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Regression Analysis 141

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 fea-

tures.) 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 depend-

ent variables and the chosen explanatory variables. Besides computing the ordi-

nary 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:

To evaluate how well this equation fits the data, we must learn how to inter-

pret the other statistics in the table.

R-SQUARED The R-squared statistic(also known as the coefficient of determi-

nation) 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 com-

puted as that is, as the sum across the data set of squared dif-

ferences 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 R^2 statistic is

computed as

[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 R^2 (11,706 2,616)/11,706  .78. This con-

firms the entry in Table 4.6. We can rewrite Equation 4.4 as

R^2  1 (SSE/TSS) [4.5]

R^2 

TSSSSE

TSS

©(QQ)^2 ,

Q28.842.12P1.03P3.09Y.

CHECK

STATION 2

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