BIVARIATE AND PARTIAL CORRELATION

BIVARIATE AND PARTIAL CORRELATION

2

3

Bivariate Correlation

Correlation Coefficient Coefficient Of Determination

Hypothesis Testing About the Linear Correlation Coefficient

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Pearson Correlation

Spearman rho correlation

Linear Correlation

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Linear Correlation Coefficient

Value of the Correlation Coefficient The value of the correlation coefficient always

lies in the range of –1 to 1; that is, -1 ≤ ρ ≤ 1 and -1 ≤ r ≤ 1

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Figure 1 Linear correlation between two variables.

(a) Perfect positive linear correlation, r = 1

r = 1

x

y

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Figure 2 Linear correlation between two variables.

(b) Perfect negative linear correlation, r = -1

r = -1

x

y

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Figure 3 Linear correlation between two variables.

(c) No linear correlation, , r ≈ 0

r ≈ 0

x

y

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Figure 4 Linear correlation between variables.

(a) Strong positive linear correlation (r is close to 1)

x

y

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Figure 5 Linear correlation between variables.

(b) Weak positive linear correlation (r is positive but close to 0)

x

y

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Figure 6 Linear correlation between variables.

(c) Strong negative linear correlation (r is close to -1)

x

y

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Figure 7Linear correlation between variables.

(d) Weak negative linear correlation (r is negative and close to 0)

x

y

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Pearson Correlation

The Pearson correlation, denoted by rxy, measures the strength of the linear relationship between two variables for a sample and is calculated as

yyxx

xyxy

SSSS

SSr

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where

and SS stands for “sum of squares”.

n

yySS

n

xx

n

yxxy

yy

xx

xy

2

2

2

2SS

SS

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Example 1

Calculate the correlation coefficient for the data on incomes and food expenditure on the seven

households given in the Table 1. Use income as an independent variable and food expenditure as a dependent variable.

Income Food Expenditure

35

49

21

39

15

28

25

9

15

7

11

5

8

9

Table 1. Incomes (in hundreds of dollars) and Food Expenditures of Seven Households

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Solution

1429.97/64/

2857.307/212/

64 212

nyy

nxx

yx

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Table.

Income

x

Food Expenditure

yxy x²

35

49

21

39

15

28

25

9

15

7

11

5

8

9

315

735

147

429

75

224

225

1225

2401

441

1521

225

784

625

Σx = 212 Σy = 64 Σxy = 2150 Σx² = 7222

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Solution 13-1

8571.60

7

)64(646

4286.8017

)212(7222SS

7143.2117

)64)(212(2150SS

22

2

22

2

n

yySS

n

xx

n

yxxy

yy

xx

xy

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Solution

.959.)8571.60)(4286.801(

7143.211

yyxx

xyxy

SSSS

SSr

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COEFFICIENT OF DETERMINATION

Coefficient of Determination The coefficient of determination, denoted by CD = r2

xy .100% represents the proportion of contribution given by variabel x to variabel y.

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Example

For the data of Table 1 on monthly incomes and food expenditures of seven households, calculate the coefficient of determination.

Solution From earlier calculations rxy = 0.959

So, CD = (0.959)2 x 100% = 91.97 %

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H0: r = 0– The correlation coefficient is zero

H1: – The correlation coefficient is positive or

negative.

0r

Hypothesis Testing About the Linear Correlation Coefficient (Pearson ‘s Correlation)

Reject H0 If tabhit rr .tabhit rr or

Do not reject H0 if tabhittab rrr

23 -rtable r table

Do not reject H0

Reject H0 Reject H0

Look for this area in the Product Moment table to find the critical values of r.

Figure

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Example 2

Using the 5% level of significance and the data from Example 1, test whether the linear correlation coefficient between incomes and food expenditures is significant ?. Assume that the populations of both variables are normally distributed.

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Solution

The value of the r table =0.754 and rxy =0,959.

rxy > r table

Hence, we reject the null hypothesis

H0: r = 0There is no correlation significant correlation between the incomes

and food expenditures .

H1: r 0 There is a correlation significant correlation between the the

incomes and food expenditures .

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Linear Correlation Coefficient(Spearman rho rank correlation coefficient)

Value of the Correlation Coefficient The value of the correlation coefficient always

lies in the range of –1 to 1; that is, -1 ≤ ρs ≤ 1 and -1 ≤ rs ≤ 1

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Calculate the value of correlation coefficient

To Calculate the value of rs, we rank the data for each variable x and y, separately and denote those ranks by u and v, respectively. Thus

)1(

61

2

2

nn

drs

22

222

.2 qp

dqprs

1.

2.

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Cont….

Where:

12

)1(

12

)1( 222 ttNN

p

12

)1(

12

)1( 222 ttNN

q

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Hypothesis about the Spearman rho rank correlation

Hypothesis about the Spearman rho rank correlation coefficient ρs , the test statistic is rs

and its observed value is calculate by using the above formula.

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Example 1.

Suppose we want to investigate the relationship between the per capita income (in 1000 of dollars) and the infant mortality rate (in persent) for different states. The following table gives data on these two variables for a random sample eight states.

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Continuu …

income (x) 29.85 19.0 19.18 31.78 25.22 16.68 23.98 26.33

FoodExpenditure(y) 8.3 10.1 10.3 7.1 9.9 11.5 8.7 9.8

a. Calculate the value of the statisticb. Can you conclude that there is no significant correlation between the per capita income and the infant mortality rates for all states? Use 05,0

ρs

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Solution

u 7 2 3 8 5 1 4 6

v 2 6 7 1 5 8 3 4

d 5 -4 -4 7 0 -7 1 2

d2 25 16 16 49 0 49 1 4

a.

1602d

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Solution

905.0504

9601

)164(8

)160(61

)1(

61

2

2

nn

drs

b. H0: ρs = 0There is no correlation significant correlation between the per capita

income and the infant mortality rates for all states.

H1: ρs 0 There is a correlation significant correlation between the per capita

income and the infant mortality rates for all states.

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Because rs=-0.905 is less than -0.738 and we reject H0. We conclude that there is a correlation significant correlation between the per capita income and the infant mortality rates for all states.

Because the value of rs from sample is negative, we can also state that as per capita income increase, infant mortality tends to decrease.

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Example 2.

X 90 60 50 70 40 30 20 80 70 60

Y 80 70 60 80 50 40 20 90 80 60

a. Calculate the value of the statisticb. Can you conclude that there is no significant correlation between the per capita income and the infant mortality rates for all states? Use 05,0

Rho Spearman.

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Solution

5,81112

)110(10

12

)1(

12

)1(

2

222

ttNN

x

805,212

)110(10

12

)1(

12

)1(

2

222

ttNN

y

22

222

.2 yx

dyxrs

957,049,161

50,154

805,812

7805,81

xrs

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EXERCISES.1.Data berikut menunjukkan urutan (rank) tingkat nilai motivasi (Rx) dan urutan tingkat prestasi (Ry) 12 orang sampel.

Rx Ry

3 12

4 10

5 1

2 2,5

1 4

6,5 5,5

8 7

11 7

8 2,5

6,5 11

8 5,5

12 7

Dari data di samping, tentukanlah:a. Koefisien korelasi antara tingkat motivasi (x) dan tingkat hasil belajar (y). b. Besar sumbangan yang diberikan x kepada y. c. uji signifikansi x dengan y.