1. M248: Analyzing data
Block D

2. Related variables Regression
Block D
UNITS D3, D2
Related variables Regression
Contents
Introduction
Section 1: Correlation
Section 2: Measures of correlation
Section 3: Regression
Section 4: Independence (Association in Contingency tables)
Terms to know and use

3. Introduction The slides are based on the following points
Correlation between two variables and the strength of this correlation.
Regression
Studying contingency tables

4. Section 1: Correlation
Two random variables are said to be related (or correlated or associated) if knowing the value of one variable tells you something about the value of the other. Alternatively we say there is a relationship between the two variables.
The first thing to do when investigating a possible relationship between two variables, is to produce a scatterplot of the data.

5. Section 1: Correlation Section 1.1: Are the variables related?
The two variables are said to be positively related if the pattern of the points in the scatterplot slopes upwards from left to right.
Example:

6. Section 1: Correlation
The two variables are said to be negatively related if the pattern of the points in the scatterplot slopes downwards from left to right.
Example:

7. Section 1: Correlation
Strong positive Linear relationship between the two variables
There is an overall downward trend but there is no suggestion about any relationship between the two variables
Strong relationship between the two variables but it is not linear

8. Section 1: Correlation
Sometimes, a relationship between two variables is more complicated and the variables cannot be classified as either positively or negatively related.
Example:
Read Examples 1.1, 1.2, 1.3 and 1.4
Solve Activity 1.1
Pattern seems to be like

9. Section 1: Correlation

10. Section 1: Correlation Section 1.2: Correlation and Causation
Causation and correlation are not equivalent. Causation means that the value of one variable is caused by the value of the other, while correlation means that there is a relationship between the two variables.

11. Section 2: Measures of correlation
There are two measures of correlations, The Pearson and the Spearman correlation. These measures are called correlation coefficients.
A correlation coefficient is a number between -1 and +1, the closer it is of those limits, the stronger the relationship between the two variables.
Correlation coefficients which measure how well a straight line can explain the relationship between two variables are called linear correlation coefficients.

12. Section 2: Measures of correlation
The range of the correlation coefficient is from 1 to 1.
If there is a strong positive linear relationship between the variables, the value of r will be close to 1.
If there is a strong negative linear relationship between the variables, the value of r will be close to 1.

13. Section 2: Measures of correlation
Section 2.1: The Pearson correlation coefficient
Two variables are said to be positively related if they increase together and decrease together, then the correlation coefficient will be positive. And if they are negatively related then it will be negative.
A correlation coefficient of 0 implies that there is no systematic linear relationship between the two variables.

14. Section 2: Measures of correlation
The Pearson correlation coefficients is a measure of how well a straight line can explain the relationship between two variables. It is only appropriate to use this coefficient if the scatterplot shows a roughly linear pattern.

15. Example 1: Car Rental Companies
Cars x
Income y
Company
(in 10,000s)
(in billions) xy x 2 y 2 A
63.0
7.0
441.00
49.00 B
29.0
3.9
113.10
841.00
15.21 C
20.8
2.1
43.68
432.64
4.41 D
19.1
2.8
53.48
364.81
7.84 E
13.4
1.4
18.76
179.56
1.96 F
8.5
1.5
2.75
72.25
2.25 Σ x = Σ y = Σ xy = Σ x 2 = Σ y 2 =
153.8
18.7
682.77
80.67

16. Example 1: Car Rental Companies
Σx = 153.8, Σy = 18.7, Σxy = , Σx2 = ,
Σy2 = 80.67, n = 6

17. Example 2: Absences/Final Grades
Number of
Final Grade
Student
absences, x
y (pct.) xy x 2 y 2 A 6 82
492 36
6,724 B 2 86
172 4
7,396 C 15 43
645
225
1,849 D 9 74
666 81
5,476 E 12 58
696
144
3,364 F 5 90
450 25
8,100 G 8 78
624 64
6,084 Σ x = Σ y = Σ xy = Σ x 2 = Σ y 2 = 57
511
3745
579
38,993

18. Example 2: Absences/Final Grades
Σx = 57, Σy = 511, Σxy = 3745, Σx2 = 579,
Σy2 = 38,993, n = 7

19. Example 3: Exercise/Milk Intake
Subject
Hours, x
Amount y xy x 2 y 2 A 3 48
144 9
2,304 B 8 64 C 2 32 64 4
1,024 D 5 64
320 25
4,096 E 8 10 80 64
100 F 5 32
160 25
1,024 G 10 56
560
100
3,136 H 2 72
144 4
5,184 I 1 48 48 1
2,304 Σ x = Σ y = Σ xy = Σ x 2 = Σ y 2 = 36
370
1,520
232
19,236

20. Example 3: Exercise/Milk Intake
Σx = 36, Σy = 370, Σxy = 1520, Σx2 = 232,
Σy2 = 19,236, n = 9

21. Section 2: Measures of correlation
Section 2.2: The Spearman rank correlation coefficient
Replacing the original data by their ranks, and measuring the strength of association between two variables by calculating the Pearson correlation coefficient with the ranks is known as the Spearman rank correlation coefficient, and is denoted by rs
The values of rs is a measure of the linearity of the relationship between the ranks.

22. Section 2: Measures of correlation
Example: GNP and adult literacy
Spearman’s Rank correlation coefficient (rs) is the best method to use.
GNP per capita
% adult literacy
Nepal
210
39.7
Sudan
290
55.5
Gambia
340
34.5
Peru
2460 89
Turkey
3160
81.4
Brazil
4570 84
Argentina
8970 97
Israel
15940 96
U.A.E.
18220
74.3
Netherlands
24760
100
Remember, Spearman’s Rank can only be used with ordinal data.
It is necessary, therefore, to rank-order the data first.

23. Section 2: Measures of correlation
Example: GNP and adult literacy

24. Section 2: Measures of correlation
Example: GNP and adult literacy
Σx = 55, Σy = 55, Σxy = 363, Σx2 = 385,
Σy2 = 385, n = 10

25. Section 2: Measures of correlation
A relationship is known as a monotonic increasing relationship if the value of rs is equal to +1. that is they have an exact curvilinear positive relationship.
Example:
Similarly a data has a Spearman rank correlation coefficient of -1 if the two variables have a monotonic decreasing relationship

26. Section 2: Measures of correlation
Section 2.3: Testing for association
The sampling distribution of the Pearson correlation:
Under the null hypothesis that there is no association between two variables, the sampling distribution of the Pearson correlation R, is such that:

27. Example : Car Rental Companies
Test the significance of the correlation coefficient found in example 1. Use r =
H0:   0 This null hypothesis means that there is no correlation between the x and y variables in the population.
H1:   0 This alternative hypothesis means that there is a significant correlation between the variables in the population.

28. Example : Car Rental Companies
Step 2: Compute the test value.
Step 3: Compute the p-value
df = 4 (use table 3 of handbook) then, < p-value < 2( )
0 < p-value < 0.002
Step 4: Decision making
Since the p-value is ≤ 0.01, then we have a strong evidence against H0
Step 5: Summarize the results.
There is a relationship between the number of cars a rental agency owns and its annual income.

29. Section 2: Measures of correlation
The approximate sampling distribution of the Spearman correlation:
For large samples, under the null hypothesis of no association, the sampling distribution of Rs, is such that:

30. Example : GNP and adult literacy
Test the significance of the correlation coefficient found in the example. Use rs = 0.73.
H0:   0 This null hypothesis means that there is no correlation between the x and y variables in the population.
H1:   0 This alternative hypothesis means that there is a significant correlation between the variables in the population.

31. Example : GNP and adult literacy
Step 2: Compute the test value.
Step 3: Compute the p-value
df = 8 (use table 3 of handbook) then
2( ) < p-value < 2(1-0.99)
0.01 < p-value < 0.02
Step 4:
Since the 0.01< p-value is ≤ 0.05, then we have a moderate evidence against H0
Step 5: Summarize the results.
There is a relationship between GNP and adult literacy.

32. Section 2: Measures of correlation
Section 2.4: Correlation using MINITAB
See MINITAB Videos on moodle

33. Regression
If the value of the correlation coefficient is significant, the next step is to determine the equation of the regression line which is the data’s line of best fit.

34. Regression
Best fit means that the sum of the squares of the vertical distance from each point to the line is at a minimum.

35. Least Square line of a linear regression
The response variable (dependent) is denoted by Y and the predicator /explanatory (independent) variable by x

36. Example : Car Rental Companies
Find the equation of the regression line for the data in Example 1.
Σx = 153.8, Σy = 18.7, Σxy = , Σx2 = ,
Σy2 = 80.67, n = 6

37. Example: Car Rental Companies
Use the equation of the regression line to predict the income of a car rental agency that has 200,000 automobiles. x = 20 corresponds to 200,000 automobiles. Hence, when a rental agency has 200,000 automobiles, its revenue will be approximately $2.52 billion.

38. Chi-Square Distributions
The chi-square variable is similar to the t variable in that its distribution is a family of curves based on the number of degrees of freedom.
The symbol for chi-square is (Greek letter chi, pronounced “ki”).
A chi-square variable cannot be negative, and the distributions are skewed to the right.

39. Chi-Square Distributions
At about 100 degrees of freedom, the chi-square distribution becomes somewhat symmetric.
The area under each chi-square distribution is equal to 1.00, or 100%.

40. The Chi-Square Test
When data can be tabulated in table form in terms of frequencies, several types of hypotheses can be tested by using the chi-square test.
The test of independence of variables is used to determine whether two variables are independent of or related to each other when a single sample is selected.
Formula for the test:
Where
d.f. = number of categories minus 1
O = observed frequency
E = expected frequency

41. Test for Independence
The chi-square test can be used to test the independence of two variables.
The hypotheses are:
H0: There is no relationship between two variables.
H1: There is a relationship between two variables.
If the null hypothesis is rejected, there is some relationship between the variables.
In order to test the null hypothesis, one must compute the expected frequencies, assuming the null hypothesis is true.

42. Contingency Tables (Association)
When data are arranged in table form for the independence test, the table is called a contingency table.
The degrees of freedom for any contingency table are d.f. = (rows – 1) (columns – 1) = (R – 1)(C – 1).

43. Example 1: College Education and Place of Residence
A sociologist wishes to see whether the number of years of college a person has completed is related to her or his place of residence. A sample of 88 people is selected and classified as shown. Can the sociologist conclude that a person’s location is dependent on the number of years of college?
Location
No College
Four-Year Degree
Advanced Degree
Total
Urban 15 12 8 35
Suburban 9 32
Rural 6 7 21 29 24 88

44. Example 1: College Education and Place of Residence
Step 1: State the hypotheses and identify the claim.
H0: A person’s place of residence is independent of the number of years of college completed.
H1: A person’s place of residence is dependent on the number of years of college completed (claim).

45. Example 1: College Education and Place of Residence
Compute the expected values.
Location
No College
Four-Year Degree
Advanced Degree
Total
Urban 15 12 8 35
Suburban 9 32
Rural 6 7 21 29 24 88
(11.53)
(13.92)
(9.55)
(10.55)
(12.73)
(8.73)
(6.92)
(8.35)
(5.73)

46. Example 1: College Education and Place of Residence
Step 2: Compute the test value.
The degrees of freedom are (3 – 1)(3 – 1) = 4.

47. Example 1: College Education and Place of Residence
Step 3: Compute the P-value.
Use Table 4 in the handbook
(1-0.5) < p-value < (1-0.1)
0.5< p- value < 0.9
Step 4: Making the decision
Since the p-value > 0.1 then we have a little evidence against H0:
Step 5: Summarize the results.
There is not enough evidence to support the claim that a person’s place of residence is dependent on the number of years of college completed.

48. Example 2: Alcohol and Gender
A researcher wishes to determine whether there is a relationship between the gender of an individual and the amount of alcohol consumed. A sample of 68 people is selected, and the following data are obtained. Can the researcher conclude that alcohol consumption is related to gender?
Gender
Alcohol Consumption
Total
Low
Moderate
High
Male 10 9 8 27
Female 13 16 12 41 23 25 20 68

49. Example 2: Alcohol and Gender
Step 1: State the hypotheses and identify the claim.
H0: The amount of alcohol that a person consumes is independent of the individual’s gender.
H1: The amount of alcohol that a person consumes is dependent on the individual’s gender (claim).

50. Example 2: Alcohol and Gender
Compute the expected values.
Gender
Alcohol Consumption
Total
Low
Moderate
High
Male 10 9 8 27
Female 13 16 12 41 23 25 20 68
(9.13)
(9.93)
(7.94)
(13.87)
(15.07)
(12.06)

51. Example 2: Alcohol and Gender
Step 2: Compute the test value.
The degrees of freedom are (2 – 1 )(3 – 1) = (1)(2) = 2.

52. Example 2: Alcohol and Gender
Step 3: Compute the P-value.
Use Table 4 in the handbook
(1-0.5) < p-value < (1-0.1)
0.5< p- value < 0.9
Step 4: Making the decision.
Since the p-value > 0.1, then we have a little evidence
against H0
Step 5: Summarize the results.
There is not enough evidence to support the claim that the amount of alcohol a person consumes is dependent on the individual’s gender.

53. Terms to know and use Related variable Monotonic relationship
Correlation
Correlation coefficient
Association
Pearson correlation coefficient
Causation
Spearman rank correlation coefficient
Positively related
Negatively related
Monotonic increasing
Monotonic decreasing

54. Terms to know and use Explanatory variables Independence
Response variable
Association
Linear regression model
Regression line
Least square line
Contingency Table