Use the coal cleansing data of Exercise 15.2 on page 629 to...

Use the coal cleansing data of Exercise 15.2 on page 629 to fit a model of the type $$ E(Y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{3} $$ where the levels are $$ \begin{aligned} &x_{1}, \quad \text { percent solids: } 8,12\\ &x_{2}, \quad \text { flow rate: } 150,250 \mathrm{gal} / \mathrm{min}\\ &x_{3}, \quad \mathrm{pH}: 5.6 \end{aligned} $$ Center and scale the variables to design units. Also conduct a test for lack of fit, and comment concerning the adequacy of the linear regression model.

Use the coal cleansing data of Exercise 15.2 on page 629 to fit a model of the type
$$
E(Y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{3}
$$
where the levels are
$$
\begin{aligned}
&x_{1}, \quad \text { percent solids: } 8,12\\
&x_{2}, \quad \text { flow rate: } 150,250 \mathrm{gal} / \mathrm{min}\\
&x_{3}, \quad \mathrm{pH}: 5.6
\end{aligned}
$$
Center and scale the variables to design units. Also conduct a test for lack of fit, and comment concerning the adequacy of the linear regression model.

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Key Concepts

Linear Regression

Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables by fitting a linear equation. It allows for estimation and inference regarding the parameters of the model, which enables predictions, hypothesis tests, and assessments of the model's overall adequacy.

Centering and Scaling

Centering and scaling are data pre-processing techniques where variables are transformed by subtracting a reference value (usually the mean or a specified baseline) and then dividing by a unit or the standard deviation. This normalization helps in interpreting the regression coefficients in terms of design units, reducing issues with multicollinearity, and stabilizing numerical computations in the analysis.

Lack-of-Fit Test

A lack-of-fit test is used to evaluate whether a chosen model appropriately describes the relationship between the variables. It compares the variability due to pure error (from replicated observations) with the additional variability that may be attributed to model mis-specification. If significant lack-of-fit is detected, it suggests that the chosen linear model may be insufficient and that a more complex model might be needed to capture the underlying relationship.

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