# In Exercise 7.7, if the true model is given by show that the expected values are given by E( 0 ) = 0

- September 24, 2021 /
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In Exercise 7.7, if the true model is given by

show that the expected values are given by

E(_{0}) = _{0}

– _{123}

E(b_{1}), E(b_{2}), E(b_{3}) as in

Exercise 7:7

Exercises 7.7 and 7.8 illustrate the effect of aliasing in

the sense of a regression model. Thus, in the case of these two exercises, the

aliasing of x_{l} with x_{2}x_{3} implies that the

coefficient b1 is not an unbiased estimator of b_{1} but is biased by _{23},

the coefficient of x_{2}x_{3}. Indeed, the coefficient of the

interaction described by the defining contrast, namely _{123},

biases the estimated intercept _{0}.

Exercise 7.7

Consider the “model inadequacy” material in Section 7.2. The

material can be used to nicely illustrate the aliasing ideas discussed in

Chapter 4. Aliasing plays an important role when one deals with fractional

factorials for fitting first-order and first-order-plus-interaction response

surface models. Suppose one fits a first-order model using a 2^{3-1}_{III}

with defining relation

I = x_{1}x_{2}x_{3}

However, suppose the true model is

Using Equation 7.2, we obtain

Use the expression

to verify the following expected values: