Regression & correlation

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  1. Course Title: Business Statistics BBA (Hons) 2nd Semester Course Instructor: Atiq ur Rehman Shah Lecturer, Federal Urdu University of Arts, Science & Technology,…
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  • 1. Course Title: Business Statistics BBA (Hons) 2nd Semester Course Instructor: Atiq ur Rehman Shah Lecturer, Federal Urdu University of Arts, Science & Technology, Islamabad +92-345-5271959 aatresh@gmail.com
  • 2. Correlation • Correlation is a LINEAR association between two random variables • Correlation is a statistical technique used to determine the degree to which two variables are related
  • 3. Scatter diagram • Rectangular coordinate • Two quantitative variables • One variable is called independent (X) and the second is called dependent (Y)
  • 4. Scatter diagram of weight and systolic blood pressure
  • 5. Scatter diagram of weight and systolic blood pressure
  • 6. Scatter plots The pattern of data is indicative of the type of relationship between your two variables: • positive relationship • negative relationship • no relationship
  • 7. Positive relationship
  • 8. Negative relationship Reliability Age of Car
  • 9. No relation
  • 10. Correlation Coefficient • The correlation coefficient (r) measures the strength and direction of relationship between two variables
  • 11. How to interpret the value of r? • r lies between -1 and 1. Values near 0 means no (linear) correlation and values near ± 1 means very strong correlation. • The negative sign means that the two variables are inversely related, that is, as one variable increases the other variable decreases.
  • 12. How to interpret the value of r?
  • 13. Pearson’s r • A 0.9 is a strong positive association (as one variable rises, so does the other) • A -0.9 is a strong negative association (as one variable rises, the other falls) r=correlation coefficient
  • 14. Coefficient of Determination Defined • Pearson’s r can be squared , r 2, to derive a coefficient of determination. • Coefficient of determination – the portion of variability in one of the variables that can be accounted for by variability in the second variable
  • 15. • Example of depression and CGPA – Pearson’s r shows negative correlation, r=-0.5 – r2=0.25 • In this example we can say that 1/4 or 0.25 of the variability in CGPA scores can be accounted for by depression (remaining 75% of variability is other factors, habits, ability, motivation, courses studied, etc)
  • 16. Coefficient of Determination and Pearson’s r • If r=0.5, then r2=0.25 • If r=0.7 then r2=0.49 • Thus while r=0.5 versus 0.7 might not look so different in terms of strength, r2 tells us that r=0.7 accounts for about twice the variability relative to r=0.5
  • 17. Example • Calculate the coefficient of correlation between the value X and Y given below: X 78 89 97 69 59 79 68 61 Y 125 137 156 112 107 136 123 108
  • 18. X Y X2 Y2 XY 78 125 6084 15625 9750 89 137 7921 18769 12193 97 156 9409 24336 15132 69 112 4761 12544 7728 59 107 3481 11449 6313 79 136 6241 18496 10744 68 123 4624 15129 8364 61 108 3721 11664 6588 Summation 600 1004 46242 128012 76812
  • 19. = 0.95 Hence the correlation co-efficient between X and Y is 0.95. ** (What does this value tells us??)**
  • 20. Regression • A statistical tool that is used to investigate the dependence of one variable (dependent variable) on one or more other variables (independent variables) • The dependent variable (Y) is the variable for which we want to make a prediction. • The independent variable (X) is the variable on the basis of which we are making predictions.
  • 21. • The linear relationship between two variables can either be positive or negative. • For instance, an increase in advertisement budget will bring more sales (positive), and increase in temperature will decrease the cooling efficiency of a room AC (negative)
  • 22. Simple Linear Regression • Positive Linear RelationshipPositive Linear Relationship yy xx Slope (b)Slope (b) is positiveis positive Regression lineRegression line InterceptIntercept (a)(a)
  • 23. Simple Linear Regression • Negative Linear RelationshipNegative Linear Relationship yy xx Slope (b)Slope (b) is negativeis negative Regression lineRegression line InterceptIntercept (a)(a)
  • 24. Simple Linear Regression • No RelationshipNo Relationship yy xx Slope (b)Slope (b) is 0is 0 Regression lineRegression line InterceptIntercept (a)(a)
  • 25. Simple Linear Regression Equation • Hence the equation for linear regression line can be written as: y= a + bx Where: y= dependent variable x= independent variable a= y-intercept (i.e value of y when x=0) b= slope
  • 26. Least-squares estimates • For a simple linear regression equation: y= a + bx We have, Where, and
  • 27. Example • Compute the least squares regression equation of Y on X for the following data. What is the regression coefficient and what does it mean?? X 5 6 8 10 12 13 15 16 17 Y 16 19 23 28 36 41 44 45 50
  • 28. X Y XY X2 5 16 80 25 6 19 114 36 8 23 184 64 10 28 280 100 12 36 432 144 13 41 533 169 15 44 660 225 16 45 720 256 17 50 850 289 Summation 102 302 3853 1308
  • 29. Now = 102/9 = 11.33 And = 302/9 = 33.56 = 9(3853) – (102) (302) 9( 1308) – (102)2 = 3873/1368 So b = 2.381
  • 30. And = 33.56 – (2.831) (11.33) = 1.47 Hence the desired estimated regression line of Y on X is y= 1.47 + 2.831x ** The estimated regression co-efficient is b=2.831, which means that yhe value of y increase by 2.831 units for a unit increase in x.
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