M15_BERE8380_12_SE_C15.6.qxd 2/21/11 8:21 PM Page Influence Analysis 1

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1 M15_BERE8380_12_SE_C15.6.qxd 2/21/11 8:21 PM Page Influence Analysis FIGURE Minitab worksheet containing computed values for the Studentized deleted residuals, the hat matrix elements, and Cook s distance statistics for the OmniPower sales data 15.6 Influence Analysis 1 In Sections 13.5 and 14.3, you used residual analysis to evaluate the regression assumptions. This section introduces several methods that measure the influence of individual observations: The hat matrix elements, The Studentized deleted residuals, t i Cook s distance statistic, Figure presents the values of these statistics computed by Minitab for the OmniPower sales data.

2 M15_BERE8380_12_SE_C15.6.qxd 2/21/11 8:21 PM Page 2 2 CHAPTER 15 Multiple Regression Model Building The Hat Matrix Elements, In Section 13.8, was defined for the simple linear regression model when constructing the confidence interval estimate of the mean response. For multiple regression models, the equation for calculating the hat matrix diagonal elements,, requires the use of matrix algebra and is beyond the scope of this text (see references 4, 5, and 7). The hat matrix diagonal element for observation i, denoted, reflects the possible influence of X i on the regression equation. If potentially influential observations are present, you may need to delete them from the model. In a regression model containing k independent variables, Hoaglin and Welsch (see reference 5) suggest the following decision rule: If 7 2(k + 1)>n, then X i is an influential observation and is a candidate for removal from the model. For the OmniPower sales data, because n = 34 and k = 2, you flag any value greater than 2(2 + 1)>34 = Referring to Figure 15.16, you see that none of the values are greater than Therefore, none of the observations are candidates for removal from the analysis. The Studentized Deleted Residuals, t i Recall from Section 13.5 that a residual is the difference between the observed value of Y and the predicted value of Y [see Equation (13.14) on page 539]. Studentized residuals are the residuals divided by the standard error of the estimate S YX and adjusted for the distance from X. The Studentized deleted residual, expressed as a t statistic in Equation (15.10), measures the difference of each Y i from the value predicted by a model that includes all observations except observation i. STUDENTIZED DELETED RESIDUAL n - k - 1 t i = e i A SSE(1 - ) - e 2 i where e i = residual for observation i k = number of independent variables SSE = error sum of squares of the regression model fitted = hat matrix diagonal element for observation i (15.10) Hoaglin and Welsch (see reference 5) suggest that if t i 7 t a>2 or t i 6 t a>2 (using a level of significance of 0.10), the observed and predicted values are so different that observation i is highly influential on the regression equation and is a candidate for removal. For the OmniPower sales data, n = 34 and k = 2. Thus, you flag any t i whose absolute value is greater than (see Table E.3). In Figure 15.16, t 14 = , t 15 = , and t 20 = are highlighted. Thus, the 14th, 15th, and 20th observations may each have an adverse effect on the model. These observations were not previously flagged according to the criterion. Since and t i measure different aspects of influence, neither criterion is sufficient by itself. When is small, t i may be large. When is large, t i may be moderate or small because the observed is consistent with the rest of the data. Y i Cook s Distance Statistic, Cook s distance statistic,, based on both and the Studentized residual, is a third criterion for identifying influential observations. To decide whether an observation flagged by either the or t i criterion is unduly affecting the model, Cook and Weisberg (see reference 4) developed Cook s statistic.

3 M15_BERE8380_12_SE_C15.6.qxd 2/21/11 8:21 PM Page Influence Analysis 3 COOK S STATISTIC where = e 2 i k MSE c (1 - ) d 2 e i = residual for observation i k = number of independent variables MSE = mean square error of the regression model fitted = hat matrix diagonal element for observation i (15.11) TABLE 15.4 Selected Critical Values of F for Cook s Statistic Cook and Weisberg suggest that if 7 F a (the critical value of the F distribution having k + 1 degrees of freedom in the numerator and n - k - 1 degrees of freedom in the denominator at a 0.50 level of significance), the observation is highly influential on the regression equation and is a candidate for removal. Table 15.4 shows critical values for Cook s statistic. A 0.50 Numerator df k 1 Denominator df n k q Source: Extracted from E. S. Pearson and H. O. Hartley, eds., Biometrika Tables for Statisticians, 3rd ed., 1966, by permission of the Biometrika Trustees. For the OmniPower sales data, since n = 34 and k = 2, there are 3 degrees of freedom in the numerator and 31 degrees of freedom in the denominator. Thus, any 7 F a, = is flagged. Referring to Figure 15.16, you see that none of the values exceed 0.187, and therefore no observations are identified as influential using Cook s statistic. Overview This section discussed three criteria for evaluating the influence of each observation on the multiple regression model. The various statistics did not lead to a consistent set of conclusions. According to both the and the criteria, none of the observations is a candidate for removal. Under such circumstances, most statisticians would conclude that there is insufficient evidence for the removal of any observation from the analysis. In addition to the three criteria presented here, there are other measures of influence (see references 1 and 6). Although different statisticians seem to prefer particular measures, currently there is no consensus as to the best measure.

4 M15_BERE8380_12_SE_C15.6.qxd 2/21/11 8:21 PM Page 4 4 CHAPTER 15 Multiple Regression Model Building Problems for Section 15.6 APPLYING THE CONCEPTS In Problem 14.4 on page 583, you used sales and number of orders to predict distribution costs at a mail-order catalog business (stored in Warecost ). Perform an influence analysis on your results and determine whether any observations the regression model after deleting these observations and compare your results In Problem 14.5 on page 583, you used horsepower and weight to predict gasoline mileage (stored in Auto2010 ). Perform an influence analysis on your results and determine whether any observations should be deleted from the analysis. If necessary, reanalyze the regression model after deleting these observations and compare your results In Problem 14.6 on page 583, you used the amount of radio advertising and newspaper advertising to predict sales (stored in Advertise ). Perform an influence analysis on your results and determine whether any observations the regression model after deleting these observations and compare your results In Problem 14.7 on page 584, you used the total staff present and remote hours to predict standby hours (stored in Standby ). Perform an influence analysis on your results and determine whether any observations should be deleted from the analysis. If necessary, reanalyze the regression model after deleting these observations and compare your results In Problem 14.8 on page 584, you used the land area of the property and age in years to predict appraised value (stored in GlenCove ). Perform an influence analysis on your results and determine whether any observations the regression model after deleting these observations and compare your results. REFERENCES 1. Andrews, D. F., and D. Pregibon, Finding the Outliers That Matter, Journal of the Royal Statistical Society 40 (Ser. B., 1978): Atkinson, A. C., Robust and Diagnostic Regression Analysis, Communications in Statistics 11 (1982): Belsley, D. A., E. Kuh, and R. Welsch, Regression Diagnostics: Identifying Influential Data and Sources of Collinearity (New York: Wiley, 1980). 4. Cook, R. D., and S. Weisberg, Residuals and Influence in Regression (New York: Chapman and Hall, 1982). 5. Hoaglin, D. C., and R. Welsch, The Hat Matrix in Regression and ANOVA, The American Statistician, 32 (1978): Hocking, R. R., Developments in Linear Regression Methodology: , Technometrics 25 (1983): Kutner, M., C. Nachtsheim, J. Neter, and W. Li, Applied Linear Statistical Models, 5th ed. (New York: McGraw- Hill/Irwin, 2005). 8. Minitab Release 16 (State College, PA: Minitab, Inc., 2010)

5 M15_BERE8380_12_SE_C15.6.qxd 2/21/11 8:21 PM Page 5 EG15.6 EXCEL GUIDE FOR INFLUENCE ANALYSIS There are no Excel Guide instructions for this section. MG15.6 MINITAB GUIDE FOR INFLUENCE ANALYSIS Use Regression to perform influence analysis. Use the Interpreting the Regression Coefficients instructions in Section MG14.1 (repeated below), replacing step 19 of those instructions with the steps 19 through 22 listed below. For example, to perform the Figure analysis of the OmniPower sales data on page, open to the OmniPower worksheet. Select Stat Regression Regression. In the Regression dialog box: 1. Double-click C1 Sales in the variables list to add Sales to the Response box. 2. Double-click C2 Price in the variables list to add Price to the Predictors box. 3. Double-click C3 Promotion in the variables list to add Promotion to the Predictors box. 4. Click Graphs. In the Regression - Graphs dialog box: 5. Click Regular and Individual Plots. 6. Check Histogram of residuals and clear all the other check boxes. 7. Click anywhere inside the Residuals versus the variables box. 8. Double-click C2 Price in the variables list to add Price in the Residuals versus the variables box. 9. Double-click C3 Promotion in the variables list to add Promotion in the Residuals versus the variables box. 10. Click OK. 11. Back in the Regression dialog box, click Results. In the Regression - Results dialog box: 12. Click In addition, the full table of fits and residuals and then click OK. 13. Back in the Regression dialog box, click Options. In the Regression - Options dialog box: 14. Check Fit Intercept. 15. Clear all the Display and Lack of Fit Test check boxes. 16. Enter 79 and 400 in the Prediction intervals for new observations box. 17. Enter 95 in the Confidence level box. 18. Click OK. 19. Back in the Regression dialog box, click Storage. In the Regression - Storage dialog box: 20. Check Deleted t residuals, Hi (leverages), and Cook s distance. 21. Click OK. 22. Back in the Regression dialog box, click OK. 5