SECTION 11.2 Summary

• The statistical model for multiple linear regression with response variable and explanatory variables is

  where . The deviations are independent Normal random variables with mean 0 and a common standard deviation . The parameters of the model are , and .

• The ’s are estimated by the coefficients of the multiple regression equation fitted to the data by the method of least squares. The parameter is estimated by the regression standard error

  where the are the residuals,

• A level confidence interval for the regression coefficient is

  where is the value for the density curve with area between and .

• Tests of the hypothesis are based on the individual statistic:

  and the distribution.

• The estimate of and the test and confidence interval for are all based on a specific multiple linear regression model. The results of all these procedures change if other explanatory variables are added to or deleted from the model.

• The ANOVA table for a multiple linear regression gives the degrees of freedom, sum of squares, and mean squares for the regression and residual sources of variation. The ANOVA statistic is the ratio MSR/MSE and is used to test the null hypothesis

  562

  If is true, this statistic has the distribution.

• The squared multiple correlation is given by the expression

  and is interpreted as the proportion of the variability in the response variable that is explained by the explanatory variables in the multiple linear regression.

• The null hypothesis that a collection of explanatory variables all have coefficients equal to zero is tested by an statistic with degrees of freedom in the numerator and degrees of freedom in the denominator. This statistic can be computed from the squared multiple correlations for the model with all the explanatory variables included () and the model with the variables deleted ():