PDF Chapter 10: Correlation and Regression Chapter 13: Nonparametric Statistics

Chapter 10: Correlation and Regression Chapter 13: Nonparametric Statistics

Objectives:

❑ Learn how to draw a scatter plot for a set of ordered pairs.

❑ Learn how to compute the correlation coefficient.

❑ Learn how to compute the equation of the regression line.

❑ Learn how to compute the Spearman rank correlation coefficient.

Overview of Chapters 10 and 13

Sec. # Title Page(s)

10 - 1 Scatter Plots and Correlation 369 – 385 13 - 6 The Spearman Rank Correlation

Coefficient 459 – 461

10 - 2 Regression 386 – 393

Remember?

Independent

variable influences Dependent

variable

At a Glance!

 Are two or more variables linearly related?

(Scatter plot and/or correlation coefficient)

 If so, what is the strength of the relationship?

(Scatter plot and/or correlation coefficient)

 What type of relationship exists?

(Scatter plot, correlation coefficient and/or regression)

 What kind of predictions can be made from the relationship?

(Regression)



A scatter plot is a graph of the ordered pairs of numbers (x, y) consisting of the independent variable x and the dependent variable y.

10 – 1: Scatter Plots and Correlation

10 – 1: Scatter Plots and Correlation (cont.)



It is a visual way to describe the nature of the relationship between the x and y. It may shows:



a positive linear relationship,



a negative linear relationship,



a curvilinear relationship,



or no relationship.



Example 10 – 1, page 372, Example 10 – 2,

page 372 – 373, Example 10 – 3, page 373.

Examples of scatter plots patterns

Correlation

 Pearson’s linear correlation coefficient, which will be denoted by 𝑟, measures the strength and the direction of a linear relationship between two quantitative variables.

Calculating 𝒓

 The linear correlation coefficient is given by

𝒓 = 𝒏∑𝒙𝒚 − (∑𝒙)(∑𝒚)

𝒏∑𝒙^𝟐 − ∑𝒙 ^𝟐 𝒏∑𝒚^𝟐 − ∑𝒚 ^𝟐

 The above coefficient is also known as Pearson product moment correlation coefficient (PPMC).

Properties of 𝒓

 The range of the correlation coefficient is from +1 to -1.

 If the value of 𝑟 is close to +1, then there is a strong positive linear relationship between the variables.

 If the value of 𝑟 is close to -1, then there is a strong negative linear relationship between the variables.

 If the value of 𝑟 is close to 0, then there is either a

weak or no linear relationship between the variables.

Properties of 𝒓

Example 10 – 4: Car Rental Companies

# of Cars (x) Revenue (y)

63 7

29 3.9

20.8 2.1

19.1 2.8

13.4 1.4

8.5 1.5

 From the left table, we obtain:

∑𝒙 = 𝟏𝟓𝟑. 𝟖,

∑𝒚 = 𝟏𝟖. 𝟕,

∑𝒙𝒚 = 𝟔𝟖𝟐. 𝟕𝟕,

∑𝒙^𝟐 = 𝟓𝟖𝟓𝟗. 𝟐𝟔,

∑𝒚^𝟐 = 𝟖𝟎. 𝟔𝟕.

Example 10 – 4 (cont.)

𝒓 = 𝟔(𝟔𝟖𝟐. 𝟕𝟕) − (𝟏𝟓𝟑. 𝟖)(𝟏𝟖. 𝟕)

𝟔(𝟓𝟖𝟓𝟗. 𝟐𝟔) − 𝟏𝟓𝟑. 𝟖 ^𝟐 𝟔(𝟖𝟎. 𝟔𝟕) − 𝟏𝟖. 𝟕 ^𝟐

= 𝟎. 𝟗𝟖𝟐

 Hence, there is a strong positive linear correlation relation between the number of rented cars and revenues.

 Example 10 – 5, page 377 (Negative correlation),

Example 10 – 6, page 378 (Weak positive correlation).

13 – 6: The Spearman Rank Correlation Coefficient

 If 𝑛 is the sample size, and 𝑑 is difference in ranks, then the Spearman rank correlation coefficient is calculated as

𝒓_𝒔 = 𝟏 − 𝟔∑𝒅^𝟐 𝒏(𝒏^𝟐 − 𝟏)

Example 13 – 7: Bank Branches and Deposits (page 459)

# of branches (X) Deposits (Y) Rank (X)

Rank (Y)

209 23 4 4

353 31 2 1

19 7 8 6

201 12 5 5

344 26 3 2

132 5 6 7

401 24 1 3

126 5 7 8

# of branches (X) Deposits (Y) Rank (X)

209 23 4

353 31 2

19 7 8

201 12 5

344 26 3

132 5 6

401 24 1

126 5 7

# of branches (X) Deposits (Y)

209 23

353 31

19 7

201 12

344 26

132 5

401 24

126 4

# of branches (X)

209 353 19 201 344 132 401 126

Example 13 – 7 (cont.)

Rank (X) Rank (Y) 𝒅 𝒅^𝟐

4 4 0 0

2 1 1 1

8 6 2 4

5 5 0 0

3 2 1 1

6 7 -1 1

1 3 -2 4

7 8 -1 1

∑ 𝟎 𝟏𝟐 = ∑𝒅^𝟐

Example 13 – 7 (cont.)

𝒓_𝒔 = 𝟏 − 𝟔∑𝒅^𝟐

𝒏 𝒏^𝟐 − 𝟏 = 𝟏 − 𝟔 ⋅ 𝟏𝟐

𝟖 𝟔𝟒 − 𝟏 = 𝟏 − 𝟕𝟐 𝟓𝟎𝟒

= 𝟎. 𝟖𝟓𝟕

 The above value indicates that we have a strong positive correlation.

 We can calculate Spearmen’s correlation if the data are ordinal-level qualitative.

10 – 2: 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.

 Best fit means that the sum of the squares of the vertical distances from each point to the line is at a minimum.

Line of best fit

Line of best fit (cont.)

Determination of the Regression Line Equation

 The equation regression line is:

𝒚^′ = 𝒂 + 𝒃 ⋅ 𝒙

 Here, 𝑎 is the intercept or the regression constant, 𝑏 is the slope or the regression coefficient, 𝑥 is the observed independent variable, and they are used to calculate 𝑦′which is the predicted dependent

variable.

Determination of the Regression Line Equation (cont.)

𝒂 = ∑𝒚 ∑𝒙^𝟐 − (∑𝒙)(∑𝒚) 𝒏 ∑𝒙^𝟐 − ∑𝒙 ^𝟐

𝒃 = 𝒏 ∑𝒙𝒚 − (∑𝒙)(∑𝒚) 𝒏 ∑𝒙^𝟐 − ∑𝒙 ^𝟐

Example 10 – 9 (page 388)

 Number of rented cars is the independent variable 𝑥, while the revenue is the dependent variable 𝑦. The regression line is found to be:

𝒚^′ = 𝟎. 𝟑𝟗𝟔 + 𝟎. 𝟏𝟎𝟔 ⋅ 𝒙

 This means that as the number of rented cars

increases by 1 as the revenue increases by 0.106 on average.

Example 10 – 10 (page 389)

 Number of absences is the independent variable 𝑥, while the final grade is the dependent variable 𝑦. The regression line is found to be:

𝒚^′ = 𝟏𝟎𝟐. 𝟒𝟗𝟑 − 𝟑. 𝟔𝟐𝟐 ⋅ 𝒙

 This means that as the number of absences

increases by 1 as the final grade decreases by 3.622 on average.

Example 10 – 11 (page 391)

 Predict the income of a car rental agency (y) that has 200,000 automobiles (x).

 Note that in the Example 10 – 1, the unit of number of rented automobiles is in ten thousands.

Therefore, 200,000 automobiles is in fact 20 ten thousand, i.e. x = 20. Hence,

𝑦^′ = 0.396 + 0.106 𝟐𝟎 = 2.516

Important Rule!

 Q. Is there any relationship between the Person’s

correlation coefficient and the regression coefficient 𝒃?

 A. The sign of the correlation coefficient and the sign of the slope of the regression line will always be the same.

Application Summary

Measure Excel only Excel + MegaStat

Scatter plot ✓

Person’s linear correlation

coefficient ✓

Spearman’s correlation

coefficient ✓

Regressions equation ✓