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1MVS 250: V. KatchMVS 250: V. Katch

STATISTICSChapter 5 Correlation/Regression

2

Overview

Paired Data

is there a relationship

if so, what is the equation

use the equation for prediction

3

Correlation

4

Definition

Correlation

exists between two variables when one of them is related to the other in some way

5

Assumptions

1. The sample of paired data (x,y) is a random sample.

2. The pairs of (x,y) data have a bivariate normal distribution.

6

Definition

Scatterplot (or scatter diagram)

is a graph in which the paired (x,y) sample data are plotted with a horizontal x axis and a vertical y axis. Each individual (x,y) pair is plotted as a single point.

7

Scatter Diagram of Paired Data

8

Scatter Diagram of Paired Data

9

Positive Linear Correlation

x x

yy y

x

Scatter Plots

(a) Positive (b) Strong positive

(c) Perfect positive

10

Negative Linear Correlation

x x

yy y

x(d) Negative (e) Strong

negative(f) Perfect

negative

Scatter Plots

11

No Linear Correlation

x x

yy

(g) No Correlation (h) Nonlinear Correlation

Scatter Plots

12

xy/n - (x/n)(y/n)(SDx) (SDy)

r =

DefinitionLinear Correlation Coefficient r

measures strength of the linear relationship between paired x and y values in a sample

Where xy/n is the mean of the cross products; (x/n) is the mean of the x variable; (y/n) is the mean of the y variable; SDx is the standard deviation of the x variable and SDy is the standard deviation of the x variable

13

Notation for the Linear Correlation Coefficient

n number of pairs of data presented

denotes the addition of the items indicated.

x/n denotes the mean of all x values.

y/n denotes the mean of all y values.

xy/n denotes the mean of the cross products [x times y, summed; divided by n]

r linear correlation coefficient for a sample

linear correlation coefficient for a population

14

Round to three decimal placesUse calculator or computer if possible

Rounding the Linear Correlation Coefficient r

15

Properties of the Linear Correlation Coefficient r

1. -1 r 1

2. Value of r does not change if all values of either variable are converted to a different scale.

3. The r is not affected by the choice of x and y. Interchange x and y and the value of r will not

change.

4. r measures strength of a linear relationship.

16

Interpreting the Linear Correlation Coefficient

If the absolute value of r exceeds the value in Sig. Table, conclude that there is a significant linear correlation.

Otherwise, there is not sufficient evidence to support the conclusion of significant linear correlation.

Remember to use n-2

17

Common Errors Involving Correlation

1. Causation: It is wrong to conclude that correlation implies causality.

2. Averages: Averages suppress individual variation and may inflate the correlation coefficient.

3. Linearity: There may be some relationship between x and y even when there is no significant linear correlation.

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0

50

100

150

200

250

0 1 2 3 4 5 6 7 8

Dis

tan

ce(f

eet)

Time (seconds)

Common Errors Involving Correlation

19

Correlation is Not Causation

A B

C

20

Correlation Calculations

Rank Order Correlation - Rho

Pearson’s - r

21

Rank Order Correlation

Hits Rank HR Rank D D2

1 10 3 8 2 4

2 9 4 7 2 4

3 8 5 6 2 4

4 7 1 10 -3 9

5 6 7 4 2 4

6 5 6 5 0 0

7 4 2 9 -5 25

8 3 10 1 2 4

9 2 9 2 0 0

10 1 8 3 2 4

22

Rank Order Correlation, cont

Hits Rank HR Rank D D2

1 10 3 8 2 4

2 9 4 7 2 4

3 8 5 6 2 4

4 7 1 10 -3 9

5 6 7 4 2 4

6 5 6 5 0 0

7 4 2 9 -5 25

8 3 10 1 2 4

9 2 9 2 0 0

10 1 8 3 2 4

Rho = 1- [6 (∑D2) / N (N2-1)]

Rho = 1- [6(58)/10(102-1)]

Rho = 1- [348 / 10 (100 -1)]

Rho = 1- [348 / 990]

Rho = 1- 0.352

Rho = 0.648

(∑D2 = 58)N=10

23

Pearson’s rHits HR xy

1 3 3

2 4 8

3 5 15

4 1 4

5 7 35

6 6 36

7 2 14

8 10 80

9 9 81

10 8 80

x/n=5.

5

x/n= 5.5

xy/n =32.86

xy/n - (x/n)(y/n)(SDx) (SDy)

r =

r = 32.86 - (5.5) (5.5)/(3.03) (3.03)

r = 35.86 - 30.25 / 9.09

r = 5.61 / 9.09

r = 0.6172

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Pearson’s r

Excel Demonstration

25

0.27

2

1.41

3

2.19

3

2.83

6

2.19

4

1.81

2

0.85

1

3.05

5

Data from the Garbage Projectx Plastic (lb)

y Household

Is there a significant linear correlation?

26

0.27

2

1.41

3

2.19

3

2.83

6

2.19

4

1.81

2

0.85

1

3.05

5


y Household


Plastic Household0.27 21.41 32.19 32.83 62.19 41.81 20.85 13.05 5

27

0.27

2

1.41

3

2.19

3

2.83

6

2.19

4

1.81

2

0.85

1

3.05

5


y Household


Plastic Garbage v Household size

R2 = 0.7096

01234567

0 0.5 1 1.5 2 2.5 3 3.5

Plastic (lbs)

Household size

r = 0.842R2 = 0.71

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n = 8 = 0.05 H0: = 0

H1 : 0

Test statistic is r = 0.842

Critical values are r = - 0.707 and 0.707(Table R with n = 8 and = 0.05)

TABLE R Critical Values of the Pearson Correlation Coefficient r

456789

101112131415161718192025303540455060708090

100

n.999.959.917.875.834.798.765.735.708.684.661.641.623.606.590.575.561.505.463.430.402.378.361.330.305.286.269.256

.950

.878

.811

.754

.707

.666

.632

.602

.576

.553

.532

.514

.497

.482

.468

.456

.444

.396

.361

.335

.312

.294

.279

.254

.236

.220

.207

.196

= .05 = .01


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0r = - 0.707 r = 0.707 1

Sample data:r = 0.842

- 1

0.842 > 0.707, That is the test statistic does fall within the critical region.

Therefore, we REJECT H0: = 0 (no correlation) and concludethere is a significant linear correlation between the weights ofdiscarded plastic and household size.


Fail to reject = 0

Reject= 0

Reject= 0

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Method 1: Test Statistic is t(follows format of earlier chapters)

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Formal Hypothesis Test

To determine whether there is a significant linear correlation between two variables

Two methods

Both methods let H0: = (no significant linear correlation)

H1: (significant linear correlation)

32

Test statistic: r

Critical values: Refer to Table R (no degrees of freedom)

Method 2: Test Statistic is r(uses fewer calculations)

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Test statistic: r

Critical values: Refer to Table A-6 (no degrees of freedom)

Method 2: Test Statistic is r(uses fewer calculations)

Fail to reject = 0

0r = - 0.811 r = 0.811 1

Sample data:r = 0.828

-1

Reject= 0

Reject= 0

34

Method 1: Test Statistic is t(follows format of earlier chapters)

Test statistic:

1 - r 2

n - 2

r

Critical values:

use Table T with degrees of freedom = n - 2

t =

35

Testing for a

Linear Correlation

If H0 is rejected conclude that thereis a significant linear correlation.

If you fail to reject H0, then there isnot sufficient evidence to conclude

that there is linear correlation.

If the absolute value of thetest statistic exceeds the

critical values, reject H0: = 0Otherwise fail to reject H0

The test statistic is

t =1 - r 2

n -2

r

Critical values of t are from Table A-3

with n -2 degrees of freedom

The test statistic is

Critical values of t are from Table A-6

r

Calculate r using

Formula 9-1

Select asignificance

level

Let H0: = 0 H1: 0

Start

METHOD 1 METHOD 2

36

Why does the critical value of r increase as sample size decreases?

A correlation by chance is more likely.

37

Coefficient of Determination(Effect Size)

r2

The part of variance of one variable that can be explained by the variance of a related variable.

38

x = 3

•••Quadrant 3

Quadrant 2 Quadrant 1

Quadrant 4

•

x

y

y = 11(x, y)

x - x = 7- 3 = 4

y - y = 23 - 11 = 12

0 1 2 3 4 5 6 70

4

8

12

16

20

24

r = (x -x) (y -y)(n -1) Sx Sy

(x, y) centroid of sample points

•(7, 23)

Justification for r Formula

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