Regression Analysis
Management believes price changes will have an immediate effect on ticket sales, but the effects of income changes will take longer (as much as three months) to play out. How would one test this effect using regression analysis?
Interpreting Regression Statistics Many computer programs are available to carry out regression analysis. (In fact, almost all of the best-selling spreadsheet programs include regression features.) These programs call for the user to specify the form of the regression equation and to input the necessary data to estimate it: values of the dependent variables and the chosen explanatory variables. Besides computing the ordinary least-squares regression coefficients, the program produces a set of statistics indicating how well the OLS equation performs. Table 4.6 lists the standard computer output for the airline’s multiple regression. The regression coefficients and constant term are listed in the third-to-last line. Using these, we obtained the regression equation: Q 28.84 2.12P 1.03P 3.09Y. To evaluate how well this equation fits the data, we must learn how to interpret the other statistics in the table. R-SQUARED The R-squared statistic (also known as the coefficient of determination) measures the proportion of the variation in the dependent variable (Q in our example) that is explained by the multiple-regression equation. Sometimes we say that it is a measure of goodness of fit, that is, how well the equation fits the data. The total variation in the dependent variable is computed as ©(Q Q)2, that is, as the sum across the data set of squared differences between the values of Q and the mean of Q. In our example, this total sum of squares (labeled TSS) happens to be 11,706. The R2 statistic is computed as
R2
TSS SSE TSS
[4.4]
The sum of squared errors, SSE, embodies the variation in Q not accounted for by the regression equation. Thus, the numerator is the amount of explained variation and R-squared is simply the ratio of explained to total variation. In our example, we can calculate that R2 (11,706 2,616)/11,706 .78. This confirms the entry in Table 4.6. We can rewrite Equation 4.4 as R2 1 (SSE/TSS)
[4.5]
CHECK STATION 2
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