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%---------------------------------------------------------------------
% 1 -----------------------------------------------------------------

\bs
  \heading{Unit 6: Fractional Factorial Experiments at Three Levels}

\bc Source : Chapter 6 (Sections 6.1 - 6.6) \ec

\vspace{.1 in}

\bi
 \item Larger-the-better and smaller-the-better problems.
 \item Basic concepts for $3^k$ full factorial designs.
 \item Analysis of $3^k$ designs using orthogonal components
 system.
 \item Design of 3-level fractional factorials.
 \item Effect aliasing, resolution and minimum aberration in $3^{k-p}$
fractional factorial designs.
%\item Analysis of 3-level designs : ANOVA using orthogonal components system.
 \item An alternative analysis method using linear-quadratic
 system.
\ei
 \es

%\end{document}
%---------------------------------------------------------------------
%  2 -----------------------------------------------------------------

\bs
  \heading{Seat Belt Experiment}

 \bi
   \item An experiment to study the effect of four factors on the
pull strength of truck seat belts.
  \item Four factors, each at three levels (Table 1).
  \item Two responses : crimp tensile strength that must be
  at least 4000 lb and flash that cannot exceed 14 mm.
  \item 27 runs were conducted; each run was replicated three times as
shown in
Table 2.
 \ei

\begin{table}[htbp]
\caption{Factors and Levels, Seat-Belt Experiment}
$ $\\
\centering \footnotesize
%\begin{center}
%\renewcommand{\baselinestretch}{1}
%\large\normalsize
\begin{tabular}{|ll|lll|} \hline
\multicolumn{2}{|c}{} &
\multicolumn{3}{|c|}{Level} \\
\multicolumn{2}{|c}{Factor} & \multicolumn{1}{|c}{0} &
\multicolumn{1}{c}{1} & \multicolumn{1}{c|}{2} \\ \hline
$A$.& pressure (psi)& 1100 & 1400 & 1700 \\
$B$.& die flat (mm) & 10.0 & 10.2 & 10.4 \\
$C$.& crimp length (mm) & 18 & 23 & 27 \\
$D$.& anchor lot (\#) & P74 & P75 & P76\\ \hline
\end{tabular}
\label{tab:ex3factor}
%\end{center}
%\renewcommand{\baselinestretch}{1.5}
%\large\normalsize
\end{table}

\es

 %---------------------------------------------------------------------
%  3 -----------------------------------------------------------------

\bs
  \heading{Design Matrix and Response Data, Seat-Belt Experiment}

\begin{table}[htbp]
\caption{Design Matrix and Response Data, Seat-Belt Experiment:
first 14 runs}
$ $\\
\centering \tiny
\begin{tabular}{|r|rrrr|rrr|rrr|} \hline
\multicolumn{1}{|c|}{} & \multicolumn{4}{c|}{Factor} &
\multicolumn{3}{c|}{} &
\multicolumn{3}{c|}{} \\
\multicolumn{1}{|c|}{Run}& \multicolumn{1}{c}{$A$} &
\multicolumn{1}{c}{$B$} & \multicolumn{1}{c}{$C$} &
\multicolumn{1}{c|}{$D$} & \multicolumn{3}{c|}{Strength} &
\multicolumn{3}{c|}{Flash} \\ \hline
1&0&0&0&0&5164&6615&5959&12.89&12.70&12.74\\
2&0&0&1&1&5356&6117&5224&12.83&12.73&13.07\\
3&0&0&2&2&3070&3773&4257&12.37&12.47&12.44\\
4&0&1&0&1&5547&6566&6320&13.29&12.86&12.70\\
5&0&1&1&2&4754&4401&5436&12.64&12.50&12.61\\
6&0&1&2&0&5524&4050&4526&12.76&12.72&12.94\\
7&0&2&0&2&5684&6251&6214&13.17&13.33&13.98\\
8&0&2&1&0&5735&6271&5843&13.02&13.11&12.67\\
9&0&2&2&1&5744&4797&5416&12.37&12.67&12.54\\
10&1&0&0&1&6843&6895&6957&13.28&13.65&13.58\\
11&1&0&1&2&6538&6328&4784&12.62&14.07&13.38\\
12&1&0&2&0&6152&5819&5963&13.19&12.94&13.15\\
13&1&1&0&2&6854&6804&6907&14.65&14.98&14.40\\
14&1&1&1&0&6799&6703&6792&13.00&13.35&12.87\\ \hline
\end{tabular}
\label{tab:ex3desdat}
\end{table}

 \es

%---------------------------------------------------------------------
% 4--------------------------------------------------------------------

\bs
  \heading{Design Matrix and Response Data, Seat-Belt Experiment (contd.)}

\begin{table}[htbp]
\caption{Design Matrix and Response Data, Seat-Belt Experiment:
last 13 runs}
$ $\\
\centering \tiny
\begin{tabular}{|r|rrrr|rrr|rrr|} \hline
\multicolumn{1}{|c|}{} & \multicolumn{4}{c|}{Factor} &
\multicolumn{3}{c|}{} &
\multicolumn{3}{c|}{} \\
\multicolumn{1}{|c|}{Run}& \multicolumn{1}{c}{$A$} &
\multicolumn{1}{c}{$B$} & \multicolumn{1}{c}{$C$} &
\multicolumn{1}{c|}{$D$} & \multicolumn{3}{c|}{Strength} &
\multicolumn{3}{c|}{Flash} \\ \hline
15&1&1&2&1&6513&6503&6568&13.13&13.40&13.80\\
16&1&2&0&0&6473&6974&6712&13.55&14.10&14.41\\
17&1&2&1&1&6832&7034&5057&14.86&13.27&13.64\\
18&1&2&2&2&4968&5684&5761&13.00&13.58&13.45\\
19&2&0&0&2&7148&6920&6220&16.70&15.85&14.90\\
20&2&0&1&0&6905&7068&7156&14.70&13.97&13.66\\
21&2&0&2&1&6933&7194&6667&13.51&13.64&13.92\\
22&2&1&0&0&7227&7170&7015&15.54&16.16&16.14\\
23&2&1&1&1&7014&7040&7200&13.97&14.09&14.52\\
24&2&1&2&2&6215&6260&6488&14.35&13.56&13.00\\
25&2&2&0&1&7145&6868&6964&15.70&16.45&15.85\\
26&2&2&1&2&7161&7263&6937&15.21&13.77&14.34\\
27&2&2&2&0&7060&7050&6950&13.51&13.42&13.07\\ \hline
\end{tabular}
\label{tab:ex3desdat}
\end{table}

 \es

%---------------------------------------------------------------------
% 5--------------------------------------------------------------------

\bs
  \heading{Larger-The-Better and Smaller-The-Better problems}
\vspace{0.1 in}
 \small
  \bi
   \item In the seat-belt experiment, the strength should be as high as
    possible and the flash as low as possible.
   \item There is no fixed nominal value for either strength or flash.
Such type of
   problems are referred to as {\bf larger-the-better} and {\bf
smaller-the-better} problems,
    respectively.
   \item For such problems increasing or decreasing the mean is
   more difficult than reducing the variation and should be done
   in the first step. (why?)
   \item Two-step procedure for larger-the-better problems:
\begin{enumerate}
 \item \emph{Find factor settings that maximize E(y).
 \item Find other factor settings that minimize Var(y).}
\end{enumerate}

\item Two-step procedure for smaller-the-better problems:
\begin{enumerate}
 \item \emph{Find factor settings that minimize E(y).
 \item Find other factor settings that minimize Var(y).}
\end{enumerate}
  \ei

 \es

%---------------------------------------------------------------------
% 6--------------------------------------------------------------------

\bs
  \heading{Situations where three-level experiments are useful}
  \vspace{0.2 in}
  \bi
   \item When there is a curvilinear relation between the response and a
quantitative factor like
   temperature. It is not possible to detect such a curvature
   effect with two levels.
   \item A qualitative factor may have three levels (e.g., three
   types of machines or three suppliers).
   \item It is common to study the effect of a factor on the
   response at its current setting $x_0$ and two settings around
   $x_0$.
  \ei

 \es

%---------------------------------------------------------------------
% 7--------------------------------------------------------------------

\bs
  \heading{Analysis of $3^k$ designs using ANOVA}
  \bi
   \item We consider a simplified version of the seat-belt experiment as a
$3^3$ full
   factorial experiment with factors $A,B,C$.
   \item Since a $3^3$ design  is a special case of a multi-way
layout, the analysis of variance method introduced in Section 3.5
can be applied to this experiment.
   \item We consider only the strength data for demonstration of the
analysis.
   \item Using analysis of variance, we can compute
the sum of squares for main effects $A$, $B$, $C$, interactions $A
\times B$, $A \times C$, $B \times C$ and $A \times B \times C$
and the residual sum of squares. Details are given in Table 4.
 \item The break-up of the degrees of freedom will be as follows:
  \bi
   \footnotesize
    \item Each main effect has two degrees of freedom because each factor
has three levels.
    \item Each two-factor interaction has $(3-1) \times (3-1)=4$ degrees
of freedom.
    \item The $A \times B \times C$ interaction has $(3-1) \times (3-1)
\times
(3-1)=8$ degrees of freedom.
    \item The residual degrees of freedom is 54$(=27\times(3-1))$,
    since there are three replicates.
   \ei
  \ei

 \es

%---------------------------------------------------------------------
% 8--------------------------------------------------------------------

\bs
  \heading{Analysis of Simplified Seat-Belt Experiment}

  \begin{table}[htbp]
\caption{ANOVA Table, Simplified Seat-Belt Experiment}
$ $\\
\centering \small
\begin{tabular}{|lccrrrrrrrrrrrrrrr|} \hline
\multicolumn{3}{|c}{}& %&
\multicolumn{3}{c}{Degrees of} & %&
\multicolumn{3}{c}{Sum of} & %&
\multicolumn{3}{c}{Mean} & %&
\multicolumn{3}{c}{} &
\multicolumn{3}{c|}{}  \\
%\multicolumn{13}{|c|}{} \\
\multicolumn{3}{|c}{Source}& %&
\multicolumn{3}{c}{Freedom} & %&
\multicolumn{3}{c}{Squares} & %&
\multicolumn{3}{c}{Squares} & %&
\multicolumn{3}{c}{$F$} & %&
\multicolumn{3}{c|}{p-value} \\ \hline
%& \multicolumn{1}{c|}{} \\ \hline

\multicolumn{2}{|c}{$A$} && \multicolumn{2}{r}{2} &&
\multicolumn{2}{r}{34621746} && \multicolumn{2}{r}{17310873} &&
\multicolumn{2}{r}{85.58} && \multicolumn{2}{r}{0.000} & \\

\multicolumn{2}{|c}{$B$} && \multicolumn{2}{r}{2} &&
\multicolumn{2}{r}{938539} && \multicolumn{2}{r}{469270} &&
\multicolumn{2}{r}{2.32} && \multicolumn{2}{r}{0.108} & \\

\multicolumn{2}{|c}{$C$} && \multicolumn{2}{r}{2} &&
\multicolumn{2}{r}{9549481} && \multicolumn{2}{r}{4774741} &&
\multicolumn{2}{r}{23.61} && \multicolumn{2}{r}{0.000} & \\

\multicolumn{2}{|c}{$A \times B$} && \multicolumn{2}{r}{4} &&
\multicolumn{2}{r}{3298246} && \multicolumn{2}{r}{824561} &&
\multicolumn{2}{r}{4.08} && \multicolumn{2}{r}{0.006} & \\

%& \multicolumn{2}{c}{$\;\;AB$} &&& \multicolumn{1}{l}{2} &&
%\multicolumn{2}{r}{2727451} && \multicolumn{2}{r}{1363725} &&
%\multicolumn{2}{r}{6.74} && \multicolumn{2}{r|}{0.002} \\

%& \multicolumn{2}{c}{$\;\;AB{}^{2}$} &&& \multicolumn{1}{l}{2} &&
%\multicolumn{2}{r}{570795} && \multicolumn{2}{r}{285397} &&
%\multicolumn{2}{r}{1.41} && \multicolumn{2}{r|}{0.253} \\

\multicolumn{2}{|c}{$A \times C$} && \multicolumn{2}{r}{4} &&
\multicolumn{2}{r}{3872179} && \multicolumn{2}{r}{968045} &&
\multicolumn{2}{r}{4.79} && \multicolumn{2}{r}{0.002} &\\

%& \multicolumn{2}{c}{$\;\;AC$} &&& \multicolumn{1}{l}{2} &&
%\multicolumn{2}{r}{2985591} && \multicolumn{2}{r}{1492796} &&
%\multicolumn{2}{r}{7.38} && \multicolumn{2}{r|}{0.001}\\

%& \multicolumn{2}{c}{$\;\;AC{}^{2}$} &&& \multicolumn{1}{l}{2} &&
%\multicolumn{2}{r}{886587} && \multicolumn{2}{r}{443294} &&
%\multicolumn{2}{r}{2.19} && \multicolumn{2}{r|}{0.122} \\

\multicolumn{2}{|c}{$B \times C$} && \multicolumn{2}{r}{4} &&
\multicolumn{2}{r}{448348} && \multicolumn{2}{r}{112087} &&
\multicolumn{2}{r}{0.55} && \multicolumn{2}{r}{0.697} & \\

%& \multicolumn{2}{c}{$\;\;BC$} &&& \multicolumn{1}{l}{2} &&
%\multicolumn{2}{r}{427214} && \multicolumn{2}{r}{213607} &&
%\multicolumn{2}{r}{1.06} && \multicolumn{2}{r|}{0.355}\\

%& \multicolumn{2}{c}{$\;\;BC^2$} &&& \multicolumn{1}{l}{2} &&
%\multicolumn{2}{r}{21134} && \multicolumn{2}{r}{10567} &&
%\multicolumn{2}{r}{0.05} && \multicolumn{2}{r|}{0.949} \\

\multicolumn{2}{|c}{$A \times B \times C$} &&
\multicolumn{2}{r}{8} && \multicolumn{2}{r}{5206919} &&
\multicolumn{2}{r}{650865} &&
\multicolumn{2}{r}{3.22} && \multicolumn{2}{r}{0.005} &\\

%& \multicolumn{2}{c}{$\;\;ABC$} &&& \multicolumn{1}{l}{2} &&
%\multicolumn{2}{r}{4492927} && \multicolumn{2}{r}{2246464} &&
%\multicolumn{2}{r}{11.11} && \multicolumn{2}{r|}{0.000} \\

%& \multicolumn{2}{c}{$\;\;ABC^2$} &&& \multicolumn{1}{l}{2} &&
%\multicolumn{2}{r}{263016} && \multicolumn{2}{r}{131508} &&
%\multicolumn{2}{r}{0.65} && \multicolumn{2}{r|}{0.526} \\

%& \multicolumn{2}{c}{$\;\;AB^2C$} &&& \multicolumn{1}{l}{2} &&
%\multicolumn{2}{r}{205537} && \multicolumn{2}{r}{102768} &&
%\multicolumn{2}{r}{0.51} && \multicolumn{2}{r|}{0.605} \\

%& \multicolumn{2}{c}{$\;\;AB^2C^2$} && & \multicolumn{1}{l}{2} &&
%\multicolumn{2}{r}{245439} && \multicolumn{2}{r}{122720} &&
%\multicolumn{2}{r}{0.61} && \multicolumn{2}{r|}{0.549} \\


\multicolumn{2}{|c}{residual} && \multicolumn{2}{r}{54} &&
\multicolumn{2}{r}{10922599} && \multicolumn{2}{r}{202270} &&&&&&&\\

\multicolumn{2}{|c}{total} && \multicolumn{2}{r}{80} &&
\multicolumn{2}{r}{68858056} && &&&&&&&& \\

\hline
\end{tabular}
\label{tab:3p3anova}
\end{table}

\es
%---------------------------------------------------------------------
% 9--------------------------------------------------------------------

\bs
  \heading{Orthogonal Components System: Decomposition of $A \times B$
Interaction}

    \bi
   \item $A \times B$ has 4 degrees of freedom.
   \item $A \times B$ has two components denoted by $AB$
and $AB^{2}$, each having 2 df.
    \item Let the levels of $A$ and $B$ be denoted by $x_1$ and $x_2$
respectively.
    \item $AB$ represents the contrasts among the response values
whose $x_{1}$ and $x_{2}$ satisfy
\begin{eqnarray*}
x_{1}+x_{2} =0,1,2 (mod \ 3) ,
\end{eqnarray*}
    \item $AB^2$ represents the contrasts among the response values
whose $x_{1}$ and $x_{2}$ satisfy
\begin{eqnarray*}
x_{1}+2x_{2} =0,1,2 (mod \ 3).
\end{eqnarray*}
  \ei
 \es

%---------------------------------------------------------------------
% 10--------------------------------------------------------------------

\bs
  \heading{Orthogonal Components System: Decomposition of $A \times B
\times C$ Interaction}

    \bi
   \item $A \times B \times C$ has 8 degrees of freedom.
   \item It can be further split up into four components denoted by
   $ABC$, $ABC^2$, $AB^2C$ and $AB^2C^2$, each having 2 df.
    \item Let the levels of $A$, $B$ and $C$ be denoted by $x_1$, $x_2$
and $x_3$ respectively.
    \item $ABC$, $ABC^2$, $AB^2C$ and $AB^2C^2$ represent the contrasts
among the three groups of $(x_1, x_2, x_3)$
satisfying each of the four systems of equations,
\begin{eqnarray*}
x_1 + x_2 +  x_3 &=& 0,1,2 (mod 3),\\
x_1 + x_2 + 2x_3 &=& 0,1,2 (mod 3),\\
x_1 + 2x_2 + x_3 &=& 0,1,2 (mod 3),\\
x_1 + 2x_2 + 2x_3 &=& 0,1,2 (mod 3).\\
\end{eqnarray*}
  \ei
 \es

%---------------------------------------------------------------------
% 11--------------------------------------------------------------------

\bs
  \heading{Uniqueness of Representation}

    \bi
\item To avoid ambiguity, {\em the convention that the coefficient
for the first nonzero factor is 1 will be used}.

\item $ABC^{2}$ is used instead of $A^{2}B^{2}C$, even though the
two are equivalent.

\item For $A^{2}B^{2}C$, there are three groups satisfying

\begin{eqnarray*}
2x_{1} + 2x_{2} + x_{3} &=& 0,1,2 (mod 3),\\
\mbox{equivalently, \hspace{0.2 in}} 2\times(2x_{1} + 2x_{2} + x_{3}) &=&
2\times (0,1,2) (mod 3),\\
\mbox{equivalently, \hspace{0.6 in}} x_{1} + x_{2} + 2x_{3} &=& 0,2,1 (mod
3),\\
\end{eqnarray*}
which corresponds to $ABC^2$ by relabeling of the groups. Hence
$ABC^2$ and $A^2B^2C$ are \emph{equivalent}.

  \ei
 \es

%---------------------------------------------------------------------
% 12--------------------------------------------------------------------

\bs
  \heading{Representation of $AB$ and $AB^2$}

\begin{table}[htbp]
\caption{Factor $A$ and $B$ Combinations ($x_1$ denotes the levels
of factor A and $x_2$ denotes the levels of factor B)} $ $
\centering \footnotesize
\begin{tabular}{|r|c|c|c|} \hline
\multicolumn{1}{|c|}{}& \multicolumn{3}{c|}{$x_{2}$}\\[-6pt]
\multicolumn{1}{|c|}{$x_{1}$}& \multicolumn{1}{c}{0}&
\multicolumn{1}{c}{1}& \multicolumn{1}{c|}{2}\\ \hline
 0 & $\alpha i \; (y_{00})$
                        & $\beta k  \; (y_{01})$
                        & $\gamma j \; (y_{02})$ \\
1 & $\beta j  \; (y_{10})$
                        & $\gamma i \; (y_{11})$
                        & $\alpha k \; (y_{12})$ \\
2 & $\gamma k \; (y_{20})$
                        & $\alpha j \; (y_{21})$
                        & $\beta i  \; (y_{22})$ \\ \hline
\end{tabular}
\label{tab:3by3}
\end{table}

\bi

\item $\alpha,\beta,\gamma$ correspond to $(x_1,x_2)$ with
$x_1+x_2=0,1,2 (mod 3)$ resp.

\item $i,j,k$ correspond to $(x_1,x_2)$ with $x_1+2x_2=0,1,2
(mod3)$ resp.

\ei

\es

%---------------------------------------------------------------------
% 12a-------------------------------------------------------------------

\bs

\heading{Connection with Graeco-Latin Square}

\bi

\item In Table 5, $(\alpha,\beta,\gamma)$ forms a Latin Square and
$(i,j,k)$ forms another Latin Square.

\item $(\alpha,\beta,\gamma)$ and $(i,j,k)$ jointly form a
Graeco-Latin Square. This implies that SS for
$(\alpha,\beta,\gamma)$ and SS for $(i,j,k)$ are
\emph{orthogonal}.

\item $SS_{AB} = 3 n [
 {( \bar{y}_{\alpha} - \bar{y}_{\cdot} )}^2
 + {( \bar{y}_{\beta} - \bar{y}_{\cdot} )}^2
 + {( \bar{y}_{\gamma} - \bar{y}_{\cdot} )}^2 ], \nonumber$
\\
where $\bar{y}_{\cdot} =  (\bar{y}_{\alpha} +
\bar{y}_{\beta} +  \bar{y}_{\gamma})/3$ and $n$ is the number of
replicates,

$\bar{y}_{\alpha} =\frac{1}{3}(y_{00}+y_{12}+y_{21})$, etc.

\item Similarly, $SS_{AB^2}=3n[{( \bar{y}_i - \bar{y}_{\cdot}
)}^2+ {( \bar{y}_j - \bar{y}_{\cdot} )}^2 + {( \bar{y}_k -
\bar{y}_{\cdot} )}^2 ]$.

\ei

\es


%---------------------------------------------------------------------
% 13--------------------------------------------------------------------

\bs
  \heading{Analysis using the Orthogonal components system}

\bi

 \item For the simplified seat-belt experiment,
$\bar{y}_{\alpha}=6024.407$, $\bar{y}_{\beta}=6177.815$ and
$\bar{y}_{\gamma}=6467.0$, so that $\bar{y}_{\cdot} = 6223.074$
and
\begin{eqnarray*}
\begin{array}{lll}
SS_{AB} &=&  (3)(9) [
 (6024.407- 6223.074)^2 +
 (6177.815- 6223.074)^2 \\
&& + (6467.0- 6223.074)^2
  ]
= 2727451 .
\end{array}
\end{eqnarray*}

\item Similarly, $SS_{AB^2}=570795$.

\item See ANOVA table on the next page.


  \ei
 \es



%---------------------------------------------------------------------
% 14--------------------------------------------------------------------

\bs
  \heading{ANOVA : Simplified Seat-Belt Experiment}

  \begin{table}[htbp]
%\caption{ANOVA Table, Simplified Seat-Belt Experiment}
$ $\\
\centering \tiny
\begin{tabular}{|lccrrrrrrrrrrrrrrr|} \hline
\multicolumn{3}{|c}{}& %&
\multicolumn{3}{c}{Degrees of} & %&
\multicolumn{3}{c}{Sum of} & %&
\multicolumn{3}{c}{Mean} & %&
\multicolumn{3}{c}{} &
\multicolumn{3}{c|}{}  \\
%\multicolumn{13}{|c|}{} \\
\multicolumn{3}{|c}{Source}& %&
\multicolumn{3}{c}{Freedom} & %&
\multicolumn{3}{c}{Squares} & %&
\multicolumn{3}{c}{Squares} & %&
\multicolumn{3}{c}{$F$} & %&
\multicolumn{3}{c|}{p-value} \\ \hline
%& \multicolumn{1}{c|}{} \\ \hline

\multicolumn{2}{|c}{$A$} && \multicolumn{2}{r}{2} &&
\multicolumn{2}{r}{34621746} && \multicolumn{2}{r}{17310873} &&
\multicolumn{2}{r}{85.58} && \multicolumn{2}{r}{0.000} & \\

\multicolumn{2}{|c}{$B$} && \multicolumn{2}{r}{2} &&
\multicolumn{2}{r}{938539} && \multicolumn{2}{r}{469270} &&
\multicolumn{2}{r}{2.32} && \multicolumn{2}{r}{0.108} & \\

\multicolumn{2}{|c}{$C$} && \multicolumn{2}{r}{2} &&
\multicolumn{2}{r}{9549481} && \multicolumn{2}{r}{4774741} &&
\multicolumn{2}{r}{23.61} && \multicolumn{2}{r}{0.000} & \\

\multicolumn{2}{|c}{$A \times B$} && \multicolumn{2}{r}{4} &&
\multicolumn{2}{r}{3298246} && \multicolumn{2}{r}{824561} &&
\multicolumn{2}{r}{4.08} && \multicolumn{2}{r}{0.006} & \\

\multicolumn{2}{|c}{$\;\;AB$} &&& \multicolumn{1}{l}{2} &&
\multicolumn{2}{r}{2727451} && \multicolumn{2}{r}{1363725} &&
\multicolumn{2}{r}{6.74} && \multicolumn{2}{r}{0.002} &\\

\multicolumn{2}{|c}{$\;\;AB{}^{2}$} &&& \multicolumn{1}{l}{2} &&
\multicolumn{2}{r}{570795} && \multicolumn{2}{r}{285397} &&
\multicolumn{2}{r}{1.41} && \multicolumn{2}{r}{0.253} &\\

\multicolumn{2}{|c}{$A \times C$} && \multicolumn{2}{r}{4} &&
\multicolumn{2}{r}{3872179} && \multicolumn{2}{r}{968045} &&
\multicolumn{2}{r}{4.79} && \multicolumn{2}{r}{0.002} &\\

\multicolumn{2}{|c}{$\;\;AC$} &&& \multicolumn{1}{l}{2} &&
\multicolumn{2}{r}{2985591} && \multicolumn{2}{r}{1492796} &&
\multicolumn{2}{r}{7.38} && \multicolumn{2}{r}{0.001} &\\

\multicolumn{2}{|c}{$\;\;AC{}^{2}$} &&& \multicolumn{1}{l}{2} &&
\multicolumn{2}{r}{886587} && \multicolumn{2}{r}{443294} &&
\multicolumn{2}{r}{2.19} && \multicolumn{2}{r}{0.122} &\\

\multicolumn{2}{|c}{$B \times C$} && \multicolumn{2}{r}{4} &&
\multicolumn{2}{r}{448348} && \multicolumn{2}{r}{112087} &&
\multicolumn{2}{r}{0.55} && \multicolumn{2}{r}{0.697} & \\

\multicolumn{2}{|c}{$\;\;BC$} &&& \multicolumn{1}{l}{2} &&
\multicolumn{2}{r}{427214} && \multicolumn{2}{r}{213607} &&
\multicolumn{2}{r}{1.06} && \multicolumn{2}{r}{0.355} &\\

\multicolumn{2}{|c}{$\;\;BC^2$} &&& \multicolumn{1}{l}{2} &&
\multicolumn{2}{r}{21134} && \multicolumn{2}{r}{10567} &&
\multicolumn{2}{r}{0.05} && \multicolumn{2}{r}{0.949} &\\

\multicolumn{2}{|c}{$A \times B \times C$} &&
\multicolumn{2}{r}{8} && \multicolumn{2}{r}{5206919} &&
\multicolumn{2}{r}{650865} &&
\multicolumn{2}{r}{3.22} && \multicolumn{2}{r}{0.005} &\\

\multicolumn{2}{|c}{$\;\;ABC$} &&& \multicolumn{1}{l}{2} &&
\multicolumn{2}{r}{4492927} && \multicolumn{2}{r}{2246464} &&
\multicolumn{2}{r}{11.11} && \multicolumn{2}{r}{0.000} & \\

\multicolumn{2}{|c}{$\;\;ABC^2$} &&& \multicolumn{1}{l}{2} &&
\multicolumn{2}{r}{263016} && \multicolumn{2}{r}{131508} &&
\multicolumn{2}{r}{0.65} && \multicolumn{2}{r}{0.526} &\\


\multicolumn{2}{|c}{$\;\;AB^2C$} &&& \multicolumn{1}{l}{2} &&
\multicolumn{2}{r}{205537} && \multicolumn{2}{r}{102768} &&
\multicolumn{2}{r}{0.51} && \multicolumn{2}{r}{0.605} &\\

\multicolumn{2}{|c}{$\;\;AB^2C^2$} && & \multicolumn{1}{l}{2} &&
\multicolumn{2}{r}{245439} && \multicolumn{2}{r}{122720} &&
\multicolumn{2}{r}{0.61} && \multicolumn{2}{r}{0.549} &\\


\multicolumn{2}{|c}{residual} && \multicolumn{2}{r}{54} &&
\multicolumn{2}{r}{10922599} && \multicolumn{2}{r}{202270} &&&&&&&\\

\multicolumn{2}{|c}{total} && \multicolumn{2}{r}{80} &&
\multicolumn{2}{r}{68858056} && &&&&&&&& \\
\hline
\end{tabular}
\label{tab:3p3anova}
\end{table}

\es




%---------------------------------------------------------------------
% 15--------------------------------------------------------------------

\bs
  \heading{Analysis of Simplified Seat-Belt Experiment (contd)}

\bi
    \item The significant main effects are $A$ and $C$.
    \item Among the interactions, $A\times B$, $A\times C$ and $A\times B \times
    C$ are significant.
    \item We have difficulty in interpretations when only one component of the interaction terms become significant.
    What is meant by ``$A\times B$ is significant''?
        \bi
        \item Here $AB$ is significant but $AB^2$ is not.
        \item Is $A\times B$ significant because of the
        significance of $AB$ alone ?
        \item For the original Seat-Belt Experiment, we have
        $AB=CD^2$.
        \ei
    \item Similarly, $AC$ is significant, but not $AC^2$. How to
    interpret the significance of $A\times C$ ?
    \item This difficulty in interpreting the significant interaction
    effects can be avoided by using Linear-Quadratic Systems.
\ei

\es



%---------------------------------------------------------------------
% 16--------------------------------------------------------------------

\bs
  \heading{Why three-level fractional factorial ? }

\bi \item Run size economy : it is not economical to use a $3^4$
design with 81 runs unless the experiment is not costly.

\item If a $3^4$ design is used for the experiment, its 81 degrees
of freedom would be allocated as follows:

\begin{table}[htbp]
%\caption{Number of Effects in a $2^5$ Design}
\centering
\begin{tabular}{|ccccc|} \hline
\multicolumn{1}{|c}{} & \multicolumn{1}{c}{Main} &
\multicolumn{3}{c|}{Interactions} \\
\multicolumn{1}{|c}{ } & \multicolumn{1}{c}{Effects} &
\multicolumn{1}{c}{2-Factor} & \multicolumn{1}{c}{3-Factor} &
\multicolumn{1}{c|}{4-Factor} \\
\hline
\# & 8 & 24 & 32 & 16 \\
\hline
\end{tabular}
\large\normalsize \label{tab:estfull}
\end{table}

\item Using effect hierarchy principle, one would argue that 3fi's
and 4fi's are not likely to be important. Out of a total of 80 df,
48 correspond to such effects !

\ei \es

%---------------------------------------------------------------------
% 17--------------------------------------------------------------------

\bs
  \heading{Defining a $3^{4-1}$ Experiment}

\bi
 \item Returning to the original seat-belt experiment, it employs
a one-third fraction of the $3^4$ design. This is denoted as a
$3^{4-1}$ design.

\item The design is constructed by choosing the column for factor
$D$ (lot \#) to be equal to ${\rm Column}\; A \;+\; {\rm Column}\;
B\; +\; {\rm Column}\; C (mod \ 3)$.

\item This relationship can be represented by the notation
\begin{eqnarray*}
D= A B C.
\end{eqnarray*}

\item If $x_{1}, \ldots, x_{4}$ are used to represent these four
columns, then
\begin{eqnarray}
 x_{4}=x_{1}+x_{2}+x_{3} (mod \ 3), \mbox{or equivalently} \nonumber\\
 x_{1}+x_{2}+x_{3}+2x_{4}=0(mod \ 3), \label{eqn:groupdef}
\end{eqnarray}
which can be represented by
\begin{eqnarray*}
{\bf I}=ABCD^{2}. \label{eqn:defrel}
\end{eqnarray*}

\ei \es

%---------------------------------------------------------------------
% 18--------------------------------------------------------------------
\bs
  \heading{Aliasing Patterns of the Seat-Belt Experiment}

\bi
 \item The aliasing patterns
can be deduced from the defining relation. For example, by adding
$2 x_1$ to both sides of (1), we have
\begin{eqnarray*}
2 x_1 = 3x_{1}+x_{2}+x_{3}+2x_{4}= x_{2}+x_{3}+2x_{4}(mod \ 3),
\end{eqnarray*}

\item This means that $A$ and $BCD^2$ are {\em aliased}. (Why?)

\item By following the same derivation, it is easy to show that
the following effects are aliased:

\tiny
\begin{equation}
\begin{array}{lllll}
A&=&BCD^{2}&=&AB^2C^2D,\\
B&=&ACD^{2}&=&AB^2CD^2, \\
C&=&ABD^{2}&=&ABC^2D^2, \\
D&=&ABC&=&ABCD,\\
AB&=&CD^{2}&=&ABC^2D, \\
AB^{2}&=&AC^{2}D&=&BC^2D, \\
AC&=&BD^{2}&=&AB^2CD, \\
AC^{2}&=&AB^{2}D &=& BC^2D^2, \\
AD&=&AB^{2}C^{2}&=&BCD, \\
AD^{2}&=&BC&=&AB^2C^2D^2, \\
BC^{2}&=&AB^{2}D^{2}&=&AC^2D^2, \\
BD&=&AB^{2}C&=&ACD, \\
CD&=&ABC^{2}&=&ABD.
\end{array}
\label{eqn:3fracalias}
\end{equation}

 \ei
\es

%---------------------------------------------------------------------
% 19--------------------------------------------------------------------

\bs
  \heading{Clear and Strongly Clear Effects}

\small

 \bi
 \item If three-factor interactions are assumed negligible, from the
aliasing relations in (2),
$A$, $B$, $C$, $D$, $AB^{2}$, $AC^{2}$, $AD$, $BC^{2}$, $BD$ and
$CD$ can be estimated.

\item These main effects or components of two-factor interactions
are called {\bf clear} because they are not aliased with any other
main effects or two-factor interaction components.

\item A two-factor interaction, say $A \times B$, is called {\em
clear} if both of its components, $AB$ and $AB^2$, are clear.

\item Note that each of the six two-factor interactions has only
one component that is clear; the other component is aliased with
one component of another two-factor interaction. For example, for
$A \times B$, $AB^{2}$ is clear but $AB$ is aliased with $CD^{2}$.

\item A main effect or two-factor interaction component is said to
be {\bf strongly clear} if it is not aliased with any other main
effects, two-factor or three-factor interaction components. A
two-factor interaction is said to be {\em strongly clear} if both
of its components are strongly clear.

\ei \es

%---------------------------------------------------------------------
% 20--------------------------------------------------------------------

\bs
  \heading{A $3^{5-2}$ Design}


 \bi
 \item 5 factors, 27 runs.
 \item The %$\frac{1}{9}$th
one-ninth fraction is defined by ${\bf I}=ABD^{2}=AB^{2}CE^{2}$,
from which two additional relations can be obtained:
\begin{eqnarray*}
{\bf I}=(ABD^{2}) (AB^{2}CE^{2})=A^{2}CD^{2}E^{2}=AC^{2}DE
\end{eqnarray*}
and
\begin{eqnarray*}
{\bf I}=(ABD^{2})(AB^{2}CE^{2})^{2}=B^{2}C^{2}D^{2}E=BCDE^{2}.
\end{eqnarray*}
%***explain why previous is same as $(ABD^{2})^2(AB^{2}CE^{2})$***
Therefore the defining contrast subgroup for this design consists
of the following defining relation:
\begin{eqnarray}
{\bf I}=ABD^{2}=AB^{2}CE^{2}=AC^{2}DE=BCDE^{2}.
\label{eqn:35m2defn1}
\end{eqnarray}

\ei \es





%---------------------------------------------------------------------
% 21 -----------------------------------------------------------------

\bs
  \heading{Resolution and Minimum Aberration}

 \bi

\item Let $A_{i}$ be to denote the number of words of length $i$
in the subgroup and $W=(A_{3},A_{4}, \ldots)$ to denote the
wordlength pattern.

\item Based on $W$, the definitions of {\bf resolution} and {\bf
minimum aberration} are the same as given before in Section~5.2.

\item The subgroup defined in (\ref{eqn:35m2defn1}) has four
words, whose lengths are 3, 4, 4, and 4. and hence $W=(1,3,0)$.
Another $3^{5-2}$ design given by $D=AB, E=AB^2$ has the defining
contrast subgroup,
\begin{eqnarray*}
{\bf I}=ABD^{2}=AB^2E^{2}=ADE =BDE^2, \label{eqn:35m2defn2}
\end{eqnarray*}
with the wordlength pattern $W=(4,0,0)$. According to the
aberration criterion, the first design has less aberration than
the second design.

\item Moreover, it can be shown that the first design has minimum
aberration.

\ei

\es





%---------------------------------------------------------------------
% 22 -----------------------------------------------------------------

\bs
  \heading{General $3^{k-p}$ Design}


 \bi

\item A $3^{k-p}$ design is a fractional factorial design with $k$
factors in $3^{k-p}$ runs.

\item It is a $3^{-p}$th fraction of the $3^{k}$ design.

\item The fractional plan is defined by $p$ independent
generators.

\item How many factors can a $3^{k-p}$ design study?
$$ (3^{n}-1)/2, \mbox{ where } n=k-p.$$
This design has $3^n$ runs with the independent generators $x_1$,
$x_2$, $\ldots$, $x_n$. We can obtain altogether $(3^n-1)/2$
orthogonal columns as different combinations of $\sum_{i=1}^n
\alpha_i x_i$ with $\alpha_i = $ 0, 1 or 2, where at least one
$\alpha_i$ should not be zero and the first nonzero $\alpha_i$
should be written as ``1'' to avoid duplication.

\item For $n$=3, the $(3^n-1)/2=13$ columns were given in
Table~6.5 of WH book.

\item A general algebraic treatment of $3^{k-p}$ designs can be
found in Kempthorne (1952).


\ei

\es

%---------------------------------------------------------------------
% 23 -----------------------------------------------------------------


\bs

\heading{Simple Analysis Methods: Plots and ANOVA}

\bi

\item Start with making a main effects plot and interaction plots
to see what effects might be important.

\item This step can be followed by a formal analysis like analysis
of variance and half-normal plots.

\ei The strength data will be considered first. The location main
effect and interaction plots are given in Figures~1 and 2. The
main effects plot suggests that factor $A$ is the most important
followed by factors $C$ and $D$. The interaction plots in
Figure~\ref{fig:strloc2fi} suggest that there may be interactions
because the lines are not parallel.

\es


%---------------------------------------------------------------------
% 24 -----------------------------------------------------------------

\bs \heading{Main Effects Plot of Strength Location}
\begin{figure}[htbp]
\centerline{\psfig{figure=picture/fig31.ps,height=2in,width=3.5in}}
\vspace{.04in} \caption{Main Effects Plot of Strength Location,
Seat-Belt Experiment}\label{fig:strlocme}
\end{figure}

\es



%---------------------------------------------------------------------
% 25 -----------------------------------------------------------------

\bs \heading{Interaction Plots of Strength Location}


\begin{figure}[htbp]
\centerline{\psfig{figure=picture/fig32.ps,height=2.75in,width=4in}}
 \caption{Interaction Plots of Strength Location, Seat-Belt Experiment}\label{fig:strloc2fi}
\end{figure}

\es




%---------------------------------------------------------------------
% 26 -----------------------------------------------------------------

\bs

\heading{ANOVA Table for Strength Location}

{\scriptsize
\begin{table}[htbp]
$ $\\
\centering
\begin{tabular}{|lrrrrr|} \hline
\multicolumn{1}{|c}{}& \multicolumn{1}{c}{Degrees of} &
\multicolumn{1}{c}{Sum of} & \multicolumn{1}{c}{Mean} &
\multicolumn{1}{c}{} &
\multicolumn{1}{c|}{} \\
\multicolumn{1}{|c}{Source}& \multicolumn{1}{c}{Freedom} &
\multicolumn{1}{c}{Squares} & \multicolumn{1}{c}{Squares} &
\multicolumn{1}{c}{$F$} & \multicolumn{1}{c|}{p-value} \\ \hline
$A$&2&34621746&17310873&85.58&0.000\\
$B$&2&938539&469270&2.32&0.108\\
$AB=CD{}^{2}$&2&2727451&1363725&6.74&0.002\\
$AB{}^{2}$&2&570795&285397&1.41&0.253\\
$C$&2&9549481&4774741&23.61&0.000\\
$AC=BD{}^{2}$&2&2985591&1492796&7.38&0.001\\
$AC^2$&2&886587&443294&2.19&0.122\\
$BC=AD{}^{2}$&2&427214&213607&1.06&0.355\\
$BC{}^{2}$&2&21134&10567&0.05&0.949\\
$D$&2&4492927&2246464&11.11&0.000\\
$AD$&2&263016&131508&0.65&0.526\\
$BD$&2&205537&102768&0.51&0.605\\
$CD$&2&245439&122720&0.61&0.549\\
residual&54&10922599&202270&&\\ \hline
\end{tabular}
\label{tab:strengthanova}
\end{table}
}

\es




%---------------------------------------------------------------------
% 27 -----------------------------------------------------------------


\bs


\heading{Analysis of Strength Location, Seat-Belt Experiment}

\bi

\item In equation (\ref{eqn:3fracalias}), the 26 degrees of
freedom in the experiment were grouped into 13 sets of effects.
The corresponding ANOVA table gives the SS values for these 13
effects.

\item Based on the p values in the ANOVA Table, clearly the factor
$A$, $C$ and $D$ main effects are significant.

\item Also two aliased sets of effects are significant,
$AB=CD{}^{2}$ and $AC=BD{}^{2}$.

\item These findings are consistent with those based on the main
effects plot and interaction plots. In particular, the
significance of $AB$ and $CD^2$ is supported by the $A \times B$
and $C \times D$ plots and the significance of $AC$ and $BD^2$ by
the $A \times C$ and $B \times D$ plots.

\ei

\es



%---------------------------------------------------------------------
% 28 -----------------------------------------------------------------


\bs


\heading{Analysis of Strength Dispersion (i.e., $\ln s^2$) Data}


The corresponding strength main effects plot and interaction plots
are displayed in Figures~\ref{fig:strdispme} and
\ref{fig:strdisp2fi}.


\begin{figure}[htbp]
\centerline{\psfig{figure=picture/fig31A.ps,height=2in,width=4in}}
 \caption{Main Effects Plot of Strength Dispersion, Seat-Belt Experiment} \label{fig:strdispme}
\end{figure}

\es





%---------------------------------------------------------------------
% 29 -----------------------------------------------------------------


\bs

\heading{Interaction Plots of Strength Dispersion}

\begin{figure}[htbp]
\centerline{\psfig{figure=picture/fig32A.ps,height=2.75in,width=4in}}
 \caption{Interaction Plots of Strength Dispersion, Seat-Belt Experiment} \label{fig:strdisp2fi}
\end{figure}

\es





%---------------------------------------------------------------------
% 30 -----------------------------------------------------------------


\bs

\heading{Half-Normal Plots}

\bi

\item Since there is no replication for the dispersion analysis,
ANOVA cannot be used to test effect significance.

\item Instead, a half-normal plot can be drawn as follows. The 26
df's can be divided into 13 groups, each having two df's. These 13
groups correspond to the 13 rows in the ANOVA table of page 27.

\item The two degrees of freedom in each group can be decomposed
further into a linear effect and a quadratic effect with the
contrast vectors $\frac{1}{\sqrt{2}}(-1,0,1)$ and
$\frac{1}{\sqrt{6}}(1,-2,1)$, respectively, where the values in
the vectors are associated with the $\ln s^2$ values at the levels
(0, 1, 2) for the group.

\item Because the linear and quadratic effects are standardized
and orthogonal to each other, these 26 effect estimates can be
plotted on the half-normal probability scale as in
Figure~\ref{fig:strdisphnp}.

\ei

\es



%---------------------------------------------------------------------
% 31 -----------------------------------------------------------------


\bs

\heading{Half-Normal Plot}

\begin{figure}[htbp]
\centerline{\psfig{figure=picture/strdisphnp.ps,height=3in,width=2in,angle=-90}}
\caption{Half-Normal Plot of Strength Dispersion Effects,
Seat-Belt Experiment} \label{fig:strdisphnp}
\end{figure}


Informal analysis of the plot suggests that the factor $A$ linear
effect may be significant. This can be confirmed by using Lenth's
method. The $t_{PSE}$ value for the $A$ linear effect is 3.99,
which has a p value of 0.003(IER) and 0.050 (EER).

\es




%---------------------------------------------------------------------
% 32 -----------------------------------------------------------------


\bs

\heading{Analysis Summary}

\bi

\item A similar analysis can be performed to identify the flash
location and dispersion effects. See Section 6.5 of WH book.

\item We can determine the optimal factor settings that maximize
the \emph{strength location} by examining the main effects plot
and interaction plots in Figures~\ref{fig:strlocme} and
\ref{fig:strloc2fi} that correspond to the significant effects
identified in the ANOVA table.


\item We can similarly determine the optimal factor settings that
minimize the \emph{strength dispersion}, \emph{the flash location}
and \emph{flash dispersion}, respectively.

\item The most obvious findings: level 2 of factor $A$ be chosen
to maximize strength while level 0 of factor $A$ be chosen to
minimize flash.

\item There is an obvious conflict in meeting the two objectives.
Trade-off strategies for handling multiple characteristics and
conflicting objectives need to be considered (See Section 6.7 of
WH).

\ei

\es





%---------------------------------------------------------------------
% 33 -----------------------------------------------------------------



\bs

\heading{An Alternative Analysis Method : Linear-Quadratic System}



In the seat-belt experiment, the factors $A, B$ and $C$ are
quantitative. The two degrees of freedom in a quantitative factor,
say $A$, can be decomposed into the linear and quadratic
components.

Letting $y_{0}$, $y_{1}$ and $y_{2}$ represent the observations at
level 0, 1 and 2, then the {\em linear effect} is defined as
\begin{eqnarray*} y_{2}-y_{0} \end{eqnarray*}
and the {\em quadratic effect} as
\begin{eqnarray*}(y_{2}+y_{0})-2y_{1}, \end{eqnarray*}
which can be re-expressed as the difference between two
consecutive linear effects $(y_{2}-y_{1})-(y_{1}-y_{0})$.


\es



%---------------------------------------------------------------------
% 34 -----------------------------------------------------------------



\bs

\heading{Linear and Quadratic Effects}


Mathematically, the linear and quadratic effects are represented
by two mutually orthogonal vectors:

\begin{equation}
\begin{array}{lll}
A_{l} & = & \frac{1}{\sqrt{2}}(-1,0,1), \\
A_{q} & = &\frac{1}{\sqrt{6}}(1,-2,1).
\end{array}
\label{eqn:linquad}
\end{equation}

\bi

\item For the sake of brevity, they are also referred to as the
$l$ and $q$ effects.

\item The scaling constants $\sqrt{2}$ and $\sqrt{6}$ yield
vectors with unit length.

\item The linear (or quadratic) effect is obtained by taking the
inner product between $A_{l}$ (or $A_{q}$) and the vector $ {\bf
y}=(y_{0}, y_{1}, y_{2})$. For factor $B, B_{l}$ and $B_{q}$ are
similarly defined.


\ei

\es



%---------------------------------------------------------------------
% 35 -----------------------------------------------------------------



\bs

\heading{Linear and Quadratic Effects (contd)}


\bi

\item Then the four degrees of freedom in the $A \times B$
interaction can be decomposed into four mutually orthogonal terms:
$(AB)_{ll}, (AB)_{lq}, (AB)_{ql}, (AB)_{qq}$, which are defined as
follows: for $i,j=0,1,2$,
\begin{equation}
\begin{array}{lll}
(AB)_{ll}(i,j)&=&A_{l}(i)B_{l}(j), \\
(AB)_{lq}(i,j)&=&A_{l}(i)B_{q}(j), \\
(AB)_{ql}(i,j)&=&A_{q}(i)B_{l}(j),  \\
(AB)_{qq}(i,j)&=&A_{q}(i)B_{q}(j).
\end{array}
\label{eqn:ab}
\end{equation}
They are called the {\bf linear-by-linear}, {\bf
linear-by-quadratic}, {\bf quadratic-by-linear} and {\bf
quadratic-by-quadratic} interaction effects. They are also
referred to as the $l \times l$, $l \times q$, $q \times l$ and $q
\times q$ effects.

\item It is easy to show that they are orthogonal to each other.

\ei


\es



%---------------------------------------------------------------------
% 36 -----------------------------------------------------------------



\bs

\heading{Linear and Quadratic Effects (contd)}

Using the nine level combinations of factors $A$ and $B$,
$y_{00},\ldots, y_{22}$ given in Table~\ref{tab:3by3}, the
contrasts $(AB)_{ll}$, $(AB)_{lq}$, $(AB)_{ql}$, $(AB)_{qq}$ can
be expressed as follows:

$(AB)_{ll}$: $\frac{1}{2} \{ (y_{22}-y_{20})-(y_{02}-y_{00})\},$

\vspace{.2cm} $(AB)_{lq}$:
 $\frac{1}{2\sqrt{3}}\{(y_{22}+y_{20}-
2y_{21})-(y_{02}+y_{00}-2y_{01})\},$

\vspace{.2cm} $(AB)_{ql}$: $\frac{1}{2\sqrt{3}}\{(y_{22}+y_{02}-
2y_{12})-(y_{20}+y_{00}-2y_{10})\},$

\vspace{.2cm} $(AB)_{qq}$: $\frac{1}{6}\{(y_{22}+y_{20}-
2y_{21})-2(y_{12}+y_{10}-2y_{11})+(y_{02}+y_{00}-2y_{01})\}.$



\bi


\item An $(AB)_{ll}$ interaction effect measures the difference
between the conditional linear $B$ effects at levels 0 and 2 of
factor $A$.

\item A significant $(AB)_{ql}$ interaction effect means that
there is curvature in the conditional linear $B$ effect over the
three levels of factor $A$.

\item The other interaction effects $(AB)_{lq}$ and $(AB)_{qq}$
can be similarly interpreted.

\ei

\es


%---------------------------------------------------------------------
% 37 -----------------------------------------------------------------



\bs

\heading{Analysis of designs with resolution at least $V$}

\bi

\item For designs of at least resolution V, all the main effects
and two-factor interactions are clear. Then, further decomposition
of these effects according to the linear-quadratic system allows
all the effects (each with one degree of freedom) to be compared
in a half-normal plot.

\item Note that for effects to be compared in a half-normal plot,
they should be uncorrelated and have the same variance.

\ei


\es


%---------------------------------------------------------------------
% 38 -----------------------------------------------------------------



\bs

\heading{Analysis of designs with resolution smaller than $V$}

\bi

\item For designs with resolution $III$ or $IV$, a more elaborate
analysis method is required to extract the maximum amount of
information from the data.

\item Consider the $3^{3-1}$ design with $C=AB$ whose design
matrix is given in Table~\ref{tab:33m1}. {\tiny
\begin{center}
\begin{table}[htbp]
\caption{Design Matrix for the $3^{3-1}$ Design}
\begin{tabular}{|r|rrr|} \hline
Run & $A$ & $B$ & $C$ \\ \hline
1 & 0 & 0 & 0 \\
2 & 0 & 1 & 1 \\
3 & 0 & 2 & 2 \\
4 & 1 & 0 & 1 \\
5 & 1 & 1 & 2 \\
6 & 1 & 2 & 0 \\
7 & 2 & 0 & 2 \\
8 & 2 & 1 & 0 \\
9 & 2 & 2 & 1 \\ \hline
\end{tabular}
\label{tab:33m1}
\end{table}
\end{center}
}

\item Its main effects and two-factor interactions have the
aliasing relations:
\begin{eqnarray}
A=BC^{2}, B=AC^{2}, C={AB}, AB^{2}=BC=AC. \label{eqn:33m1alias}
\end{eqnarray}

\ei


\es


%---------------------------------------------------------------------
% 39 -----------------------------------------------------------------



\bs

\heading{Analysis of designs with resolution $III$ (contd)}

\bi

\item In addition to estimating the six degrees of freedom in the
main effects $A$, $B$ and $C$, there are two degrees of freedom
left for estimating the three aliased effects $AB^{2}$, $BC$ and
$AC$, which, as discussed before, are difficult to interpret.

\item Instead, consider using the remaining two degrees of freedom
to estimate any pair of the $l \times l, l \times q, q \times l$
or $q \times q$ effects between $A,B$ and $C$.

\item Suppose that the two interaction effects taken are
$(AB)_{ll}$ and $(AB)_{lq}$. Then the eight degrees of freedom can
be represented by the model matrix given in
Table~\ref{tab:3linquad}.

\begin{table}[htbp]
\caption{A System of Contrasts for the $3^{3-1}$ Design}
\centering\tiny
\begin{tabular}{|c|rrrrrrrr|} \hline
Run & $A_{l}$ & $A_{q}$ & $B_{l}$ & $B_{q}$ & $C_{l}$ & $C_{q}$ &
$(AB)_{ll}$ & $(AB)_{lq}$ \\ \hline
1 & -1 & 1 & -1 & 1 & -1 & 1 & 1 & -1 \\
2 & -1 & 1  & 0 &-2 & 0 & -2 & 0 & 2 \\
3 & -1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 \\
4 & 0 & -2 & -1 & 1 & 0 & -2 & 0 & 0\\
5 & 0 & -2 & 0 & -2 & 1 & 1 & 0 & 0 \\
6 & 0 & -2 & 1 & 1 & -1 & 1 & 0 & 0 \\
7 & 1 & 1 & -1 & 1 & 1 & 1 & -1 & 1 \\
8 & 1 & 1 & 0 & -2 & -1 & 1 & 0 & -2 \\
9 & 1 & 1 & 1 & 1 & 0 & -2 & 1 & 1 \\ \hline
\end{tabular}
\label{tab:3linquad}
\end{table}

\ei


\es


%---------------------------------------------------------------------
% 40 -----------------------------------------------------------------



\bs

\heading{Analysis of designs with resolution $III$ (contd)}

\bi

\item Because any component of $A \times B$ is orthogonal to $A$
and to $B$, there are only four non-orthogonal pairs of columns
whose correlations are:
\begin{equation}
\begin{array}{lll}
Corr((AB)_{ll},C_{l})&=&-\sqrt{\frac{3}{8}},\\
Corr((AB)_{ll},C_{q})&=&-\frac{1}{\sqrt{8}}, \\
Corr((AB)_{lq},C_{l})&=& \;\;\; \frac{1}{\sqrt{8}}, \\
Corr((AB)_{lq},C_{q})&=&-\sqrt{\frac{3}{8}} .
\end{array}
\label{eqn:ABdefn}
\end{equation}

\item Obviously, $(AB)_{ll}$ and $(AB)_{lq}$ can be estimated in
addition to the three main effects.

\item Because the last four columns are not mutually orthogonal,
they cannot be estimated with full efficiency.

\item The estimability of  $(AB)_{ll}$ and $(AB)_{lq}$
demonstrates an advantage of the linear-quadratic system over the
orthogonal components system. For the same design, the $AB$
interaction component cannot be estimated because it is aliased
with the main effect $C$.

\ei


\es


%---------------------------------------------------------------------
% 41 -----------------------------------------------------------------



\bs

\heading{Analysis Strategy for Qualitative Factors}

\bi

\item For a qualitative factor like factor $D$ (lot number) in the
seat-belt experiment, the linear contrast $(-1,0,+1)$ may make
sense because it represents the comparison between levels 0 and 2.

\item On the other hand, the ``quadratic'' contrast $(+1,-2,+1)$,
which compares level 1 with the average of levels 0 and 2, makes
sense only if such a comparison is of practical interest. For
example, if levels 0 and 2 represent two internal suppliers, then
the ``quadratic'' contrast measures the difference between
internal and external suppliers.


\ei


\es


%---------------------------------------------------------------------
% 42 -----------------------------------------------------------------



\bs

\heading{Analysis Strategy for Qualitative Factors (contd)}

\bi

\item When the quadratic contrast makes no sense, two out of the
following three contrasts can be chosen to represent the two
degrees of freedom for the main effect of a qualitative factor:

{\footnotesize
\begin{eqnarray*} D_{01}=\left\{ \begin{array}{rlll}
-1 & & 0 & \\
1 & {\rm for \ level} & 1 & {\rm  of \ factor} \ D,\\
0 & & 2 & \end{array} \right. \end{eqnarray*}
\begin{eqnarray*} D_{02}=\left\{ \begin{array}{rlll}
-1 & & 0 & \\
0  & {\rm for \ level} & 1 & {\rm  of \ factor} \ D,\\
1 & & 2 & \end{array} \right. \end{eqnarray*}
\begin{eqnarray*} D_{12}=\left\{ \begin{array}{rlll}
0  & & 0 & \\
-1 & {\rm for \ level} & 1 & {\rm  of \ factor} \ D,\\
1 & & 2 & \end{array} \right. \label{eqn:D01D02D12}
\end{eqnarray*}
} \ei


\es


%---------------------------------------------------------------------
% 43 -----------------------------------------------------------------



\bs

\heading{Analysis Strategy for Qualitative Factors (contd)}

\vspace{0.25 in}


\bi


\item Mathematically, they are represented by the standardized
vectors:
$$D_{01} = \frac{1}{\sqrt{2}} (-1, 1, 0), D_{02} =
\frac{1}{\sqrt{2}} (-1, 0, 1), D_{12} = \frac{1}{\sqrt{2}} ( 0,
-1, 1) .$$



\item These contrasts are not orthogonal to each other and have
pairwise correlations of $1/2$ or $-1/2$.

\item On the other hand, each of them is readily interpretable as
a comparison between two of the three levels.

\item The two contrasts should be chosen to be of interest to the
investigator. For example, if level 0 is the main supplier and
levels 1 and 2 are minor suppliers, then $D_{01}$ and $D_{02}$
should be used.

\ei


\vspace{0.25 in}


\es


%---------------------------------------------------------------------
% 44 -----------------------------------------------------------------



\bs

\heading{Qualitative and Quantitative Factors}

\vspace{0.25 in}

\bi

\item The interaction between a quantitative factor and a
qualitative factor, say $A \times D$, can be decomposed into four
effects.

\item As in (\ref{eqn:ab}), we define the four interaction effects
as follows:
\begin{equation}
\begin{array}{l}
(AD)_{l,01}(i,j) = A_l (i) D_{01}(j), \\
(AD)_{l,02}(i,j) = A_l (i) D_{02}(j), \\
(AD)_{q,01}(i,j) = A_q (i) D_{01}(j), \\
(AD)_{q,02}(i,j) = A_q (i) D_{02}(j) .
\end{array}
\label{eqn:effectlqx0102}
\end{equation}

\ei

\vspace{0.5 in}

\es


%---------------------------------------------------------------------
% 45 -----------------------------------------------------------------



\bs

\heading{Variable Selection Strategy}

Since many of these contrasts are not mutually orthogonal, a
general purpose analysis strategy cannot be based on the
orthogonality assumption. Therefore, the following variable
selection strategy is recommended.

{\footnotesize
\begin{description}
\item [(i)] For a quantitative factor, say $A$, use $A_{l}$ and
$A_{q}$ for the $A$ main effect.

\item [(ii)] For a qualitative factor, say $D$, use $D_{l}$ and
$D_{q}$ if $D_{q}$ is interpretable; otherwise, select two
contrasts from $D_{01}, D_{02}$, and $D_{12}$ for the $D$ main
effect.

\item [(iii)] For a pair of factors, say $X$ and $Y$, use the
products of the two contrasts of $X$ and the two contrasts of $Y$
(chosen in (i) or (ii)) as defined in (\ref{eqn:ab}) or
(\ref{eqn:effectlqx0102}) to represent the four degrees of freedom
in the interaction $X \times Y$.

\item [(iv)] Using the contrasts defined in (i)-(iii) for all the
factors and their two-factor interactions as candidate variables,
perform a stepwise regression or subset selection procedure to
identify a suitable model. To avoid incompatible models, use the
effect heredity principle to rule out interactions whose parent
factors are both not significant.

\item [(v)] If all the factors are quantitative, use the original
scale, say $x_{A}$, to represent the linear effect of $A$,
$x_{A}^{2}$ the quadratic effect and $x_{A}^{i}x_{B}^{j}$ the
interaction between $x_{A}^{i}$ and $x_{B}^{j}$. This works
particularly well if some factors have unevenly spaced levels.

\end{description}
}

\es


%---------------------------------------------------------------------
% 45 -----------------------------------------------------------------



\bs

\heading{Analysis of Seat-Belt Experiment}

\bi

\item Returning to the seat-belt experiment, although the original
design has resolution IV, its capacity for estimating two-factor
interactions is much better than what the definition of resolution
IV would suggest.

\item After estimating the four main effects, there are still 18
degrees of freedom available for estimating some components of the
two-factor interactions.

\item From (\ref{eqn:3fracalias}), $A$, $B$, $C$ and $D$ are
estimable and only one of the two components in each of the six
interactions $A \times B$, $A \times C$, $A \times D$, $B \times
C$, $B \times D$ and $C \times D$ is estimable.

\item Because of the difficulty of providing a physical
interpretation of an interaction component, a simple and efficient
modeling strategy that does not throw away the information in the
interactions is to consider the contrasts $(A_{l}, A_{q})$,
$(B_{l}, B_{q})$, $(C_{l}, C_{q})$ and $(D_{01}, D_{02}, D_{12})$
for the main effects and the 30 products between these four groups
of contrasts for the interactions.

\ei

\es

%---------------------------------------------------------------------
% 46 -----------------------------------------------------------------



\bs

\heading{Analysis of Seat-Belt Experiment (contd)}

\bi


\item Using these 39 contrasts as the candidate variables, the
variable selection procedure was applied to the data.

\item Performing a stepwise regression on the strength data
(response $y_1$), the following model with an $R^2$ of 0.811 was
identified:
\begin{equation}
\begin{array}{lll}
\hat{y}_1 &=&  6223.0741 + 1116.2859 A_l  -190.2437 A_q + 178.6885 B_l \\
          & &   -589.5437 C_l +   294.2883 (AB)_{ql} +  627.9444 (AC)_{ll} \\
          & &  -191.2855 D_{01} -468.4190 D_{12} -486.4444 (CD)_{l,12}
\end{array}
\label{eqn:predstrengthloc}
\end{equation}

\item Note that this model obeys effect heredity. The $A$, $B$,
$C$ and $D$ main effects and $A \times B$, $A \times C$ and $C
\times D$ interactions are significant. In contrast, the simple
analysis from the previous section identified the $A$, $C$ and $D$
main effects and the $AC (= BD^2)$ and $AB (= CD^2)$ interaction
components as significant.

\ei

\es

%---------------------------------------------------------------------
% 47 -----------------------------------------------------------------



\bs

\heading{Analysis of Seat-Belt Experiment (contd)}

\bi

\item Performing a stepwise regression on the flash data (response
$y_2$), the following model with an $R^2$ of 0.857 was identified:
\begin{equation}
\begin{array}{lll}
\hat{y}_2 &=&   13.6657 +  1.2408 A_l + 0.1857 B_l \\
          & &    -0.8551 C_l +  0.2043 C_q   -0.9406 (AC)_{ll} \\
          & &    -0.3775 (AC)_{ql} -0.3765 (BC)_{qq}  -0.2978 (CD)_{l,12}
\end{array}
\label{eqn:predflashloc}
\end{equation}

\item Again, the identified model obeys effect heredity. The $A$,
$B$, and $C$ main effects and $A \times C$, $B \times C$ and $C
\times D$ interactions are significant. In contrast, the simple
analysis from the previous section identified the $A$ and $C$ main
effects and the $AC (= BD^2)$, $AC^2$ and $BC^2$ interaction
components as significant.

\ei

\es


%---------------------------------------------------------------------
% last ---------------------------------------------------------------

\bs
  \heading{Comments on Board}

 \vspace{3 in}
\es



\end{document}
