Correlation and Regression

Transcription

1 Correlation and Regression 1

2 Overview Introduction Scatter Plots Correlation Regression Coefficient of Determination 2

3 Objectives of the topic 1. Draw a scatter plot for a set of ordered pairs. 2. Compute the correlation coefficient. 3. Test the hypothesis H 0 : ρ = Compute the equation of the regression line. 5. Compute the coefficient of determination. 6. Have a working idea of the concept of multiple regression ( it is not examinable) 3

4 Introduction Correlation is a statistical method used to determine whether a linear relationship between variables exists. In this course reference will be made only to pair wise correlation. i.e the correlation between only two variables Regression is a statistical method used to describe and estimate the nature of the relationship between variables that is, positive or negative, linear or nonlinear. 4

5 introduction If there exists a linear relationship, then regression will be used to estimate the equation for the linear relationship Correlation and regression therefore work hand in hand. They are complements not substitutes. 5

6 Introduction to correlation and regression The purpose of this topic is to answer these questions statistically: 1. Are two variables related?- is there a particular observable or logical relationship between two variables? If one variable increases, what happens to the other variable? Does it increase or decrease? Consider sales and revenue; motivation and worker performance; exchange rate and imports; smoking and lung cancer; weight and blood pressure; interest rate and investment; study time and score. 6

7 Introduction to correlation and regression 2. If there is a relationship, what type of relationship exists? Is it positive or negative? 3. What is the strength of the linear relationship? Is it a weak relationship or a strong relationship? 4. What kind of predictions can be made from the relationship? 7

8 Introduction To answer question 1,2 and 3,the correlation coefficient, a numerical measure to determine whether two variables are related and to determine the strength of the relationship between the variables is used. To answer question 3, regression will be used. 8

9 Scatter Plots and Correlation The first step in correlation is to have a conceptual or logical understanding of the relation between the two variables. What do you think is the relationship between exchange rate and imports. The next step is to construct a scatter plot for the data to confirm the conceptual relationship A scatter plot is a graph of the ordered pairs (x, y) of numbers consisting of the independent variable x and the dependent variable y. 9

10 Scatter plots and correlation Correlation is symmetric- i.e it does not matter which of the two variables is dependent and which is independent 10

11 Example1: Car Rental Companies Construct a scatter plot for the data shown for car rental companies in the United States for a recent year. Step 1: Draw and label the x and y axes and denote one of the variables by X and the other by Y. Step 2: Plot each point on the graph. 11

12 Example 1 What do you think is the conceptual or logical relationship between the number of cars given out to hire and the sales revenue made. Positive? Negative? No relationship? As more cars are given out to hire does the sales revenue of the firm decrease or increase? The scatter plot is shown in the following diagram 12

13 Example : Car Rental Companies Positive Relationship cars (in ten thousands) 13

14 Example 2: Absences/Final Grades Construct a scatter plot for the data obtained in a study on the number of absences and the final grades of seven randomly selected students from a statistics class. Step 1: Draw and label the x and y axes and denote one of the variables by X and the other by Y Step 2: Plot each point on the graph. 14

15 Example -2: Absences/Final Grades Negative Relationship number of absentees 15

16 Correlation-interpretation After the scatter plot, the next step is to numerically calculate the correlation coefficient between the variables and interpret the coefficient The correlation coefficient computed from the sample data measures the strength and direction of a linear relationship between two variables. There are several types of correlation coefficients. Two will be explained in this course; The two explained in this course are the Pearson product moment correlation coefficient (PPMC) and the Spearman s Rank correlation coefficient 16

17 Correlation-interpretation The symbol for the sample correlation coefficient is r. The symbol for the population correlation coefficient is. 17

18 Correlation-interpretation 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. 18

19 Correlation-interpretation If r=1, then there exists a perfect positive relationship between the variables If r=-1, then there exists a perfect negative relationship between the variables If r=0, then there exists no relationship between the variables If 0<r<0.5, then there exists a weak positive relationship between the variables If 0.5<r<1, then there exists a strong positive relationship between the variables 19

20 Correlation-interpretation If -0.5<r<0, then there exists a weak negative relationship between the variables If -0.5<r<-1, then there exists a strong negative relationship between the variables 20

21 Correlation-interpretation 21

22 Pearson s Product Moment Correlation Coefficient (PPMCC) 22

23 Correlation Coefficient (PPMCC) The formula for the correlation coefficient (PPMCC) is given by r n xy x y n x x n y y where n is the number of data pairs. 23

24 Correlation coefficient (PPMCC) It can also be expressed in notation form as: r S S xx xy S Where S n xy x y ; xy yy 2 2 Syy n y y S 2 2 xx n x x 24

25 PPMCC- alternative/easier formula Alternatively, the formula can be written as: r XY n XY X nx Y ny Y Where and X are mean of Y and X respectively 25

26 Example 1: Car Rental Companies Compute the correlation coefficient for the data in Example 1. Company A B C D E F Cars x (in 10,000s) Σx = Income y (in billions) xy x 2 y Σy = Σxy = Σx 2 = Σy 2 =

27 Example 1: Car Rental Companies Compute the correlation coefficient for the data in Example 1. Σx = 153.8, Σy = 18.7, Σxy = , Σx 2 = , Σy 2 = 80.67, n = 6 r r r n xy x y n x x n y y (strong positive relationship) 27

28 Example 10-5: Absences/Final Grades Compute the correlation coefficient for the data in Example Student A B C D E F Number of absences, x Final Grade y (pct.) xy x 2 y ,724 7,396 1,849 5,476 3,364 8,100 G ,084 Σx = 57 Σy = 511 Σxy = 3745 Σx 2 = 579 Σy 2 = 38,993 28

29 Alternative calculation We can calculate for the mean of Y and X. the mean of X r and Y are given as and respectively ( )(3.1167) ( ( ) 2 ( (3.1167) 2 r Can you interpret the correlation coefficient? There is a strong positive correlation between car rentals and sales revenue 29

30 Example 2: Absences/Final Grades Compute the correlation coefficient for the data in Example 2. Σx = 57, Σy = 511, Σxy = 3745, Σx 2 = 579, Σy 2 = 38,993, n = 7 r r r n xy x y n x x n y y , (strong negative relationship) 30

31 Spearman s Rank correlation coefficient 31

32 Formula for spearman s rank correlation coefficient The formulas is given by: r s 1 6 nn d 2 2 ( 1) Where n is the number of observations. d is the difference in ranks between X and Y. The interpretation for the spearman s correlation is the same as for the PPMCC 32

33 STEP BY STEP APPROACH 1. Rank X and Y in an ascending or descending order 2. Calculate the difference between the ranks 3. Find the square of the difference and calculate 33

34 Example-Step by step approach Consider the data from example 1 COMPANY CARS(in 10,000s) Income (in billions) A 63 7 B C D E F

35 Example-Step by step approach Represent one of the variable by X and the other by Y. Remember that correlation is symmetric Order the data in ascending order Find the difference in the ranks Calculate the correlation coefficient 35

36 solution company Cars(X) Income (y) Rank of X Rank of Y A B C D E F d 4 d 2 36

37 solution The spearman s rank correlation coefficient is given as: 6(4) r s 2 6(6 1) Note: the answer will not necessarily be the same but the interpretation will be the same. In this case, there exists a strong positive correlation between car rentals and revenue 37

38 Special cases How do we rank the data when two or more data points have the same rank? When two scores tie in rank, both are given the mean of the two ranks they would occupy and the next rank is eliminated to keep n (the number of observations) consistent. For example, if two data points tied for 4 th place, both would receive a rank of 4.5 ((4 + 5) 2), and the next data point would be ranked number 6. if three points tied for 4 th place, the three would receive a rank of 5 (( ) 3), and the next school would be ranked number 7. 38

39 example Consider the following example company Cars(X) Income (y) Rank of X Rank of Y A B C D E F d d

40 solution The spearman s rank correlation coefficient is given as: 6(10.5) r s 2 6(6 1) 40