Read the Decision Dilemma found on pages 477-487 in your text. Consider the questions asked at the end of that discussion in light of the ethical considerations presented on page 498. What variables might be used? What variables should not be used? To what degree can someone depend on the results of the regression analysis? Why?

Alternative question: Find an article in the news that presents statistical results. Consider whether the study was done ethically and whether the presentation of results is appropriate.

Ethical Considerations Multiple regression analysis can be used either intentionally or unintentionally in questionable or unethical ways. When degrees of freedom are small, an inated value of R2 can be obtained, leading to overenthusiastic expectations about the predictability of a regression model. To prevent this type of reliance, a business analyst should take into account the nature of the data, the variables, and the value of the adjusted R2. Another misleading aspect of multiple regression can be the tendency of analysts to assume causeandeffect relationships between the dependent variable and predictors. Just because independent variables produce a signicant R2 does not necessarily mean those variables are causing the deviation of the g values. Indeed, some other force not in the model may be driving both the independent variables and the dependent variable over the range of values being studied. Some people use the estimates of the regression coefcients to compare the worth of the predictor variables; the larger the coefficient, the greater is its worth. At least two problems can be found in this approach. The first is that most variables are measured in different units. Thus, regression coefcient weights are partly a function of the unit of measurement of the variable. Second, if multicollinearity (discussed in pter 14,] is present, the interpretation of the regression co efcients is questionable. In addition, the presence of multicollinearity raises several issues about the interpretation of other regression output. Analysts who ignore this problem are at risk of presenting spurious results. Another danger in using regression analysis is in the extrapolation of the model to values beyond the range of values used to derive the model. A regression model that ts data within a given range does not necessarily t data outside that range. One of the uses of regression analysis is in the area of forecasting. Users need to be aware that what has occurred in the past is not guaranteed to continue to occur in the future. Unscrupulous and sometimes even wellintentioned business decision makers can use regression models to project conclusions about the future that have little or no basis. The receiver of such messages should be cautioned that regression models may lack validity outside the range of values in which the models were developed. Summary Multiple regression analysis is a statistical tool in which a mathematical model is developed in an attempt to predict a dependent variable by two or more independent variables or in which at least one predictor is nonlinear. Because doing multiple regression analysis by hand is extremely tedious and time-consuming, it is almost always done on a computer. The standard output from a multiple regression analysis is similar to that of simple regression analysis. A regression equation is produced with a constant that is analogous to the y-intercept in simple regression and with estimates of the regression coefficients that are analogous to the estimate of the slope in simple regression. An F test for the overall model is computed to determine whether at least one of the regression coefficients is significantly different from zero. This F value is usually displayed in an ANOVA table, which is part of the regression output. The ANOVA table also contains the sum of squares of error and sum of squares of regression, which are used to compute other statistics in the model. Most multiple regression computer output contains t values, which are used to determine the significance of the regression coefficients. Using these t values, business analysts can make decisions about including or excluding variables from the model. Residuals, standard error of the estimate, and R" are also standard computer regression output with multiple regression. The coefficient of determination for simple regression models is denoted 2, whereas for multiple regression it is R". The interpretation of residuals, standard error of the estimate, and R2 in multiple regression is similar to that in simple regression. Because R2 can be inflated with nonsignificant variables in the mix, an adjusted R2 is often computed. Unlike 2, adjusted R2 takes into account the degrees of freedom and the number of observations