1. BIVARIATE AND PARTIAL CORRELATION

3. Bivariate Correlation
Correlation Coefficient
Coefficient Of Determination
Hypothesis Testing About the Linear Correlation Coefficient

4. Linear Correlation
Pearson Correlation
Spearman rho correlation

5. 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

6. Figure 1 Linear correlation between two variables.
(a) Perfect positive linear correlation, r = 1 y
r = 1 x

7. Figure 2 Linear correlation between two variables.
(b) Perfect negative linear correlation, r = -1 y
r = -1 x

8. Figure 3 Linear correlation between two variables.
(c) No linear correlation, , r ≈ 0 y
r ≈ 0 x

9. Figure 4 Linear correlation between variables.y
(a) Strong positive linear correlation (r is close to 1) x

10. Figure 5 Linear correlation between variables.
(b) Weak positive linear correlation (r is positive but close to 0) y x

11. Figure 6 Linear correlation between variables.
(c) Strong negative linear correlation (r is close to -1) y x

12. Figure 7Linear correlation between variables.
(d) Weak negative linear correlation (r is negative and close to 0) y x

13. 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

14. where
and SS stands for “sum of squares”.

15. 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.
Table 1. Incomes (in hundreds of dollars) and Food Expenditures of Seven Households
Income
Food Expenditure 35 49 21 39 15 28 25 9 7 11 5 8

16. Solution

17. Table. Income x Food Expenditure y xy x² 35 49 21 39 15 28 25 9 7 11 5
315
735
147
429 75
224
225
1225
2401
441
1521
784
625
Σx = 212
Σy = 64
Σxy = 2150
Σx² = 7222

18. Solution 13-1

19. Solution

20. COEFFICIENT OF DETERMINATION
The coefficient of determination, denoted by
CD = r2xy .100%
represents the proportion of contribution given by
variabel x to variabel y.

21. 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% = %

22. Hypothesis Testing About the Linear Correlation Coefficient (Pearson ‘s Correlation)
H0: r = 0
The correlation coefficient is zero
H1:
The correlation coefficient is positive or negative.
Reject H0 If or
Do not reject H0 if

23. Figure
Look for this area in the Product Moment table to find the critical values of r.
Reject H0
Reject H0
Do not reject H0
-rtable r table

24. 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.

25. Solution
H0: r = 0
There 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 .
The value of the r table =0.754 and rxy =0,959.
rxy > r table
Hence, we reject the null hypothesis

26. Value of the Correlation Coefficient
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

27. 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. 2.

28. Cont….
Where:

29. 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.

30. 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.

31. Continuu …
income (x)
29.85
19.0
19.18
31.78
25.22
16.68
23.98
26.33
Food
Expenditure
(y)
8.3
10.1
10.3
7.1
9.9
11.5
8.7
9.8
Calculate the value of the statistic
Can you conclude that there is no significant correlation between
the per capita income and the infant mortality rates for all states?
Use ρs

32. Solution a. u 7 2 3 8 5 1 4 6 v d -4 -7 d2 25 16 49

33. Solution
H0: ρs = 0
There 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. b.

34. 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.

35. Example 2. X 90 60 50 70 40 30 20 80 Y
Calculate the value of the statistic
Can you conclude that there is no significant correlation between
the per capita income and the infant mortality rates for all states?
Use
Rho Spearman.

36. Solution

37. EXERCISES. Dari data di samping, tentukanlah:
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
6,5
5,5 8 7 11
Dari data di samping, tentukanlah:
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.