Download - BIVARIATE AND PARTIAL CORRELATION
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BIVARIATE AND PARTIAL CORRELATION
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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
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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.