone

Measured

Vary at unknown rate

Effect

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Example: What affects a students

arrival to class?Variables: Type of School

FSPPP, Business School, FSKM

Type of Student Gender? CGPA?

Class Time Morning, Afternoon, Evening

Mode of Transportation Motorcycle, Car, UiTM bus

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Scatter Plot (scatter diagram) A scatter plot is used to showthe relationship

between two variables.

The scatter plot is a visual way to describe the nature ofthe relationshipbetween the independentvariable (x) and the dependent variable (y).

Interpreting scatter plots:

Positivelinear relationship

Negativelinear relationship

Nonlinearrelationship

No relationship

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Scatter Plot Examples

y

x

y

x

y

y

x

x

Linear relationships Nonlinear (Curvilinear)

relationships

Positive

Negative

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y

x

y

x

y

y

x

x

Strong relationships Weak relationships

(continued)

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y

x

y

x

No relationship

(continued)

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Example 1 (pg. 134) Draw a scatter diagram for the following data and state

the type of relationship between the variables.

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x 1 3 5 7 9 13 17

y 0 5 11 14 19 22 30

Correlation [email protected]

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Correlation coefficient measures the strength and directionof

a LINEAR relationship between a pair of random variables.

The POPULATIONcorrelation coefficient (rho) measures the

strength of the association between the variables.

The samplecorrelation coefficient ror s is an estimate of and is used to measure the strength of the linear relationship

in the sample observations.

Correlation Coefficient r or s indicates

strengthof relationship (strong, weak, or none)

directionof relationship

positive (direct) variables move in same direction

negative (inverse) variables move in oppositedirections

rranges in value from1.0 to +1.0.

Very Strong No Strong VeryStrong Relationship Strong

-1.0 -0.8 -0.5 0.0 +0.5 +0.8 +1.0

Moderate Weak Weak ModerateNegative Positive

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-vePerfect

+vePerfect

Do Variables Relate to One Another?

Is teachers pay related to performance?

Is exercise related to illness?

Is CO2 related to global warming?

Is TV viewing related to shoe size?

Is shoe size related to height?

Is height related to IQ?

Is cigarettes smoked per day related to

lung capacity?

Positive

Negative

PositiveZero

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Positive correlation

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Two variables move in the same direction

Negative correlation

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Two variables tend to go in the opposite direction

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Methods for CalculatingCorrelation Coefficient, rors

Pearson Product-Moment Correlation

Coefficient

Spearman RankCorrelation Coefficient

Pearson Coefficient of Correlation

Both variables must be quantitative and normallydistributed.

Calculation for r:

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Example 2

Refer to Example 1. Compute Pearson coefficientof correlation and interpret the result.

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The Spearman rank correlation coefficient

Spearmans rank correlation coefficient is a measure of associationbetween two variables that are at least of ordinal scale (suitable forqualitative data).

Can also be applied to quantitative databut the variables must firstsbe rankedand then only it is calculated based on these rankings.

where:

d = difference between two ranks

n = number of pairs of observations

NOTE: Be careful with tied observations22

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How to calculate Spearmans rank

correlation coefficient?

1. List each set of scores in a column.

2. Rank the two sets of scores.

3. Place the appropriate rank beside each score.

4. Head a column dand determine the differencein rank for

each pair of scores.

(Note:Sum of the dcolumn should always be 0)

5. Square each number in the dcolumn and sum the

values (d 2).

6. Use the formula to calculate the correlation coefficient.

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Refer Example 5 pg. 140

StudentSubject d d

2

Statistics Computer

A 1 3

B 2 1

C 3 4

D 4 2

E 5 5

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Five students A, B, C, D, E are rankedin two subjects, statistics and

computer programming with the following results.

Calculate the Spearmans rank correlation coefficient.

)1(61

2

2

nnd

s


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Refer Example 6 pg. 141x y Rank of x,

Rx

Rank of y,Ry

d=Rx-Ry d2

6.0 80

6.2 80

6.5 78

6.8 75

7.0 70

7.2 60

7.5 60

7.8 55

8.0 50

8.2 48

8.4 45

8.7 40

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The Regression Line Regression indicates the degree to which the variation in one

variable X, is related to or can be explained by the variation in

another variable Y

Once you know there is a significant linear correlation, youcan write an equation describing the relationship between

thexandyvariables.

This equation is called the line of regression or least squares

line.

The equation of a line may be written as:

where bis the slope of the line and ais the y-intercept.

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Regression line Creates a line of best fit running through the data

Analyze the relationship between the two quantitativevariables,Xand Y

a-intercept: if x= 0 is in the range, then ais the meanof the distribution

of the response y, when x= 0; if x= 0 is not in the range, then ahas no practical

interpretation

b-slope: change in the mean of the distribution of the response

produced by a unit change inx

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Dependentvariable

Independentvariable

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The Least Squares Regression Line

The values of a and bin the regression line y = a+ bxcan be calculated by using the least squares method (ormethod of least squares), given by the following formula:

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x y8 78

2 92

5 90

12 58

15 43

9 746 81

Absences

Final

Grade

Example 3: Application

959085807570656055

4540

50

0 2 4 6 8 10 12 14 16

FinalGrade

X

Absences

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Calculate aand b.

Write the equation of the

line of regression with

x= number of absences

and y= final grade.

The line of regression is:

60848464

8100

3364

1849

54766561

624184

450

696

645

666486

57 516 3751 579 39898

1 8 78

2 2 92

3 5 90

4 12 585 15 43

6 9 747 6 81

644

25

144

225

8136

xy x2 y2x y

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0 2 4 6 8 10 12 14 16

40455055606570758085

9095

Absences

Final

Grade

The line of regression is: y = -3.924x + 105.667

Note that the point = (8.143, 73.714) is on the line.

The Line of Regression31

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The regression line can be used to predict values of y

for values of xfalling within the range of the data.

The regression equation for number of times absent and final

grade is:

Use this equation to predictthe expected grade for a student with

(a) 3 absences (b) 12 absences

Predicting yValues

(a) y =3.924(3) + 105.667 = 93.895

(b) y =3.924(12) + 105.667 = 58.579

y =

3.924x

+ 105.667

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Coefficient of Determination

The coefficient of determination, r2,measures thestrength of the association and is the ratio of explained

variation in yto the total variation in y.

Interpretation :proportion of the variation iny that is explained by the variation in x

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The correlation coefficient of number of times absent and final

grade is r=0.975. The coefficient of determination is

r2 = (0.975)2 = 0.9506.

Interpretation:About 95.06%of the variation in final grades can be

explained by the number of times a student is absent.

Note:The other 4.94%is unexplained and can be due to sampling

error or other variables such as intelligence, amount of time

studied, etc.

Recall Example 3

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