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Regression analysis

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regression analysis in statistics

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Regression analysis

  1. 1. B Y A N W E S H B I S W A S ( 1 7 M B 4 0 0 9 ) A M A A N A L I ( 1 7 M B 4 0 2 2 ) REGRESSION ANALYSIS AND ITS APPLICATION IN BUSINESS
  2. 2. Regression Analysis. . .  It is the study of the relationship between variables.  It is one of the most commonly used tools for business analysis.  It is easy to use and applies to many situations.
  3. 3. TYPES OF REGRSSION…  Simple Regression: single explanatory variable  Multiple Regression: includes any number of explanatory variables.
  4. 4.  Dependent variable: the single variable being explained/ predicted by the regression model  Independent variable: The explanatory variable(s) used to predict the dependent variable.  Coefficients (β): values, computed by the regression tool, reflecting explanatory to dependent variable relationships.  Residuals (ε): the portion of the dependent variable that isn’t explained by the model; the model under and over predictions.
  5. 5. TYPES OF REGRESSION ANALYSIS…  Linear Regression: straight-line relationship Form:y=mx+b  Non-linear: implies curved relationships logarithmic relationships  Cross Sectional: data gathered from the same time period  Time Series: Involves data observed over equally spaced points in time.
  6. 6. Simple Linear Regression Model. . .  Only one independent variable, x  Relationship between x andy is described by a linear function  Changes in y are assumed to be caused by changes in x
  7. 7. ASSUMPTIONS  Linear relationship  Multivariate normality  No or little multicollinearity  No auto-correlation
  8. 8. TYPES
  9. 9. Estimated Regression Model. . . The sample regression line provides an estimate of the population regression line
  10. 10. EXAMPLE (USING EXCEL)  On a Friday, 22 students in a class were asked to record the numbers of hours they spent studying for a test on Monday and the numbers of hours they spent watching television. The results are shown below.  Book2.xlsx MARKS HOURS 40 1 44 1 51 2 58 3 49 3 48 4 64 4 55 5 69 5 58 5 75 5 68 6 63 6 93 6 84 7 67 7 90 8 76 8 95 9 72 9 85 9 98 10
  11. 11. GRAPHICAL REPRESENTATION y = 5.639x + 36.745 0 20 40 60 80 100 120 0 2 4 6 8 10 12 MARKSOBTAINED HOURS STUDIED MARKS MARKS Linear (MARKS)
  12. 12. ACTUAL ANALYSIS SUMMARY OUTPUT Regression Statistics Multiple R 0.86107 R Square 0.741442 Adjusted R Square0.728514 Standard Error8.976161 Observations 22 ANOVA df SS MS F Significance F Regression 1 4620.934 4620.934 57.35199 2.69E-07 Residual 20 1611.429 80.57147 Total 21 6232.364 CoefficientsStandard Error t Stat P-value Lower 95%Upper 95%Lower 95.0%Upper 95.0% Intercept 36.74539 4.58186 8.019753 1.12E-07 27.18779 46.30298 27.18779 46.30298 HOURS 5.639037 0.744613 7.57311 2.69E-07 4.085801 7.192272 4.085801 7.192272
  13. 13. Interpretation of the Intercept,b0  Marks Obtained   5.639 (hours studied)36.745 b0 is the estimated average value of Y when the value of X is zero (if x = 0 is in the range of observed x values) 36.745 just indicates that, for marks within the range of sizes observed, 36.745 is the portion of the marks not explained by hours studied.
  14. 14. Interpretation of the Slope Coefficient, b1  Marks Obtained 36.745  (hours studied)5.639 b1 measures the estimated change in the average value of Y as a result of a one- unit change in X – Here, b1 = 5.639 tells us that the average value of marks increases by 5.639 , on average, for each additional one hour studied.
  15. 15. Coefficient of Determination, R2 Note: In the single independent variable case, the coefficient of determination is R2  r2 where: R2 = Coefficient of determination r = Simple correlation coefficient
  16. 16. Examples of Approximate R2 Values R2 = +1 y x y x R2 = -1 R2 = +-1 Perfect linear relationship between x and y: 100% of the variation in y is explained by variation in x
  17. 17. Examples of Approximate R2 Values R2 = 0 No linear relationship between x and y: The value of Y does not depend on x. (None of the variation in y is explained by variation in x) y x R2 = 0
  18. 18. OUTPUT R2  SSR  4620.9341  0.7414 SST 6232.3636 R Square 0.741441681 Adjusted R Square 0.728513765 Standard Error 8.976161388 Observations 22 ANOVA df SS Regression 1 4620.934171 Residual 20 1611.429465 Total 21 6232.363636 THIS MEANS THAT 74.14% OF VARIATION IN MARKS CAN BE EXPLAINED BY VARIATION IN STUDY HOURS
  19. 19. Standard Error of Estimate. . .  The standard deviation of the variation of observations around the regression line is estimated by  n  k 1 ESS s  Where ESS = ERROR Sum of squares n = Sample size k = number of independent variables in the model
  20. 20. OUTPUT R Square 0.741441681 Adjusted R Square 0.728513765 Standard Error 8.976161388 Observations 22 ANOVA df SS Regression 1 4620.934171 Residual 20 1611.429465 Total 21 6232.363636 sε  8.9761
  21. 21. THANK YOU

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