CORRELATION ANALYSIS. Dr. Anulawathie Menike Dept. of Economics

Transcription

1 CORRELATION ANALYSIS Dr. Anulawathie Menike Dept. of Economics 1

2 What is Correlation The correlation is one of the most common and most useful statistics. It is a term used to describe the relationship of two variables. Where one variable has an effect on the other. The correlation analysis enable us to have an idea about the strength/degree and direction of the relationship between two variables. It refers to the techniques used in measuring the closeness of the relationship between the variables. 2

3 What is Correlation Cont.. Eg: 1. monthly earnings of households and monthly expenses on entertainment. when monthly earnings increase, there is a increase on entertainment expenses. (Positive correlation) 2. price of a commodity and the demand for it. when price increases, there is a decline in its demand. (Negative correlation) 3. Temperature and Ice-cream sales. When temperature increases, Ice-cream sales go up. (Positive correlation) - Such relations are studied in correlation - 3

4 Examples of Correlation Fertilizer used and yield of rice per acre. Number of police round and number of crime. Monthly income and consumption expenditure. Age of car and selling price. Height and weight of individuals. Advertising expenditure and sales of a product. Calories eaten per day and IQ. No. of hours spent studying and final grade. Price and supply of a commodity. Number of dependents and monthly expenditure. 4

5 Types of Correlation There are different types of correlation. Some of the most important types are: 1. Positive and Negative Correlation. 2. Simple, Partial and Multiple Correlation. 5

6 Positive Correlation If the two variables increase or decrease together, they have a positive (direct) correlation. (If the values of two variables changing with same direction) Eg: The height and weight of a growing child. Price and supply of a commodity. Study time and grades. Statistics score and GPA. 6

7 Negative Correlation If one variable is increasing and the other is decreasing, they have a negative (inverse/indirect) correlation. (If the values of two variables changing with opposite direction) Eg: Price and demand of a commodity. Alcohol consumption and driving ability. Women's education and fertility. Age of the car and selling price. 7

8 Simple, partial and Multiple Correlation The distinction between these three types of correlation depends upon the number of variables studied. Simple correlation When only two variables are studied, it is a problem of simple correlation. Eg: Relationship between fertilizer used and yield of rice per acre. 8

9 Partial Correlation In partial correlation we recognize more than two variables. But consider only two variables, i.e. it is a study of the relationship between one dependent variable and one independent variable by keeping the other independent variables constant. 9

10 Partial Correlation Cont. Eg: X 1 = Sales of a product X 2 = Advertising expenditure X 3 = No. of sales representatives When we study the relationship between X 1 and X 3, keeping an advertising expenditure (X 2 ) as constant. 10

11 Multiple Correlation In multiple correlation 3 or more variables are studied simultaneously (all together). Eg: Relationship between the yield of rice per acre and both the amount of rainfall and the amount of fertilizers used. 11

12 Methods to Study Correlation 1. Scatter Diagram Method/Dot Diagram Method/ Scatter plot Method 2. Karl Pearson s Coefficient of Correlation Method 3. Spearman's Rank Correlation Method 4. Two-Way Frequency Table Method 12

13 Scatter Diagram The relationship between any two variables can be portrayed graphically on an x- and y- axis. It is a graph of plotted points where each point represents the values of X and Y as a coordinate. This diagram may indicate both degree and the type (nature) of correlation. 13

14 Scatter Plots The pattern of data reveals the type of relationship between our two variables. 1. Positive relationship 2. Negative relationship 3. No relationship 14

15 Positive Relationship

16 Negative Relationship Y Final Grades X Absences 16

17 No Relationship

18 Advantages of Scatter Diagram Simple & Non Mathematical method. Not influenced by the size of extreme item. First step in investing the relationship between two variables. 18

19 Different Types of Correlation Correlations that follow a line It is called Linear correlations 19

20 Disadvantages of scatter diagram Can not adopt the exact degree of correlation 20

21 Simple correlation coefficient (r) (Karl Pearson s Correlation Coefficient) The measure of correlation called the correlation coefficient. It is also called Pearson s Correlation Coefficient or Product Moment Correlation Coefficient. It measures the direction and strength of linear (straight line) relationship between two variables of the quantitative type. The numerical value of r ranges from -1 to +1 (-1 r +1 ) 21

22 Interpretation of correlation coefficient (r) The numerical value of r ranges from -1 to +1. (-1 r +1 ) If the correlation coefficient higher, there is a strong relationship of two variables. In general, r > 0 indicates positive correlation r < 0 indicates negative correlation r = 0 indicates no relationship between two variables (Eg: weight and pulse rate, weight and hair color, shoe sizes and test scores, height and savings). 22

23 Interpretation of correlation coefficient (r) cont.. If r = +1, X and Y are perfectly positive linear relationship. If r = -1, X and Y are perfectly negative linear relationship. If r = 0, there is absolutely no association between the variables The closer to 1, the stronger the negative linear relationship The closer to +1, the stronger the positive linear relationship The closer to 0, the weaker any positive linear relationship 23

24 correlation coefficient (r) Pearson s correlation coefficient describes strength(degree) and direction(positive or negative) of linear relationship between two variables. 24

25 What is Strength/degree of the Relationship? Think of strength as to how closely the data points fall on and near a line drawn through the data. 25

26 What is Direction of the Relationship? (The sign + or - ) of r denotes the direction of association. If the sign is +ve, this means the relationship is positive or direct (Both variables rise or fall together in the same direction). If the sign is -ve, this means the relationship is negative (inverse) or indirect (A value of one variable increase, value of other variable decrease) 26

27 Linear Correlation (r) If X and Y indicate the two variables under consideration, a scatter diagram shows the location of points (X, Y) on a rectangular coordinate system. If all points in this scatter diagram seem to lie near a line, as in fig. 1 and 2, the correlation is called linear. Fig. 1 Fig. 2 Positive Linear Correlation Negative Linear Correlation No Correlation 27

28 Advantages of Pearson s Correlation Coefficient It summarizes in one value, the degree of correlation & direction of correlation also. 28

29 Limitations of Pearson s Correlation Coefficient Always assume linear relationship. Interpreting the value of r is difficult. Value of Correlation Coefficient is affected by the extreme values. Time consuming methods. 29

30 Y Final Grades Computation of correlation coefficient Scatter Diagram X Absences Absences Final Grade x y

31 Computation of correlation coefficient x y Method 1 xy x 2 y There is a strong negative correlation. 31

32 Exercises Calculate the correlation coefficient between the amount of radio time (X in minutes) and number of electrical appliances sold (Y). X Y Hint : X= 160, X 2 = 4480, Y= 116, Y 2 = 2462, XY= 3275 r = = 0.84 This shows a high degree of correlation (strong positive). 32

33 Computation of correlation coefficient Method 2 (Mean deviation Method) Calculate the correlation coefficient between the price(x) and supply(y) from the given data. X Y (X - X) (Y - Y) X Y x y x2 y2 xy r = = = Two variables X and Y have almost perfect positive correlation. 33

34 Exercises Height of sisters (cm) Height of brothers(cm) x x 0, y 0, x 28, y 34, xy 25 r = Σxy Σx 2 Σy 2 34

35 Computation of correlation coefficient Method 3 (Assume mean method) A company manufactures different types of electrical appliances. It has been using radio for advertising its products. The following data shows amounts of radio time (X, in minutes) and the number of electrical appliances sold (Y) over the last six weeks. Calculate the coefficient of correlation between the two series. X Y

36 36

37 Coefficient of Determination (r 2 ) This indicates the percentage of the contribution that the factor makes towards the relationship between X and Y. In the first example, r = , then the coefficient of determination r 2 = This means that about 95% of the relationship is due to the absences. 37

38 Spearman's Rank Correlation Method It is a non-parametric measure of correlation. It is applied to the problems which data cannot be measured quantitatively. Eg: honesty, beauty, character, intelligence, morality etc. This method is based on the ranks (order) of the observations. 38

39 Spearman's Rank Correlation Cont. Spearman Rank correlation coefficient could be computed in the following cases: Both variables are quantitative. Both variables are qualitative ordinal. One variable is quantitative and the other is qualitative ordinal. 39

40 Computation of Rank Correlation Coefficient 1. When actual ranks are given In this situation the following steps are involved. I. Compute d (the difference of ranks). II. Compute d 2. III. Obtain the sum d 2. IV. Use the formula r s = 1-6Σd 2 n n

41 Example Suppose that 10 salesman employed by a company were given a month s training, they took a test and were ranked on the basis of their performance. They were then posted to their respective areas. At the end of six months, they were rated in respect of their sales performance. These ranks are shown below. calculate the coefficient of Rank Correlation. 41

42 Example Cont.. Salesman Ranks obtained in training Ranks based on sales performance d (R 1 - R 2 ) d d 2 = 24 r s = 1-6Σd 2 n n 2 1 = =

43 Computation of Rank Correlation Coefficient 2. When the ranks are not given If the ranks are not given, then we need to assign ranks to the data series. The lowest value in the series can be assigned rank 1 or the highest value in the series can be assigned rank 1. We need to follow the same scheme of ranking for the other series. Then calculate the rank correlation coefficient in similar way as we do when the ranks are given. 43

44 Procedure Rank the values of X from 1 to n. (n is the no. of pairs of values of X and Y in the sample). Rank the values of Y from 1 to n. Compute the value of d i (Subtracting the rank of Y from the rank of X). Square each d i squared values). Apply the formula. r s = 1 - and compute d 2 (Sum of the 6Σd 2 n(n 2 1) 44

45 Example For the following data, calculate the coefficient of Rank Correlation. X: Y:

46 Computation of Rank Correlation Coefficient X Rank X Y Rank Y d D r s = 1-6Σd 2 n n 2 1 = D 2 = 4 = =

47 Computation of Rank Correlation Coefficient 3. When equal ranks/repeated ranks are given In such cases average ranks should be assigned to each individual. 47

48 Example A psychologist wanted to compare two methods A and B of teaching. He selected a random sample of 22 students. He grouped them into 11 pairs so that the students in a pair have approximately equal scores on an intelligence test. In each pair one student was taught by method A and the other by method B and examined after the course. The marks obtained by them are tabulated below. Pair : A : B : Find the rank correlation coefficient. Let X denotes Method A and Y denotes Method B. 48

49 Computation X Y Rank X Rank Y d = X - Y d d 2 =

50 Computation r = 1-6 Σd m3 1 m m3 2 m m3 3 m 3 + n n 2 1 r = = = =

51 Correlation Coefficient for Grouped Data (Two Way Frequency Table) A bivariate distribution refers to data concerning two variables X and Y. If in a bivariate distribution data, the number of pairs of observation is large, then it becomes very difficult to calculate the coefficient of correlation (R). Therefore, to facilitate the calculation of R in such a large pairs of observations, the data are summarized in the form of a Two way Frequency Table. It is also called Bivariate Frequency Table or Contingency Table or Correlation Table, or simply Two way Table. 51

52 Example Below table shows the frequency distribution of the final grades of 100 students in mathematics and Economics. Referring to this table, find: 1. The coefficient of correlation (r). 2. The number of students who received grades of in Mathematics and in Economics. 3 The percentage of students with mathematics grades below The number of students who received a grade of 70 or more in Economics and of less than 80 in Mathematics. 5. The percentage of students who passed at least one of the subjects; assume that the minimum passing grade is

53 Economics Grades (Y) Correlation Table Mathematics Grades (X) Total Total

54 54

55 Computation: fd x d y = Sum of corner numbers in each column/row. Eg: fd x d y = = 32 (column) fd x d y = = 33 (row) f x = = 7 f y = = 12 f x d x = -2 x 7 = -14 f y d y = -3 x 12 = -36 f x d x2 = -2 2 x 7 = 28 f y d y2 = -3 2 x 12 =

56 Computation of Correlation Coefficient r = r = n Σfd x dy Σf x d x Σf y d y 2 n Σf x d x Σfx d x n Σf y d y Σfy d y (64)( 55) ( 55) 2 r = r =

57 Cont.. r = r = r = r =

58 1. R = Answers = 47, the percentage is 47% = The number of students with grades below 60 in both subjects are = 17. Thus the number of students with grades 60 or over in either Economics or Mathematics or in both is = 83, the percentage is 83% 58

59 Exercises Calculate the coefficient of correlation between the ages of husbands and wives from the following table. Age of Husbands Age of Wives Total T0tal

60 Age of Husbands X m dx Y m dy Age of Wives fx N = 53 fxdx= fxdx 10 fxdx 2 fxdx 2 = 98 fdxdy= fdxdy 86 fy fydy fydy 2 fdxdy fydy =16 fydy 2 = 92 fdxdy = 86 60