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Correlation and regression Dr. Ghada Abo-Zaid

Correlation and regression Dr. Ghada Abo-Zaid. Correlation and regression. Outline. Once you have finished studying this chapter, you will be able to: Draw a scatter diagram, and explain the relationship between two variables from the plot. Understand the definition of covariance.

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Correlation and regression Dr. Ghada Abo-Zaid

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  1. Correlation and regressionDr. GhadaAbo-Zaid Correlation and regression

  2. Outline Once you have finished studying this chapter, you will be able to: • Draw a scatter diagram, and explain the relationship between two variables from the plot. • Understand the definition of covariance. • Calculate the covariance, and interpret the results. • Calculate the coefficient of correlation and interpret the results. • Clarify the difference between the covariance and correlation.

  3. Outline • Identify the assumptions and limitations of correlation coefficient. • Test the hypothesis of coefficient of correlation. • Understand the definition of Spearman's rank correlation coefficient. • Calculate Spearman's rank correlation coefficient and interpret the results. • Identify the difference between the correlation coefficient and Spearman's rank correlation coefficient. • Test the hypothesis of Spearman's rank correlation coefficient.

  4. Scatter Plot • Exploring the dataset before starting any statistical analysis is considered currently as one of the most important steps in the statistical analysis, especially in social science research. • A scatter plot or scatter diagram might be used for examining initially whether there is an association between two variables, and shows the direction of this association.

  5. Possible scatter plots association between X and Y variables.

  6. Negative association

  7. No association and non-linear association

  8. Covariance • Basically, covariance is used for detecting the direction of an association between two random variables. • If the two variables are moved at the same direction, it is named as a positive covariance. • If the two variables are moved at the reverse directions, it is named as a negative covariance.

  9. Covariance In other words, A covariance is a positive or a negative single number that help in detecting the association between two variables by its sign. For example if the single number is minus, this refers to an indirect association between two variables and vice versa. Covariance is denoted as Cov (X,Y).

  10. Sampling Covariance The sample covariance between X and Y is defined by two formulas: • Second: Short calculation formula where and are the sampling means for X and Y respectively.

  11. Example • In the stock market, the interest of the analyst is to select stocks that: a) reduce the risk taken for the same amount of return. b) select the stocks that are working well together. • Table 4.1 shows the daily returns for two stocks using the closing prices, say NSGB bank, , and Sidikrier petroleum, , in 2014, for a sequence of 10 days.

  12. Table 4.1: Gives the daily returns for two stocks using the closing prices. Calculate the covariance between X and Yby using a) Long calculation formula b) Short calculation formula

  13. Short calculation formula

  14. Short calculation formula

  15. Short calculation formula • Interpretation: The result indicates to a positive relationship between the two variables (return of the two stocks) X and Y.

  16. Correlation • Though a covariance measure gives the direct association between two variables, it is still not capable of measuring the size or strength of an association. • A correlation is a statistical measure that determines the strength of an association between two variables and detect their direction. • It is also named as Pearson's correlation coefficient in honour of Karl Person (1857 -1936)

  17. Short calculation formula

  18. Note that the coefficient of correlation lies between +1 and -1. • if r = +1 this indicates a perfect positivecorrelation between X and Y. • if r = -1 , this indicates a perfect negative association between X and Y. • If r = 0 , this is an indicator of no correlation between X and Y.

  19. Assumptions of Person's Correlation • The variables X and Y must be continuous random variables. • The data for X and Y variables must tend to a normal distribution ( bell shape).

  20. Example A sample of 8 students was selected randomly to examine the association between the number of hours a student spent studying for an exam (X) and the score that a student obtained on that exam (Y). The data are given below

  21. Example Find the linear correlation coefficient between the number of hours a student spent in studying and the score a student obtained in the exam, and interpret the result.

  22. Solution by using short calculation formula

  23. Solution by using short calculation formula

  24. Solution by using short calculation formula

  25. Solution by using short calculation formula • Interpretation: This indicates there is a very strong positive association between the number of hours a student spent in studying and the number of the score on the exam. • Interpretation: This means that the more hours a student spent in studying, the better score he or she will obtain.

  26. Hypothesis Test for a Linear Correlation Coefficient • Hypothesis test for a linear correlation coefficient is basically used to detect whether the sample correlation coefficient r is the estimator of population correlation coefficient r (rho) or not by using the Student t distribution. • The student t statistic formula is given below: which is distributed as with degree of freedom

  27. listed the steps of the hypothesis t- test for a linear correlation coefficient

  28. the steps of the hypothesis t- test for a linear correlation coefficient

  29. Example • A sample of 7 observations was taken randomly to examine the association between the income per thousand pounds, X, and the number of breads con consumed for person per day, Y

  30. Example • Find the linear correlation coefficient between X and Y and interpret the result • Test the significant of the linear correlation coefficient at significant level, , equals 5%

  31. Solution • Find the linear correlation coefficient between X and Y and interpret the result

  32. Solution

  33. Interpretation: This indicates there is a very strong negative association between the income and the number of bread consumed for person. This means that the more income a person earns the less money spent on bread.

  34. Test the significant of the linear correlation coefficient at significant level, , equals 5% • Step 1: Let • Step 2 :

  35. Test the significant of the linear correlation coefficient at significant level, , equals 5%

  36. Test the significant of the linear correlation coefficient at significant level, , equals 5% We conclude that there is a sufficient evidence to support that there is a linear correlation coefficient between the two variables.

  37. Rank Correlation • Coefficient of correlation is used to measure the association between two variables, but this is under certain conditions. • One of these conditions is that X and Y random variables should be continuous. • In addition, the data of X and Y variables are underlying the normal distribution. • What happen if one of those conditions is not achieved?

  38. Rank Correlation • this basically lead to think of another measure of correlation called rank correlation coefficient • It is also named as Spearman's rank correlation coefficient. • Spearman's rank correlation coefficient, is a non-parametric statistics measure that is equivalent to Pearson's correlation coefficient, r.

  39. It is also undertaken to measure the association between two variables, even if these variables do not underlying normal distribution or they are not continuous variables. • Spearman's Rank correlation coefficient is undertaken if the data are in orders or can be ranked in orders.

  40. Rank Correlation • The formula of Spearman's Rank correlation coefficient, is given as:

  41. Steps for calculating Spearman's Rank Correlation Coefficient

  42. Steps for calculating Spearman's Rank Correlation Coefficient

  43. Example • The following table gives the grades of 8 students in linear algebra course , X, and probability course, Y • where : E , V.G, G, and P are excellent, very good, good, and pass respectively. Find the correlation between X and Y.

  44. Solution

  45. Solution Interpretation: This indicates that there is a moderate positive association between the evaluation grades of linear algebra and probability course.

  46. Hypothesis Significant test of Spearman's Rank Correlation Coefficient • Hypothesis test is also undertaken for a Spearman's Rank Correlation Coefficientto detect whether the sample rank correlation coefficient is an estimator of population correlation coefficient r (rho) or not by using the Student t distribution.

  47. the steps of the hypothesis t- test for Spearman's Rank Correlation Coefficient

  48. the steps of the hypothesis t- test for Spearman's Rank Correlation Coefficient

  49. Example • In the previous Example test the significance of Spearman's Rank Correlation Coefficient

  50. Solution

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