In Myers and Montgomery (2002), a data set. is discussed in...

In Myers and Montgomery (2002), a data set. is discussed in which a $2^{3}$ factorial is used by an engineer to study the effects of cutting speed $(A)$, tool geometry (B). and cutting angle $(C)$ on the life (in hours) of a machine tool. Two levels of each factor are chosen, and duplicates were run at, each design point with the: order of the runs being random. The data are presented here. (a) Calculate all seven effects. Which appear, based on their magnitude, to be important? (b) Do an analysis of variance and observe $P$ -values. (c) Do your results in (a) and (b) agree? (d) The engineer felt confident that cutting speed and cutting angle should interact. If this interaction is significant, draw an interaction plot and discuss the engineering meaning of the interaction. $$ \begin{array}{ccccc} & A & B & C & \text { Life } \\ \hline(1) & - & - & - & 22.31 \\ a & + & - & - & 32,43 \\ b & - & -i & - & 35.34 \\ a b & + & + & - & 35.47 \\ c & - & - & + & 44,45 \\ a c & + & - & - & 40.37 \\ b c & - & + & + & 60,50 \\ a b c & - & - & + & 39,41 \end{array} $$

In Myers and Montgomery (2002), a data set. is discussed in which a $2^{3}$ factorial is used by an engineer to study the effects of cutting speed $(A)$, tool geometry
(B). and cutting angle $(C)$ on the life (in hours) of a machine tool. Two levels of each factor are chosen, and duplicates were run at, each design point with the: order of the runs being random. The data are presented here.
(a) Calculate all seven effects. Which appear, based on their magnitude, to be important?
(b) Do an analysis of variance and observe $P$ -values.
(c) Do your results in (a) and (b) agree?
(d) The engineer felt confident that cutting speed and cutting angle should interact. If this interaction is significant, draw an interaction plot and discuss the engineering meaning of the interaction.
$$
\begin{array}{ccccc}
& A & B & C & \text { Life } \\
\hline(1) & - & - & - & 22.31 \\
a & + & - & - & 32,43 \\
b & - & -i & - & 35.34 \\
a b & + & + & - & 35.47 \\
c & - & - & + & 44,45 \\
a c & + & - & - & 40.37 \\
b c & - & + & + & 60,50 \\
a b c & - & - & + & 39,41
\end{array}
$$

Key Concepts

Factorial Design

A factorial design is an experimental setup that involves multiple factors, each at a number of levels, where all possible combinations of the factor levels are investigated. This type of design allows for the study of both individual (main) effects of each factor and the interactions between factors, making it efficient in exploring how different variables collectively influence an outcome.

Main Effects

Main effects refer to the independent impact that changing one factor has on the response variable, averaged over the levels of other factors. By isolating the effect of a single variable, main effects help in understanding which factors have a significant direct influence on the overall performance or outcome of the system under experimental investigation.

Interaction Effects

Interaction effects occur when the effect of one factor on the response variable differs depending on the level of another factor. This means that the combined effect of two or more factors is not simply additive, and recognizing these interactions is crucial in understanding the complete dynamics of the system, particularly when factors influence each other.

Effects Estimation

Effects estimation involves calculating the magnitude of main and interaction effects from experimental data. This process typically uses contrasts or differences between treatment means to determine how significantly each factor or combination of factors alters the response variable, providing an insight into which factors are most influential.

Analysis of Variance (ANOVA)

Analysis of Variance (ANOVA) is a statistical methodology used to compare the variances among different groups to test the significance of the factors and their interactions in an experiment. ANOVA helps determine whether the observed differences in the response variable are likely due to true underlying effects or simply random variation.

Interaction Plot

An interaction plot is a graphical tool used to visualize the interaction between two or more factors. By plotting the response variable against one factor at different levels of another factor, these plots help in identifying whether the effect of one variable changes across the levels of another, thereby offering a visual representation of any interaction that may exist.

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