Introduction to Linear Regression Analysis. Douglas C. Montgomery

Introduction to Linear Regression Analysis

(2.17)

But

is just the corrected sum of squares of the response observations, so

The residual sum of squares has n − 2 degrees of freedom, because two degrees of freedom are associated with the estimates in21-1 and in21-2 involved in obtaining in21-3 . Section C.3 shows that the expected value of SS_Res is E(SS_Res) = (n − 2)σ², so an unbiased estimator of σ2 is

(2.19)

The quantity MS_Res is called the residual mean square. The square root of in21-4 is sometimes called the standard error of regression, and it has the same units as the response variable y.

Because in21-5 depends on the residual sum of squares, any violation of the assumptions on the model errors or any misspecification of the model form may seriously damage the usefulness of in21-6 as an estimate of σ². Because in21-7 is computed from the regression model residuals, we say that it is a model-dependent estimate of σ².

Example 2.2 The Rocket Propellant Data

To estimate σ² for the rocket propellant data in Example 2.1, first find

From Eq. (2.18) the residual sum of squares is

Therefore, the estimate of σ² is computed from Eq. (2.19) as

Remember that this estimate of σ² is model dependent. Note that this differs slightly from the value given in the Minitab output (Table 2.3) because of rounding.

2.2.4 Alternate Form of the Model

There is an alternate form of the simple linear regression model that is occasionally useful. Suppose that we redefine the regressor variable xi as the deviation from its own average, say in22-1 . The regression model then becomes

(2.20)

Note that redefining the regressor variable in Eq. (2.20) has shifted the origin of the x’s from zero to in22-2 . In order to keep the fitted values the same in both the original and transformed models, it is necessary to modify the original intercept. The relationship between the original and transformed intercept is

(2.21)

It is easy to show that the least-squares estimator of the transformed intercept is in22-3 . The estimator of the slope is unaffected by the transformation. This alternate form of the model has some advantages. First, the least-squares estimators in22-4 and in22-5 are uncorrelated, that is, in22-6 . This will make some applications of the model easier, such as finding confidence intervals on the mean of y (see Section 2.4.2). Finally, the fitted model is

(2.22)

Although Eqs. (2.22) and (2.8) are equivalent (they both produce the same value of in22-7 for the same value of x), Eq. (2.22) directly reminds the analyst that the regression model is only valid over the range of x in the original data. This region is centered at in22-8 .

2.3 HYPOTHESIS TESTING ON THE SLOPE AND INTERCEPT

We are often interested in testing hypotheses and constructing confidence intervals about the model parameters. Hypothesis testing is discussed in this section, and Section 2.4 deals with confidence intervals. These procedures require that we make the additional assumption that the model errors εi are normally distributed. Thus, the complete assumptions are that the errors are normally and independently distributed with mean 0 and variance σ², abbreviated NID(0, σ²). In Chapter 4 we discuss how these assumptions can be checked through residual analysis.

2.3.1 Use of t Tests

Suppose that we wish to test the hypothesis that the slope equals a constant, say β₁₀. The appropriate hypotheses are

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