SECTION 11.1 SUMMARY

• • Data for multiple linear regression consist of the values of a response variable y and p explanatory variables x₁, x₂, … , x_p for n cases. We write the data and enter them into software in the form

	Variables
Individual	y	x1	x2	…	xp
1	y1	x11	x12	…	x1p
2	y2	x21	x22	…	x2p
n	yn	xn1	xn2	…

• • The statistical model for multiple linear regression with response variable y and p explanatory variables x₁, x₂, … , x_p is

  where i = 1, 2, … , n. The ϵ_i are assumed to be independent and Normally distributed with mean 0 and standard deviation σ. The parameters of the model are β₀, β₁, β₂, … , β_p, and σ.

• • The multiple regression equation predicts the response variable by a linear relationship with all the explanatory variables:

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• • The β’s are estimated by b₀, b₁, b₂, … , b_p, which are obtained by the method of least squares. The model standard deviation σ is estimated by

  where the e_i are the residuals,

• • A level C confidence interval for β_j is

• where t* is the value for the t(n − p − 1) density curve with area C between −t* and t*.

• • The test of the hypothesis H₀: β_j = 0 is based on the t statistic

  and the t(n − p − 1) distribution.

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

  H₀: β₁ = β₂ = … = β_p = 0

  If H₀ is true, this statistic has an F(p, n − p − 1) distribution.

• • The squared multiple correlation is given by the expression

  and is interpreted as the proportion of the variability in the response variable y that is explained by the explanatory variables x₁, x₂, … , x_p in the multiple linear regression.