Business Research Methods, 9th ed.Chapter 23

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William G. Zikmund, Barry J. Babin, Jon C. Carr, Mitch Griffin

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Business Research Methods, 9th ed.Chapter 23

  1. 1. Business Research Methods William G. Zikmund Chapter 23 Bivariate Analysis: Measures of Associations
  2. 2. Measures of Association • A general term that refers to a number of bivariate statistical techniques used to measure the strength of a relationship between two variables.
  3. 3. Relationships Among Variables • Correlation analysis • Bivariate regression analysis
  4. 4. Type of Measurement Measure of Association Interval and Ratio Scales Correlation Coefficient Bivariate Regression
  5. 5. Type of Measurement Measure of Association Ordinal Scales Chi-square Rank Correlation
  6. 6. Type of Measurement Measure of Association Nominal Chi-Square Phi Coefficient Contingency Coefficient
  7. 7. Correlation Coefficient • A statistical measure of the covariation or association between two variables. • Are dollar sales associated with advertising dollar expenditures?
  8. 8. The Correlation coefficient for two variables, X and Y is xyr .
  9. 9. Correlation Coefficient • r • r ranges from +1 to -1 • r = +1 a perfect positive linear relationship • r = -1 a perfect negative linear relationship • r = 0 indicates no correlation
  10. 10. ( )( ) ( ) ( )∑ ∑ ∑ −− −− == 22 YYiXXi YYXX rr ii yxxy Simple Correlation Coefficient
  11. 11. 22 yx xy yxxy rr σσ σ == Simple Correlation Coefficient
  12. 12. = Variance of X = Variance of Y = Covariance of X and Y 2 xσ 2 yσ xyσ Simple Correlation Coefficient Alternative Method
  13. 13. X Y NO CORRELATION . Correlation Patterns
  14. 14. X Y PERFECT NEGATIVE CORRELATION - r= -1.0 . Correlation Patterns
  15. 15. X Y A HIGH POSITIVE CORRELATION r = +.98 . Correlation Patterns
  16. 16. Pg 629 ( )( )589.5837.17 3389.6− =r 712.99 3389.6− = 635.−= Calculation of r
  17. 17. Coefficient of Determination Variance variance2 Total Explained r =
  18. 18. Correlation Does Not Mean Causation • High correlation • Rooster’s crow and the rising of the sun – Rooster does not cause the sun to rise. • Teachers’ salaries and the consumption of liquor – Covary because they are both influenced by a third variable
  19. 19. Correlation Matrix • The standard form for reporting correlational results.
  20. 20. Correlation Matrix Var1 Var2 Var3 Var1 1.0 0.45 0.31 Var2 0.45 1.0 0.10 Var3 0.31 0.10 1.0
  21. 21. Walkup’s First Laws of Statistics • Law No. 1 – Everything correlates with everything, especially when the same individual defines the variables to be correlated. • Law No. 2 – It won’t help very much to find a good correlation between the variable you are interested in and some other variable that you don’t understand any better.
  22. 22. • Law No. 3 – Unless you can think of a logical reason why two variables should be connected as cause and effect, it doesn’t help much to find a correlation between them. In Columbus, Ohio, the mean monthly rainfall correlates very nicely with the number of letters in the names of the months! Walkup’s First Laws of Statistics
  23. 23. Going back to previous conditionsGoing back to previous conditions Tall men’s sonsTall men’s sons DICTIONARYDICTIONARY DEFINITIONDEFINITION GOING ORGOING OR MOVINGMOVING BACKWARDBACKWARD Regression
  24. 24. Bivariate Regression • A measure of linear association that investigates a straight line relationship • Useful in forecasting
  25. 25. Bivariate Linear Regression • A measure of linear association that investigates a straight-line relationship • Y = a + bX • where • Y is the dependent variable • X is the independent variable • a and b are two constants to be estimated
  26. 26. Y intercept • a • An intercepted segment of a line • The point at which a regression line intercepts the Y-axis
  27. 27. Slope • b • The inclination of a regression line as compared to a base line • Rise over run • D - notation for “a change in”
  28. 28. Y 160 150 140 130 120 110 100 90 80 70 80 90 100 110 120 130 140 150 160 170 180 190 X My line Your line . Scatter Diagram and Eyeball Forecast
  29. 29. 130 120 110 100 90 80 80 90 100 110 120 130 140 150 160 170 180 190 X Y . XaY βˆˆˆ += X∆ Yˆ∆ Regression Line and Slope
  30. 30. X Y 160 150 140 130 120 110 100 90 80 70 80 90 100 110 120 130 140 150 160 170 180 190 Y “hat” for Dealer 3 Actual Y for Dealer 7 Y “hat” for Dealer 7 Actual Y for Dealer 3 Least-Squares Regression Line
  31. 31. 130 120 110 100 90 80 80 90 100 110 120 130 140 150 160 170 180 190 X Y } } { Deviation not explained Total deviation Deviation explained by the regression Y . Scatter Diagram of Explained and Unexplained Variation
  32. 32. The Least-Square Method • Uses the criterion of attempting to make the least amount of total error in prediction of Y from X. More technically, the procedure used in the least-squares method generates a straight line that minimizes the sum of squared deviations of the actual values from this predicted regression line.
  33. 33. The Least-Square Method • A relatively simple mathematical technique that ensures that the straight line will most closely represent the relationship between X and Y.
  34. 34. Regression - Least-Square Method ∑= n i ie 1 2 minimumis
  35. 35. = - (The “residual”) = actual value of the dependent variable = estimated value of the dependent variable (Y hat) n = number of observations i = number of the observation ie iY iYˆ iY iYˆ
  36. 36. The Logic behind the Least- Squares Technique • No straight line can completely represent every dot in the scatter diagram • There will be a discrepancy between most of the actual scores (each dot) and the predicted score • Uses the criterion of attempting to make the least amount of total error in prediction of Y from X
  37. 37. XYa βˆˆ −= Bivariate Regression
  38. 38. ( ) ( )( ) ( ) ( )22 ˆ ∑∑ ∑∑∑ − − = XXn YXXYn β Bivariate Regression
  39. 39. = estimated slope of the line (the “regression coefficient”) = estimated intercept of the y axis = dependent variable = mean of the dependent variable = independent variable = mean of the independent variable = number of observations βˆ X Y n aˆ Y X
  40. 40. ( ) ( ) 625,515,3759,24515 875,806,2345,19315ˆ − − =β 625,515,3385,686,3 875,806,2175,900,2 − − = 760,170 300,93 = 54638.=
  41. 41. ( )12554638.8.99ˆ −=a 3.688.99 −= 5.31=
  42. 42. ( )12554638.8.99ˆ −=a 3.688.99 −= 5.31=
  43. 43. ( )XY 546.5.31ˆ += ( )89546.5.31 += 6.485.31 += 1.80=
  44. 44. ( )XY 546.5.31ˆ += ( )89546.5.31 += 6.485.31 += 1.80=
  45. 45. ( )165546.5.31ˆ 129)valueY(Actual7Dealer 7 += = Y 6.121= ( )95546.5.31ˆ )80valueY(Actual3Dealer 3 += = Y 4.83=
  46. 46. 99 ˆYYei −= 5.9697 −= 5.0=
  47. 47. ( )165546.5.31ˆ 129)valueY(Actual7Dealer 7 += = Y 6.121= ( )95546.5.31ˆ )80valueY(Actual3Dealer 3 += = Y 4.83=
  48. 48. 99 ˆYYei −= 5.9697 −= 5.0=
  49. 49. ( )119546.5.31ˆ 9 +=Y
  50. 50. F-Test (Regression) • A procedure to determine whether there is more variability explained by the regression or unexplained by the regression. • Analysis of variance summary table
  51. 51. Total Deviation can be Partitioned into Two Parts • Total deviation equals • Deviation explained by the regression plus • Deviation unexplained by the regression
  52. 52. “We are always acting on what has just finished happening. It happened at least 1/30th of a second ago.We think we’re in the present, but we aren’t. The present we know is only a movie of the past.” Tom Wolfe in The Electric Kool-Aid Acid Test .
  53. 53. ( ) ( ) ( )iiii YYYYYY ˆˆ −+−=− Partitioning the Variance Total deviation = Deviation explained by the regression Deviation unexplained by the regression (Residual error) +
  54. 54. = Mean of the total group = Value predicted with regression equation = Actual value Y Yˆ iY
  55. 55. ( ) ( ) ( )∑∑∑ −+−=− 222 ˆˆ iiii YYYYYY Total variation explained = Explained variation Unexplained variation (residual) +
  56. 56. SSeSSrSSt += Sum of Squares
  57. 57. Coefficient of Determination r2 • The proportion of variance in Y that is explained by X (or vice versa) • A measure obtained by squaring the correlation coefficient; that proportion of the total variance of a variable that is accounted for by knowing the value of another variable
  58. 58. Coefficient of Determination r2 SSt SSe SSt SSr r −== 12
  59. 59. Source of Variation • Explained by Regression • Degrees of Freedom – k-1 where k= number of estimated constants (variables) • Sum of Squares – SSr • Mean Squared – SSr/k-1
  60. 60. Source of Variation • Unexplained by Regression • Degrees of Freedom – n-k where n=number of observations • Sum of Squares – SSe • Mean Squared – SSe/n-k
  61. 61. r2 in the Example 875. 4.882,3 49.398,32 ==r
  62. 62. Multiple Regression • Extension of Bivariate Regression • Multidimensional when three or more variables are involved • Simultaneously investigates the effect of two or more variables on a single dependent variable • Discussed in Chapter 24
  63. 63. Correlation Coefficient, r = .75 Correlation: Player Salary and Ticket Price -20 -10 0 10 20 30 1995 1996 1997 1998 1999 2000 2001 Change in Ticket Price Change in Player Salary

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