1. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-1 Chapter 12 Simple Linear Regression Statistics for Managers Using Microsoft ® Excel 4 th Edition

2. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-2 Chapter Goals After completing this chapter, you should be able to:  Explain the simple linear regression model  Obtain and interpret the simple linear regression equation for a set of data  Evaluate regression residuals for aptness of the fitted model  Understand the assumptions behind regression analysis  Explain measures of variation and determine whether the independent variable is significant

3. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-3 Chapter Goals After completing this chapter, you should be able to:  Calculate and interpret confidence intervals for the regression coefficients  Use the Durbin-Watson statistic to check for autocorrelation  Form confidence and prediction intervals around an estimated Y value for a given X  Recognize some potential problems if regression analysis is used incorrectly (continued)

4. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-4 Correlation vs. Regression  A scatter plot (or scatter diagram) can be used to show the relationship between two variables  Correlation analysis is used to measure strength of the association (linear relationship) between two variables  Correlation is only concerned with strength of the relationship  No causal effect is implied with correlation  Correlation was first presented in Chapter 3

5. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-5 Regression Analysis  Regression analysis is used to:  Predict the value of a dependent variable based on the value of at least one independent variable  Explain the impact of changes in an independent variable on the dependent variable Dependent variable: the variable we wish to explain Independent variable: the variable used to explain the dependent variable

6. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-6 Simple Linear Regression Model  Only one independent variable, X  Relationship between X and Y is described by a linear function  Changes in Y are assumed to be caused by changes in X

7. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-7 Types of Relationships Y X Y X Y Y X X Linear relationshipsCurvilinear relationships

8. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-8 Types of Relationships Y X Y X Y Y X X Strong relationshipsWeak relationships (continued)

9. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-9 Types of Relationships Y X Y X No relationship (continued)

10. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-10 Linear component Population Linear Regression Equation The population regression model: Population Y intercept Population Slope Coefficient Random Error term, or residual Dependent Variable Independent Variable Random Error component

11. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-11 The simple linear regression equation provides an estimate of the population regression line Sample Linear Regression Equation Estimate of the regression intercept Estimate of the regression slope Estimated (or predicted) Y value for observation i Value of X for observation i The individual random error terms e i have a mean of zero

12. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-12 Finding the Least Squares Equation  The coefficients b 0 and b 1, and other regression results in this chapter, will be found using Excel Formulas are shown in the text at the end of the chapter for those who are interested

13. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-13 Simple Linear Regression Example  A real estate agent wishes to examine the relationship between the selling price of a home and its size (measured in square feet)  A random sample of 10 houses is selected  Dependent variable (Y) = house price in $1000s  Independent variable (X) = square feet

14. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-14 Sample Data for House Price Model House Price in $1000s (Y) Square Feet (X) 2451400 3121600 2791700 3081875 1991100 2191550 4052350 3242450 3191425 2551700

15. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-15 Graphical Presentation  House price model: scatter plot

16. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-16 Regression Using Excel  Tools / Data Analysis / Regression

17. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-17 Excel Output Regression Statistics Multiple R0.76211 R Square0.58082 Adjusted R Square0.52842 Standard Error41.33032 Observations10 ANOVA dfSSMSFSignificance F Regression118934.9348 11.08480.01039 Residual813665.56521708.1957 Total932600.5000 CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept98.2483358.033481.692960.12892-35.57720232.07386 Square Feet0.109770.032973.329380.010390.033740.18580 The regression equation is:

18. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-18 Graphical Presentation  House price model: scatter plot and regression line Slope = 0.10977 Intercept = 98.248

19. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-19 Interpretation of the Intercept, b 0  b 0 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)  Here, no houses had 0 square feet, so b 0 = 98.24833 just indicates that, for houses within the range of sizes observed, $98,248.33 is the portion of the house price not explained by square feet

20. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-20  b 1 measures the estimated change in the average value of Y as a result of a one-unit change in X  Here, b 1 =.10977 tells us that the average value of a house increases by.10977($1000) = $109.77, on average, for each additional one square foot of size Interpretation of the Slope Coefficient, b 1

21. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-21 Predict the price for a house with 2000 square feet: The predicted price for a house with 2000 square feet is 317.85($1,000s) = $317,850 Predictions using Regression Analysis

22. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-22 Interpolation vs. Extrapolation  When using a regression model for prediction, only predict within the relevant range of data Relevant range for interpolation Do not try to extrapolate beyond the range of observed X’s

23. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-23  The coefficient of determination is the portion of the total variation in the dependent variable that is explained by variation in the independent variable  The coefficient of determination is also called r-squared and is denoted as r 2 Coefficient of Determination, r 2 note:

24. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-24 Excel Output Regression Statistics Multiple R0.76211 R Square0.58082 Adjusted R Square0.52842 Standard Error41.33032 Observations10 ANOVA dfSSMSFSignificance F Regression118934.9348 11.08480.01039 Residual813665.56521708.1957 Total932600.5000 CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept98.2483358.033481.692960.12892-35.57720232.07386 Square Feet0.109770.032973.329380.010390.033740.18580 58.08% of the variation in house prices is explained by variation in square feet

25. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-25 Standard Error of Estimate  The standard deviation of the variation of observations around the regression line is estimated by Where SSE = error sum of squares n = sample size

26. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-26 Excel Output Regression Statistics Multiple R0.76211 R Square0.58082 Adjusted R Square0.52842 Standard Error41.33032 Observations10 ANOVA dfSSMSFSignificance F Regression118934.9348 11.08480.01039 Residual813665.56521708.1957 Total932600.5000 CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept98.2483358.033481.692960.12892-35.57720232.07386 Square Feet0.109770.032973.329380.010390.033740.18580

27. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-27 Comparing Standard Errors YY X X S YX is a measure of the variation of observed Y values from the regression line The magnitude of S YX should always be judged relative to the size of the Y values in the sample data i.e., S YX = $41.33K is moderately small relative to house prices in the $200 - $300K range

28. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-28 Assumptions of Regression  Normality of Error  Error values (ε) are normally distributed for any given value of X  Homoscedasticity  The probability distribution of the errors has constant variance  Independence of Errors  Error values are statistically independent

29. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-29 Residual Analysis for Linearity Not Linear Linear x residuals x Y x Y x

30. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-30 Residual Analysis for Homoscedasticity Non-constant variance Constant variance xx Y x x Y residuals

31. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-31 Residual Analysis for Independence Not Independent Independent X X residuals X

32. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-32 Excel Residual Output RESIDUAL OUTPUT Predicted House PriceResiduals 1251.92316-6.923162 2273.8767138.12329 3284.85348-5.853484 4304.062843.937162 5218.99284-19.99284 6268.38832-49.38832 7356.2025148.79749 8367.17929-43.17929 9254.667464.33264 10284.85348-29.85348 Does not appear to violate any regression assumptions

33. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-33  Used when data is collected over time to detect if autocorrelation is present  Autocorrelation exists if residuals in one time period are related to residuals in another period Measuring Autocorrelation: The Durbin-Watson Statistic

34. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-34 Autocorrelation  Autocorrelation is correlation of the errors (residuals) over time  Violates the regression assumption that residuals are random and independent  Here, residuals show a cyclic pattern, not random

35. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-35 The Durbin-Watson Statistic  The possible range is 0 ≤ D ≤ 4  D should be close to 2 if H 0 is true  D less than 2 may signal positive autocorrelation, D greater than 2 may signal negative autocorrelation  The Durbin-Watson statistic is used to test for autocorrelation H 0 : residuals are not correlated H 1 : autocorrelation is present

36. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-36 Testing for Positive Autocorrelation  Calculate the Durbin-Watson test statistic = D (The Durbin-Watson Statistic can be found using PHStat) Decision rule: reject H 0 if D < d L H 0 : positive autocorrelation does not exist H 1 : positive autocorrelation is present 0dUdU 2dLdL Reject H 0 Do not reject H 0  Find the value d L from the Durbin-Watson table (for sample size n and number of independent variables k) Inconclusive

37. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-37  Example with n = 25: Durbin-Watson Calculations Sum of Squared Difference of Residuals3296.18 Sum of Squared Residuals3279.98 Durbin-Watson Statistic1.00494 Testing for Positive Autocorrelation (continued) Excel/PHStat output:

38. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-38  Here, n = 25 and there is k = 1 one independent variable  Using the Durbin-Watson table, d L = 1.29  D = 1.00494 < d L = 1.29, so reject H 0 and conclude that significant positive autocorrelation exists  Therefore the linear model is not the appropriate model to forecast sales Testing for Positive Autocorrelation (continued) Decision: reject H 0 since D = 1.00494 < d L 0 d U =1.45 2 d L =1.29 Reject H 0 Do not reject H 0 Inconclusive

39. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-39 Inference about the Slope: t Test  t test for a population slope  Is there a linear relationship between X and Y?  Null and alternative hypotheses H 0 : β 1 = 0(no linear relationship) H 1 : β 1  0(linear relationship does exist)  Test statistic where: b 1 = regression slope coefficient β 1 = hypothesized slope S b1 = standard error of the slope

40. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-40 House Price in $1000s (y) Square Feet (x) 2451400 3121600 2791700 3081875 1991100 2191550 4052350 3242450 3191425 2551700 Estimated Regression Equation: The slope of this model is 0.1098 Does square footage of the house affect its sales price? Inference about the Slope: t Test (continued)

41. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-41 Inferences about the Slope: t Test Example H 0 : β 1 = 0 H 1 : β 1  0 From Excel output: CoefficientsStandard Errort StatP-value Intercept98.2483358.033481.692960.12892 Square Feet0.109770.032973.329380.01039 b1b1

42. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-42 Inferences about the Slope: t Test Example H 0 : β 1 = 0 H 1 : β 1  0 Test Statistic: t = 3.329 There is sufficient evidence that square footage affects house price From Excel output: Reject H 0 at  =.05 CoefficientsStandard Errort StatP-value Intercept98.2483358.033481.692960.12892 Square Feet0.109770.032973.329380.01039 pb1b1 Decision: Conclusion: Reject H 0  /2=.025 Do not reject H 0 0  /2=.025 3.329 d.f. = 10-2 = 8 (continued)

43. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-43 Confidence Interval Estimate for the Slope Confidence Interval Estimate of the Slope: Excel Printout for House Prices: At 95% level of confidence, the confidence interval for the slope is (0.0337, 0.1858) CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept98.2483358.033481.692960.12892-35.57720232.07386 Square Feet0.109770.032973.329380.010390.033740.18580 d.f. = n - 2

44. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-44 Since the units of the house price variable is $1000s, we are 95% confident that the average impact on sales price is between $33.70 and $185.80 per square foot of house size CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept98.2483358.033481.692960.12892-35.57720232.07386 Square Feet0.109770.032973.329380.010390.033740.18580 This 95% confidence interval does not include 0. Conclusion: There is a significant relationship between house price and square feet at the.05 level of significance Confidence Interval Estimate for the Slope (continued)

45. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-45 Confidence Interval for the Average Y, Given X Confidence interval estimate for the mean value of Y given a particular X i Size of interval varies according to distance away from mean, X

46. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-46 Prediction Interval for an Individual Y, Given X Confidence interval estimate for an Individual value of Y given a particular X i This extra term adds to the interval width to reflect the added uncertainty for an individual case

47. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-47 Estimation of Mean Values: Example Find the 95% confidence interval for the mean price of 2,000 square-foot houses Predicted Price Y i = 317.85 ($1,000s)  Confidence Interval Estimate for μ Y|X=X i The confidence interval endpoints are 280.66 -- 354.90, or from $280,660 -- $354,900

48. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-48 Estimation of Individual Values: Example Find the 95% confidence interval for an individual house with 2,000 square feet Predicted Price Y i = 317.85 ($1,000s)  Prediction Interval Estimate for Y X=X i The prediction interval endpoints are 215.50 -- 420.07, or from $215,500 -- $420,070

49. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-49 Finding Confidence and Prediction Intervals in Excel  In Excel, use PHStat | regression | simple linear regression …  Check the “confidence and prediction interval for X=” box and enter the X-value and confidence level desired

50. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-50 Input values Finding Confidence and Prediction Intervals in Excel (continued) Confidence Interval Estimate for μ Y|X=Xi Prediction Interval Estimate for Y X=Xi Y 

51. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-51 Pitfalls of Regression Analysis  Lacking an awareness of the assumptions underlying least-squares regression  Not knowing how to evaluate the assumptions  Not knowing the alternatives to least-squares regression if a particular assumption is violated  Using a regression model without knowledge of the subject matter  Extrapolating outside the relevant range

52. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-52 Chapter Summary  Introduced types of regression models  Reviewed assumptions of regression and correlation  Discussed determining the simple linear regression equation  Described measures of variation  Discussed residual analysis  Addressed measuring autocorrelation

53. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-53 Chapter Summary  Described inference about the slope  Addressed estimation of mean values and prediction of individual values  Discussed possible pitfalls in regression and recommended strategies to avoid them (continued)