Multiple Regression

In this chapter, you will learn how to interpret the results of a multiple regression analysis. The prediction equation (also called the regression equation) is estimated from your data, and the resulting regression coefficients tell you the effect of each X variable on Y while holding the other X variables fixed (we call this “adjusting for” and “controlling for” the other X variables, and this adjustment happens automatically whenever you run a multiple regression). Just as we did in the previous chapter, we measure the quality of the regression using both the standard error of estimate (indicating the approximate size of prediction errors or residuals) and the coefficient of determination (indicating the percentage of the variability of Y that is explained by all of the X variables taken together). Statistical inference in multiple regression is based on the linear model and will begin with the F test (an overall test of whether the X variables together have a significant effect on Y ), which produces a p -value and may be interpreted as a test of whether the R -squared (percent variance explained) is large enough to be considered statistically significant. If the F test is significant, you may proceed to the t tests, one for each of the X variables (testing the effect of that X variable on Y while controlling for all of the other X variables). These t tests may be performed using any of the methods we have covered: the p -value, the confidence interval (for the regression coefficient), or the t statistic.