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\bs
  \heading{Unit 4: Full Factorial Experiments at Two Levels}

\bc Source : Chapter~4 (sections 4.1-4.4, 4.6-4.12, 4.15). \ec

\bi \small
 \item An Epitaxial Layer Growth Experiment (Section 4.1).
 \item Basic concepts for $2^k$ designs (Section 4.2).
 \item Factorial effects and plots (Section 4.3).
 \item Using Regression to Compute Factorial Effects (Section 4.4).
 \item Fundamental principles (Section 4.6).
 \item Comparisons with "one-factor-at-a-time" approach (Section 4.7).
 \item Normal and Half-normal plots for detecting effect significance (Section 4.8).
 \item Lenth's Method (Section 4.9).
 \item Nominal-the best problem, quadratic loss function (Section 4.10).
 \item Use of Log Sample Variance for Dispersion Analysis (Section 4.11).
 \item Analysis of Location and Dispersion: Epitaxial Growth
 Experiment (Section 4.12).
 \item Blocking in $2^k$ design (Section 4.15).
\ei
 \es
 \Large



%---------------------------------------------------------------------
%  2 -----------------------------------------------------------------

\bs
  \heading{Epitaxial Layer Growth Experiment}
\bi
 \item An AT\&T experiment based on $2^4$ design; four factors each at two
 levels. There are 6 replicates for each of the 16 (=$2^4$) level
 combinations; data given on the next page.
\begin{table}[htbp]
\caption{Factors and Levels, Adapted Epitaxial Layer Growth
Experiment}
\begin{center}
%\renewcommand{\baselinestretch}{1}
%\large\normalsize
\begin{tabular}{|ll|cc|} \hline
 \multicolumn{2}{|c}{} & \multicolumn{2}{|c|}{Level} \\[-12pt]
 \multicolumn{2}{|c}{Factor} & \multicolumn{1}{|c}{$-$} & \multicolumn{1}{c|}{$+$} \\ \hline
$A$.&susceptor-rotation method& continuous& oscillating \\
$B$.&nozzle position& 2 & 6\\
$C$.&deposition temperature ($^{\circ}$C)& 1210& 1220\\
$D$.&deposition time& low& high\\ \hline
\end{tabular}
\end{center}
%\renewcommand{\baselinestretch}{1.5}
%\large\normalsize
\label{tab:ex1factor}
\end{table}

\item {\bf Objective :} Reduce variation of $y$ (=layer thickness)
around its target 14.5 $\mu$m by changing factor level
combinations.
 \ei

 \es
%---------------------------------------------------------------------
%  3 -----------------------------------------------------------------

\bs
  \heading{Data from Epitaxial Layer Growth Experiment}


\begin{table}[htbp]
\caption{Design Matrix and Thickness Data, Adapted Epitaxial Layer
Growth Experiment}
$ $\\
%\begin{center}
%\renewcommand{\baselinestretch}{1}
%\large\normalsize
\centering\tiny
\begin{tabular}{|r|rrrr|rrrrrr|r|rr|} \hline
\multicolumn{1}{|c}{} & \multicolumn{4}{|c|}{Factor} &
\multicolumn{6}{c|}{} & \multicolumn{1}{c|}{} &
\multicolumn{1}{c}{} &
\multicolumn{1}{c|}{}\\
\multicolumn{1}{|c}{Run} & \multicolumn{1}{|c}{$A$} &
\multicolumn{1}{c}{$B$} & \multicolumn{1}{c}{$C$} &
\multicolumn{1}{c|}{$D$} & \multicolumn{6}{c|}{Thickness} &
\multicolumn{1}{c|}{$\bar{y}$} & \multicolumn{1}{c}{$s^2$} &
\multicolumn{1}{c|}{$\ln{s^2}$} \\ \hline
1&$-$&$-$&$-$&$+$&14.506&14.153&14.134&14.339&14.953&15.455&14.59&0.270&-1.309\\
2&$-$&$-$&$-$&$-$&12.886&12.963&13.669&13.869&14.145&14.007&13.59&0.291&-1.234\\
3&$-$&$-$&$+$&$+$&13.926&14.052&14.392&14.428&13.568&15.074&14.24&0.268&-1.317\\
4&$-$&$-$&$+$&$-$&13.758&13.992&14.808&13.554&14.283&13.904&14.05&0.197&-1.625\\
5&$-$&$+$&$-$&$+$&14.629&13.940&14.466&14.538&15.281&15.046&14.65&0.221&-1.510\\
6&$-$&$+$&$-$&$-$&14.059&13.989&13.666&14.706&13.863&13.357&13.94&0.205&-1.585\\
7&$-$&$+$&$+$&$+$&13.800&13.896&14.887&14.902&14.461&14.454&14.40&0.222&-1.505\\
8&$-$&$+$&$+$&$-$&13.707&13.623&14.210&14.042&14.881&14.378&14.14&0.215&-1.537\\
9&$+$&$-$&$-$&$+$&15.050&14.361&13.916&14.431&14.968&15.294&14.67&0.269&-1.313\\
10&$+$&$-$&$-$&$-$&14.249&13.900&13.065&13.143&13.708&14.255&13.72&0.272&-1.302\\
11&$+$&$-$&$+$&$+$&13.327&13.457&14.368&14.405&13.932&13.552&13.84&0.220&-1.514\\
12&$+$&$-$&$+$&$-$&13.605&13.190&13.695&14.259&14.428&14.223&13.90&0.229&-1.474\\
13&$+$&$+$&$-$&$+$&14.274&13.904&14.317&14.754&15.188&14.923&14.56&0.227&-1.483\\
14&$+$&$+$&$-$&$-$&13.775&14.586&14.379&13.775&13.382&13.382&13.88&0.253&-1.374\\
15&$+$&$+$&$+$&$+$&13.723&13.914&14.913&14.808&14.469&13.973&14.30&0.250&-1.386\\
16&$+$&$+$&$+$&$-$&14.031&14.467&14.675&14.252&13.658&13.578&14.11&0.192&-1.650\\
\hline
\end{tabular}
%\end{center}
%\renewcommand{\baselinestretch}{1}
%\large\normalsize
\label{tab:ex1desdat}
\end{table}
\tiny

 \es

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

\bs
  \heading{$2^k$ Designs: A General discussion}
\vspace{0.3 in}
 \bi
  \item $2 \times 2 \times \ldots \times 2 = 2^k$ design.
  \item Planning matrix vs model matrix (see Tables 4.3, 4.5).
  \item Run order and restricted randomization (see Table 4.4).
  \item \emph{Balance}: each factor level appears the same number
  of times in the design.
  \item \emph{Orthogonality} : for any pair of factors, each possible
  level combination appears the same number of times in the
  design.
  \item Replicated vs unreplicated experiment.
\ei \es



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

\bs
  \heading{Main effects and Plots}
 \bi
  \item Main effect of factor A:
  \begin{eqnarray*}
{\rm ME}(A) = \bar{z}(A+)-\bar{z}(A-). \label{eqn:medef}
\end{eqnarray*}
\item Advantages of factorial designs (R.A.Fisher):
$reproducibility$ and \emph{wider inductive basis} for inference.
\item Informal analysis using the {\bf main effects plot} can be
powerful.


\begin{figure}[htbp]
%\centerline{\psfig{figure=ex1me.ps,width=7in,height=6in}}
\centerline{\psfig{figure=picture/ex1me.ps,height=1.6in,width=2.6in}}
%\vspace{2.5in}
\caption{\footnotesize Main Effects Plot, Adapted Epitaxial Layer
Growth Experiment} \label{fig:ex1me}
\end{figure}



 \ei
\es



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

\bs
  \heading{Interaction Effects}

\bi
 \item \textbf{Conditional main effect} of $B$ at $+$ level of $A$:
\begin{eqnarray*}
{\rm ME}(B|A+) = \bar{z}(B+|A+)-\bar{z}(B-|A+). \label{eqn:CondME}
\end{eqnarray*}

\item \textbf{Two-factor interaction effect} between $A$ and $B$:
\begin{eqnarray}
INT(A,B) &=& \frac{1}{2}\{{\rm ME}(B|A+) - {\rm ME}(B|A-) \}\nonumber \\
&=& \frac{1}{2}\{{\rm ME}(A|B+) - {\rm ME}(A|B-)\} \nonumber \\
&=&\frac{1}{2} \{ \bar{z}(A+|B+)+\bar{z}(A-|B-) \}
 -\frac{1}{2} \{ \bar{z}(A+|B-)+\bar{z}(A-|B+) \} , \nonumber \\
\end{eqnarray}

The first two definitions in (1) give more insight on the term
"interaction" than the third one in (1). The latter is commonly
used in standard texts.
 \ei \es




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

\bs
  \heading{Interaction Effect Plots}

\begin{figure}[htbp]
%\centerline{\psfig{figure=ex1int.ps,width=7in,height=6in}}
\centerline{\psfig{figure=picture/ex1int.ps,height=2.5in,width=4.5in,
angle=0}}
%\vspace{2.5in}
\caption{Interaction Plots, Adapted Epitaxial Layer Growth
Experiment } \label{fig:ex1int}
\end{figure}

 \es



%---------------------------------------------------------------------
%  9 -----------------------------------------------------------------

\bs
  \heading{Synergistic and Antagonistic Plots}
\bi \item An $A$-against-$B$ plot is \emph{synergystic} if ${\rm
ME}(B|A+){\rm ME}(B|A-) > 0$ \\ and  $antagonistic$ if
%\begin{eqnarray*}
${\rm ME}(B|A+){\rm ME}(B|A-) <0 $.
%\end{eqnarray*}

An antagonistic plot suggests a more \emph{complex} underlying
relationship than what the data reveal.

\begin{figure}[htbp]
%\centerline{\psfig{figure=ex1cddc.ps,width=7in,height=6in}}
\centerline{\psfig{figure=picture/ex1cddc.ps,height=1.5in,width=4.5in}}
%\vspace{2.5in}
\caption{$C$-against-$D$ and $D$-against-$C$ Plots, Adapted
Epitaxial Layer Growth Experiment} \label{fig:ex1cddc}
\end{figure}
 \ei
 \es


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

\bs
  \heading{More on Factorial Effects}

%\tiny
\begin{eqnarray*}
{\rm INT}(A,B,C) &=& \frac{1}{2} {\rm INT}(A,B|C+)-\frac{1}{2}
{\rm INT}(A,B|C-)
%\label{eqn:intabc1} \\
  =\frac{1}{2} {\rm INT}(A,C|B+) \nonumber \\
 &&-\frac{1}{2} {\rm INT}(A,C|B-) = \frac{1}{2} {\rm INT}(B,C|A+)-\frac{1}{2} {\rm INT}(B,C|A-) .  \nonumber
\end{eqnarray*}
\begin{eqnarray*}
{\rm INT}(A_1 , A_2 , \ldots , A_k )& = & \frac{1}{2} {\rm
INT}(A_1 , A_2 , \ldots , A_{k-1}| A_{k}+ )
% \nonumber \\
- \frac{1}{2} {\rm INT}(A_1 , A_2 , \ldots , A_{k-1}| A_{k}- ) .
\label{eqn:kfi}
\end{eqnarray*}

%\normalsize
\bi
 \item A general factorial effect
\begin{eqnarray*} \hat{\theta} = \bar{z}_{+} - \bar{z}_{-}, \end{eqnarray*}
where $\bar{z}_+$ and $\bar{z}_-$ are averages of one half of the
observations. If $N$ is the total number of observations,
\begin{eqnarray*}
Var(\hat{\theta})=\frac{\sigma^{2}}{N/2}+\frac{\sigma^{2}}{N/2}=
\frac{4}{N}\sigma^{2}, \label{eqn:varthetahatforFE}
\end{eqnarray*}
$\sigma^2$ = variance of an observation.

\ei \es

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

\bs
  \heading{Using Regression Analysis to Compute Factorial Effects}

Consider the $2^3$ design for factors $A$, $B$ and $C$, whose
columns are denoted by $\bf{x_1}$, $\bf{x_2}$ and $\bf{x_3}$ (=1 or -1).\\
The interactions $AB$, $AC$, $BC$, $ABC$ are then equal to

\hspace{.8 in} $\bf{x_4} = \bf{x_1x_2}, \bf{x_5} = \bf{x_1x_3},
\bf{x_6} =
\bf{x_2x_3}, \bf{x_7} = \bf{x_1x_2x_3}$ (see Table~3).\\
Use the regression model
\begin{eqnarray*}
z_i = \beta_0 + \sum_{j=1}^{7} \beta_j x_{ij} + \epsilon_i ,
\end{eqnarray*}
where $i = i^{th}$ observation.\\
The regression (i.e., least squares) estimate of $\beta_j$ is
\begin{eqnarray*}
\hat{\beta_j} &=& \frac{1}{1-(-1)}(\bar{z}(x_{ij}= +1)
-\bar{z}(x_{ij}= -1)) \label{eqn:lseconnection}
\end{eqnarray*}
\hspace{1.23 in}= $\frac{1}{2}$(factorial effect of variable
$x_j$)

\es

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

\bs
  \heading{Model Matrix for $2^3$ Design}

\begin{table}[htbp]
\caption{Model Matrix for $2^3$ Design}
%\begin{center}
%\renewcommand{\baselinestretch}{1}
%\large\normalsize
$ $\\
\centering\small
\begin{tabular}{|ccccccc|} \hline
1 & 2 & 3 & 12 & 13 & 23 & 123\\
$A$ & $B$ & $C$ & $AB$ & $AC$ & $BC$ & $ABC$\\
\hline
$-$&$-$&$-$&$+$& $+$&$+$&$-$\\
$-$&$-$&$+$&$+$& $-$&$-$&$+$\\
$-$&$+$&$-$&$-$& $+$&$-$&$+$\\
$-$&$+$&$+$&$-$& $-$&$+$&$-$\\
$+$&$-$&$-$&$-$& $-$&$+$&$+$\\
$+$&$-$&$+$&$-$& $+$&$-$&$-$\\
$+$&$+$&$-$&$+$& $-$&$-$&$-$\\
$+$&$+$&$+$&$+$& $+$&$+$&$+$\\ \hline
\end{tabular}
%\end{center}
\label{tab:2e3data} \label{tab:modelmatrix}
\end{table}

\es

%---------------------------------------------------------------------
% 14 ---------------------------------------------------------------
\bs
  \heading{Factorial Effects, Adapted Epi-Layer Growth Experiment}
\begin{table}[htbp]
\caption{Factorial Effects, Adapted Epitaxial Layer Growth
Experiment}
$ $\\
\centering\tiny
\begin{tabular}{|lrr|} \hline
\multicolumn{1}{|c}{Effect} & \multicolumn{1}{c}{$\bar{y}$} &
\multicolumn{1}{c|}{$\ln{ s^2 }$} \\ \hline
$A$&-0.078&0.016\\
$B$&0.173&-0.118\\
$C$&-0.078&-0.112\\
$D$& 0.490& 0.056\\
$AB$&0.008&0.045\\
$AC$&-0.093&-0.026\\
$AD$&-0.050&-0.029\\
$BC$&0.058&0.080\\
$BD$&-0.030& 0.010\\
$CD$&-0.345& 0.085\\
$ABC$ &0.098&-0.032\\
$ABD$ & 0.025& 0.042\\
$ACD$ &-0.030&0.000\\
$BCD$ & 0.110&-0.003\\
$ABCD$ & 0.020& 0.103\\ \hline
\end{tabular}
\label{tab:ex1eff}
\end{table}
\tiny

\es

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

\bs
  \heading{Fundamental Principles in Factorial Design}

\bi

\item \textbf{Effect Hierarchy Principle} \\
(i) Lower order effects are more likely to be important than
higher order effects. \\
(ii) Effects of the same order are equally likely to be important.

\item \textbf{Effect Sparsity principle} (Box-Meyer)\\
The number of relatively important effects in a factorial
experiment is small.

This is similar to the \emph{Pareto Principle} in quality
investigation.

Effect hierarchy and sparsity principles are more
effective/relevant for \emph{screening experiments}.

\item \textbf{Effect Heredity Principle} (Hamada-Wu)\\
In order for an interaction to be significant, at least one of its
parent factors should be significant.

For modeling, McCullagh and Nelder called it the \emph{Marginality
Principle}.

\ei

\es


%---------------------------------------------------------------------
% 15 ---------------------------------------------------------------
\bs
 \heading{One-Factor-At-A-Time (ofat) Approach}

\begin{table}[htbp]
\caption{ Planning Matrix for $2^3$ Design and Response Data For
Comparison with One-Factor-At-A-Time Approach}
$ $\\
%\begin{center}
%\renewcommand{\baselinestretch}{1}
%\large\normalsize
\centering\small
\begin{tabular}{|rrr|r|} \hline
\multicolumn{3}{|c|}{Factor} &
\multicolumn{1}{c|}{Percent} \\
\multicolumn{1}{|c}{$P$} & \multicolumn{1}{c}{$R$} &
\multicolumn{1}{c|}{$S$} & \multicolumn{1}{c|}{Burned} \\ \hline
$1200$&$0.3$&slow& 11 \\
$1200$&$0.3$&fast& 17  \\
$1200$&$0.6$&slow& 25 \\
$1200$&$0.6$&fast& 29 \\
$1400$&$0.3$&slow& 02 \\
$1400$&$0.3$&fast& 09 \\
$1400$&$0.6$&slow& 37 \\
$1400$&$0.6$&fast& 40 \\ \hline
\end{tabular}
%\end{center}
\label{tab:desofat}
\end{table}

\es

%---------------------------------------------------------------------
% 16 ---------------------------------------------------------------
\bs
 \heading{One-Factor-At-A-Time (ofat) Approach (Contd.)}

\begin{figure}[htbp]
%%%
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%%%
%%  End of Graph.
%%%
%\centerline{\psfig{figure=ch1/figures/cheng1.ps,width=4.5in,height=3in}}
%\vspace*{0.025in}
\caption{The Path of a One-Factor-At-A-Time Plan}
\label{fig:ofatpath}
\end{figure}

The three steps of ofat as illustrated in the arrows in Figure 4
are detailed in steps 1-3 on page 174 of WH. \es



%---------------------------------------------------------------------
% 17 ---------------------------------------------------------------
\bs
 \heading{Disadvantages of ofat Approach Relative to Factorial Approach}
 \begin{enumerate}
\item \emph{It requires more runs for the same precision in effect
estimation.} In the example, the $2^3$ design requires 8 runs. For
ofat to have the same precision, each of the 4 corners on the ofat
path needs to have 4 runs, totaling 16 runs. In general, to be
comparable to a $2^k$ design, ofat would require $2^{k-1}$ runs at
each of the $k+1$ corners on its path, totaling $(k+1)2^{k-1}$.
The ratio is $(k+1)2^{k-1}/2^k = (k+1)/2$.

\item \emph{It cannot estimate some interactions}.

\item \emph{Conclusions for analysis not as general}.

\item It can miss optimal settings.\\


For points $2-4$, see Figure~4.

 \end{enumerate}

 \es

%---------------------------------------------------------------------
% 18 ---------------------------------------------------------------
\bs
 \heading{Why Experimenters Continue to Use ofat?}
 \bi
\item Most physical laws are taught by varying one factor at a
time. Easier to think and focus on one factor each time.

\item Experimenters often have good intuition about the problem
when thinking in this mode.

\item No exposure to statistical design of experiments.

\item \textbf{Challenges} for DOE researchers: To combine the
factorial approach with the good intuition rendered by the the
ofat approach. Needs a new outlook.

\ei
 \es


%---------------------------------------------------------------------
% 21 -----------------------------------------------------------------
\bs
 \heading{Normal Plot of Factorial Effects}
 \bi

\item Suppose $\hat{\theta}_i, i =1,\cdots,I$, are the factorial
effect estimates (example in Table 4). Order them as
$\hat{\theta}_{(1)} \le \cdots \le \hat{\theta}_{(I)}$. {\bf
Normal probability plot} (see Unit 2):

\vspace{0.2 in}

 \bc $\hat{\theta}_i$ (vertical) vs. $
\Phi^{-1}([i-0.5]/I)$ (horizontal) \ec
\begin{figure}[htbp]
%\centerline{\psfig{figure=ex1me.ps,width=7in,height=6in}}
\centerline{\psfig{figure=picture/ex1npm.ps,height=2.6in,width=1.6in,angle=270}}
%\vspace{2.5in}
\caption{\footnotesize {Normal Plot of Location Effects, Adapted
Epitaxial Layer Growth Experiment}
} %\label{fig:ex1me}
\end{figure}

\ei \es
%----------------------------------------------------------------
% 22--------------------------------------------------------------
\bs
 \heading{Use of Normal Plot to Detect Effect Significance}
 \bi
 \item \textbf{Deduction Step.} Null hypothesis $H_0:$ all
 factorial effects = 0 . Under $H_0$, $\hat{\theta}_i \sim N(0, \sigma^2)$
and the resulting normal plot should follow a \textit{straight
line}.

\item \textbf{Induction Step.} By fitting a straight line to the
middle group of points (around 0) in the normal plot, \textit{any
effect whose corresponding point falls off the line is declared
significant} (Daniel, 1959).

\item Unlike $t$ or $F$ test, no estimate of $\sigma^2$ is
required. Method is especially suitable for \textit{unreplicated}
experiments. In $t$ test, $s^2$ is the \textit{reference
quantity}. For unreplicated experiments, Daniel's idea is to use
the \textit{normal curve as the reference distribution}.


\item In Figure 5, $D$, $CD$ (and possibly $B$?) are significant.
Method is informal and judgemental.

\iffalse

\item Unlike $t$ or $F$ test, no estimate of $\sigma^2$ is
required. Method is especially suitable for \textit{unreplicated}
experiments. In $t$ test, $s^2$ is the \textit{reference
quantity}. For unreplicated experiments, Daniel's idea use the
\textit{normal curve as the reference distribution}.

\fi

 \ei \es



%----------------------------------------------------------------
% 23--------------------------------------------------------------
\bs
 \heading{Normal and Half Normal Plots}

 \begin{figure}[htbp]
%\centerline{\psfig{figure=ex1me.ps,width=7in,height=6in}}
\centerline{\psfig{figure=picture/hnpvsnp.ps,height=2.75in,width=2.6in,angle=0}}
%\vspace{2.5in}
 \caption{\small Comparison of Normal and Half-Normal Plots}
\label{fig:hnpvsnp}
 %\label{fig:ex1me}
\end{figure}
\es



%----------------------------------------------------------------
% 24--------------------------------------------------------------
\bs

\heading{Visual Misjudgement with Normal Plot}

\vspace{0.3 in}
 \noindent {\bf Potential misuse of normal plot :}


\noindent In Figure 6 (top), by following the procedure for
detecting effect significance, one may declare $C$, $K$ and $I$
are significant, because they ``deviate'' from the middle straight
line. This is \textit{wrong} because it ignores the obvious fact
that $K$ and $I$ are smaller than $G$ and $O$ in magnitude. This
points to a potential visual misjudgement and misuse with the
normal plot.



\es


%----------------------------------------------------------------
% 25--------------------------------------------------------------
\bs

\heading{Half-Normal Plot}
\bi

\item {\bf Idea: } Order the absolute $\hat{\theta}_{(i)}$ values
as $|\hat{\theta}|_{(1)} \le \cdots
 |\hat{\theta}|_{(I)}$ and plot them on the positive axis of the normal
distribution (thus the term ``half-normal''). This would avoid the
potential misjudgement between the positive and negative values.

\item
The {\bf half-normal probability plot} consists of the points\\
\begin{equation}
(\Phi^{-1}(0.5 + 0.5[i-0.5]/I ), {|\hat{\theta}|}_{(i)}),  \;
\mbox{{\rm for}} \;  i=1, \ldots, 2^k-1.
\label{eqn:HlafNormalPlotPoints}
\end{equation}

\item In Figure 6 (bottom), only $C$ is declared significant.
Notice that $K$ and $I$ no longer stand out in terms of the
absolute values.

\item For the rest of the book, \textbf{half-normal plots will be
used for detecting effect significance}

\ei

\es


%----------------------------------------------------------------
% 35--------------------------------------------------------------
\bs

\heading{A Formal Test of Effect Significance : Lenth's Method}

\bi \item Sometimes it is desirable to have a formal test that can
assign p values to the effects. The following method is also
available in packages like SAS.

\item Lenth's Method

\begin{enumerate}

\item

Compute the pseudo standard error
\begin{eqnarray*}
PSE =1.5 \cdot \mbox{median}_{ \{|\hat{\theta}_i| < 2.5 s_0\}} |
\hat{\theta}_i | , \label{eqn:PSE}
\end{eqnarray*}
where the median is computed among the $|\hat{\theta}_i|$ with
$|\hat{\theta}_i| < 2.5 s_0$ and
\begin{eqnarray*}
s_{0} = 1.5 \cdot \mbox{median} | \hat{\theta}_i | .
\label{eqn:so}
\end{eqnarray*}

(Justification : If $\theta_i = 0$ and error is normal, $s_0$ is a
$consistent$ estimate of the standard deviation of
$\hat{\theta}_i$. Use of median gives ``robustness'' to outlying
values.)

\end{enumerate}\ei
\es


%----------------------------------------------------------------
% 36--------------------------------------------------------------
\bs


\heading{A Formal Test of Effect Significance (Contd.)}


\bi
 \item[2.] Compute
\begin{eqnarray*} t_{PSE,i} =
\frac{\hat{\theta}_i}{PSE}, \;\;\mbox{for each $i$}.
\label{eqn:lenth}
\end{eqnarray*}
%Then, the $t$ values in
%(\ref{eqn:lenth})
%can be compared against
%the critical values given in
%Table~\ref{tab:LenthCriticalValues}.


If $|t_{PSE,i}|$ exceeds the critical value given in Appendix H
(or from software),  $\hat{\theta}_i$ is declared significant.

\item Two versions of the critical values are considered next.


\ei
 \es




%----------------------------------------------------------------
% 37--------------------------------------------------------------
\bs

\heading{Two Versions of Lenth's Method}

\bi \item {\bf Individual Error Rate (IER)}

$H_0 :$ all $\theta_i$'s $=0$, normal error.

IER$_{\alpha}$  at level $\alpha$ is determined by
\begin{eqnarray*}
\begin{array}{lll}
Prob(|t_{PSE,i}| > {\rm IER}_\alpha | H_0 )&=& \alpha ,
\;\;\;\mbox{for $i = 1, \cdots, I$.}
\end{array}
\end{eqnarray*}


(Note : Because $\theta_i=0$, $t_{PSE,i}$ has the $same$
distribution under $H_0$ for all $i$.)

 \item {\bf Experiment-wise Error Rate (EER)}
\begin{eqnarray*}
Prob(|t_{PSE,i}| > {\rm EER}_\alpha\; {\rm \; for \; at \; least \; one }\; i, i=1, \ldots, I | H_0 )   \nonumber \\
= Prob( \max_{1 \leq i \leq I} |t_{PSE,i}| > {\rm EER}_\alpha |
H_0 ) = \alpha .
\end{eqnarray*}
\item EER accounts for the number of tests done in the experiment
but often gives conservative results (less powerful). In screening
experiments, IER is more powerful and preferable because many of
the $\theta_i$'s are negligible (recall the effect sparsity
principle). The EER critical values can be inflated by considering
many $\theta_i$ values. (Why?)


 \ei

\es

%----------------------------------------------------------------
% 38--------------------------------------------------------------
\bs

\heading{Illustration with Adapted Epi-Layer Growth Experiment}

\begin{enumerate}

\item In Table 4, $\mbox{median} | \hat{\theta}_i |$ = 0.078, $s_0
= 1.5 \times 0.078$ =  0.117. \\ Trimming constant $2.5 s_0 = 2.5
\times 0.117 =  0.292$, which eliminates 0.490 ($D$) and 0.345 ($CD$).\\
Then $\mbox{median}_{\{|\hat{\theta}_i| < 2.5 s_0\}} |
\hat{\theta}_i |$ = 0.058, $PSE = 1.5 \times 0.058 = 0.087$. \\The
corresponding $|t_{PSE}|$ values appear in Table 6.

\item For $\alpha = 0.01$, $\mbox{IER}_{0.01} = 3.63$ for $I =
15$. By comparing with the $|t_{PSE}|$ values, $D$ and $CD$ are
significant at 0.01 level. Use of $\mbox{EER}_{0.01} = 6.45$ (for
$I = 15$) will not detect any effect significance. Analysis of the
$|t_{PSE}|$ values for $\ln s^2$ (Table 6) detects no significant
effect (details on page 182 of WH), thus confirming the
half-normal plot analysis in Figure 4.10 of section 4.8.

\end{enumerate}

\bi \item p values of effects can be obtained from packages or by
interpolating the critical values in the tables in appendix H.
(See page 182 for illustration). \ei

\es

%----------------------------------------------------------------
% 39--------------------------------------------------------------
\bs

\heading{$|t_{PSE}|$ Values for Adapted Epi-Layer Growth
Experiment}

\begin{table}[htbp]
\caption{$|t_{PSE}|$ Values, Adapted Epitaxial Layer Growth
Experiment}
$ $\\
\centering\tiny
\begin{tabular}{|lrr|} \hline
\multicolumn{1}{|c}{Effect} & \multicolumn{1}{c}{$\bar{y}$} &
\multicolumn{1}{c|}{$\ln{ s^2 }$} \\ \hline
$A$& 0.90& 0.25 \\
$B$& 1.99& 1.87 \\
$C$& 0.90& 1.78 \\
$D$& 5.63& 0.89 \\
$AB$& 0.09& 0.71 \\
$AC$& 1.07& 0.41 \\
$AD$& 0.57& 0.46 \\
$BC$& 0.67& 1.27 \\
$BD$& 0.34& 0.16 \\
$CD$& 3.97& 1.35 \\
$ABC$& 1.13& 0.51 \\
$ABD$& 0.29& 0.67 \\
$ACD$& 0.34& 0.00 \\
$BCD$& 1.26& 0.05 \\
$ABCD$& 0.23& 1.63 \\ \hline
\end{tabular}
\label{tab:LayerTPSEvalues}
\end{table}
\tiny

\es

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

\bs
  \heading{Nominal-the-Best Problem}

 \bi
   \item There is a nominal or target value $t$ based on engineering
   design requirements. Define a quantitative loss due to
   deviation of $y$ from $t$.

   {\bf Quadratic loss :} $L(y,t) = c(y-t)^2$. \\
   $E(L(y,t)) = c Var(y)+ c(E(y)-t)^2$.

   \item \textbf{Two-step procedure for nominal-the-best problem}: \\
   (i) Select levels of some factors to minimize $Var(y)$.\\
   (ii)Select the level of a factor not in (i) to move $E(y)$
   closer to $t$.\\
   A factor in step (ii) is an $adjustment$ factor if it has a
   significant effect on $E(y)$ but not on $Var(y)$. Procedure is
   effective only if an adjustment factor can be found. This is
   often done on engineering ground. (Examples of adjustment
   factors : deposition time in surface film deposition process, mold size in tile fabrication, location and spacing of markings on the dial of a weighing scale).
\ei
 \vspace{3 in} \es

%---------------------------------------------------------------------
% 19 ---------------------------------------------------------------
\bs
 \heading{Why Take $\ln$ $s^2$ ?}
 \bi
\item It maps $s^2$ over (0,$\infty$) to $\ln$ $s^2$ over
($-\infty, \infty$). Regression and ANOVA assume the responses are
nearly normal, i.e. over ($-\infty, \infty$).

\item Better for variance prediction. Suppose $z = \ln s^2$.
$\hat{z}$ = predicted value of $\ln \sigma^2$, then $e^{\hat{Z}}$
= predicted variance of $\sigma^2$, always nonnegative.

\item Most physical laws have a multiplicative component. Log
converts {\it multiplicity} into {\it additivity}.

\item Variance stabilizing property: next page.

\ei
 \es

%----------------------------------------------------------------------
% 20 -----------------------------------------------------------------
\bs
 \heading{$\ln$ $s^2$ as a Variance Stabilizing Transformation}
 \bi

\item Assume $y_{ij} \sim N(0,\sigma^2_i)$. Then $(n_i-1)s_i^2 =
\sum_{j=1}^{n_i}(y_{ij}-\bar{y}_i)^2 \sim
\sigma^2_i\chi^{2}_{n_i-1}$,
\begin{equation} \ln s_i^2 = \ln \sigma_i^2 +
\ln(\chi^2_{n_i-1}/(n_i-1)). \end{equation}

\item $X$ a random variable, $h$ a smooth function,
\begin{equation}
\mbox{Var}(h(X)) \approx [h'(E(X))]^2 \mbox{Var}(X)
\end{equation}


\item Take $X =\frac{\chi_{\nu}^2}{\nu}$ and $h = \ln$. Then
$E(X)=1$ and $\mbox{Var}(X)=\frac{2}{\nu}$.

\item Applying (3) to $X =\frac{\chi_{\nu}^2}{\nu}$ leads to
$$\mbox{Var}(\mbox{ln}(X)) \approx [h'(1)]^2 \frac{2}{\nu}=\frac{2}{\nu}.$$

In (2), $\nu = n_i-1$, $\ln s_i^2 \sim N(\ln \sigma_i^2,
2(n_i-1)^{-1})$. The variance of $\ln s_i^2$, $2(n_i-1)^{-1}$, is
nearly constant for $n_i-1 \geq 9$.


%\item Variance stabilizing property: next page.

\ei
 \es

%----------------------------------------------------------------
% 26--------------------------------------------------------------
\bs

\heading{Epi-layer Growth Experiment Revisited}

\begin{center} {Original data from Shoemaker, Tsui and Wu (1991)}.
\end{center}

\begin{table}[htbp]
\caption{Design Matrix and Thickness Data, Original Epitaxial
Layer Growth Experiment}
$ $\\
%\begin{center}
%\renewcommand{\baselinestretch}{1}
%\large\normalsize
\centering\tiny
\begin{tabular}{|rrrr|rrrrrr|r|rr|} \hline
\multicolumn{4}{|c|}{Design} & \multicolumn{6}{c|}{} &
\multicolumn{1}{c|}{} & \multicolumn{1}{c}{} &
\multicolumn{1}{c|}{}\\
\multicolumn{1}{|c}{$$A} & \multicolumn{1}{c}{$B$} &
\multicolumn{1}{c}{$C$} & \multicolumn{1}{c|}{$D$} &
\multicolumn{6}{c|}{Thickness} & \multicolumn{1}{c|}{$\bar{y}$} &
\multicolumn{1}{c}{$s^2$} & \multicolumn{1}{c|}{$\ln{s^2}$} \\
\hline
$-$&$-$&$-$&$+$&14.812&14.774&14.772&14.794&14.860&14.914&14.821&0.003&-5.771\\
$-$&$-$&$-$&$-$&13.768&13.778&13.870&13.896&13.932&13.914&13.860&0.005&-5.311\\
$-$&$-$&$+$&$+$&14.722&14.736&14.774&14.778&14.682&14.850&14.757&0.003&-5.704\\
$-$&$-$&$+$&$-$&13.860&13.876&13.932&13.846&13.896&13.870&13.880&0.001&-6.984\\
$-$&$+$&$-$&$+$&14.886&14.810&14.868&14.876&14.958&14.932&14.888&0.003&-5.917\\
$-$&$+$&$-$&$-$&14.182&14.172&14.126&14.274&14.154&14.082&14.165&0.004&-5.485\\
$-$&$+$&$+$&$+$&14.758&14.784&15.054&15.058&14.938&14.936&14.921&0.016&-4.107\\
$-$&$+$&$+$&$-$&13.996&13.988&14.044&14.028&14.108&14.060&14.037&0.002&-6.237\\
$+$&$-$&$-$&$+$&15.272&14.656&14.258&14.718&15.198&15.490&14.932&0.215&-1.538\\
$+$&$-$&$-$&$-$&14.324&14.092&13.536&13.588&13.964&14.328&13.972&0.121&-2.116\\
$+$&$-$&$+$&$+$&13.918&14.044&14.926&14.962&14.504&14.136&14.415&0.206&-1.579\\
$+$&$-$&$+$&$-$&13.614&13.202&13.704&14.264&14.432&14.228&13.907&0.226&-1.487\\
$+$&$+$&$-$&$+$&14.648&14.350&14.682&15.034&15.384&15.170&14.878&0.147&-1.916\\
$+$&$+$&$-$&$-$&13.970&14.448&14.326&13.970&13.738&13.738&14.032&0.088&-2.430\\
$+$&$+$&$+$&$+$&14.184&14.402&15.544&15.424&15.036&14.470&14.843&0.327&-1.118\\
$+$&$+$&$+$&$-$&13.866&14.130&14.256&14.000&13.640&13.592&13.914&0.070&-2.653\\
\hline
\end{tabular}
%\end{center}
\label{tab:ex2desdat}
\end{table}
\tiny

\es


%----------------------------------------------------------------
% 27--------------------------------------------------------------
\bs

\heading{Epi-layer Growth Experiment: Effect Estimates}

\begin{table}[htbp]
\caption{Factorial Effects, Original Epitaxial Layer Growth
Experiment}
$ $\\
%\begin{center}
%\renewcommand{\baselinestretch}{1}
%\large\normalsize
\centering\tiny
\begin{tabular}{|lrr|} \hline
\multicolumn{1}{|c}{Effect} & \multicolumn{1}{c}{$\bar{y}$} &
\multicolumn{1}{c|}{$\ln{ s^2 }$} \\ \hline
$A$&-0.055&3.834\\
$B$&0.142&0.078\\
$C$&-0.109&0.077\\
$D$& 0.836& 0.632\\
$AB$&-0.032&-0.428\\
$AC$&-0.074&0.214\\
$AD$&-0.025& 0.002\\
$BC$&0.047&0.331\\
$BD$& 0.010& 0.305\\
$CD$&-0.037& 0.582\\
$ABC$&0.060&-0.335\\
$ABD$& 0.067& 0.086\\
$ACD$&-0.056&-0.494\\
$BCD$& 0.098& 0.314\\
$ABCD$& 0.036& 0.109\\ \hline
\end{tabular}
%\end{center}
\label{tab:ex2eff}
\end{table}
\tiny

\es

%----------------------------------------------------------------
% 28--------------------------------------------------------------
\bs


\heading{Epi-layer Growth Experiment: Half-Normal Plots}
\begin{center}
\begin{tabular}{cc}
\psfig{figure=picture/ex2hnpm.ps,height=2.5in,width=2.in,angle=0}
&
\psfig{figure=picture/ex2hnps2.ps,height=2.5in,width=2.in,angle=0} \\
Figure~7 : Location effects &
Figure~8 : Dispersion effects \\
\end{tabular}
\end{center}

\es

%----------------------------------------------------------------
% 29--------------------------------------------------------------
\bs

\heading{Epi-layer Growth Experiment: Analysis and Optimization}

\bi

\item From the two plots, $D$ is significant for $\bar{y}$ and $A$
is significant for $z = \ln s^2$. \\ $D$ is an adjustment factor.
Fitted models :
\begin{eqnarray*}
\hat{y}&=&\hat{\alpha}+\hat{\beta}_{D}x_{D}=14.389+0.418x_D,\\
\hat{z}&=&\hat{\gamma}_{0}+\hat{\gamma}_{A}x_{A} =
-3.772+1.917x_{A},
\end{eqnarray*}

\item Two-step procedure: \\
(i) Choose $A$ at $-$ level (continuous rotation). \\
(ii)Choose $x_D = 0.266$ to satisfy $14.5=14.389+0.418x_D$\\
If $x_D$ = 30 and 40 seconds for $D$ = $-$ and $+$, $x_D$ = 0.266
would correspond to 35+0.266(5) = 36.33 seconds.

\item Predicted variance
\begin{eqnarray*}
{\hat{\sigma}}^2 = \exp (-3.772 + 1.917(-1)) = {(0.058)}^2.
\end{eqnarray*}
This is too optimistic! Predicted values should be validated with
a \emph{confirmation experiment}. \ei \es

%----------------------------------------------------------------
% 30--------------------------------------------------------------
\bs

\heading{$2^k$ Designs in $2^q$ Blocks}

\bi

\item Example: Arranging a $2^3$ design in 2 blocks (of size 4).
Use the 123 column in Table~9 to define the blocking scheme: block
I if 123 = $-$ and block II if 123 = $+$. Therefore the block
effect estimate $\bar{y}(II) - \bar{y}(I)$ is identical to the
estimate of the 123 interaction $\bar{y}(123=+) - \bar{y}(123=-)$.
The block effect $B$ and the interaction 123 are called {\bf
confounded}. Notationally,

\[
\begin{array}{lll}
{\bf B} &=& {\bf 123}.
\end{array}
\]

\item By giving up the ability to estimate 123, this blocking
scheme increases the precision in the estimates of main effects
and 2fi's by arranging 8 runs in two $homogeneous$ blocks.

\item Why sacrificing 123?\\
{\bf ans}: Effect hierarchy principle.

 \ei \es

%----------------------------------------------------------------
% 31--------------------------------------------------------------
\bs

\heading{Arrangement of $2^3$ Design in $2$ Blocks}

\begin{table}[htbp]
\caption{Arranging a $2^3$ Design in Two Blocks of Size Four }
$ $\\
%\begin{center}
%\renewcommand{\baselinestretch}{1}
%\large\normalsize
\centering\small
\begin{tabular}{|r|rrrrrrrc|} \hline
\multicolumn{1}{|c}{Run} & \multicolumn{1}{|c}{{\bf 1}} &
\multicolumn{1}{c}{{\bf 2}} & \multicolumn{1}{c}{{\bf 3}} &
\multicolumn{1}{c}{{\bf 12}} & \multicolumn{1}{c}{{\bf 13}} &
\multicolumn{1}{c}{{\bf 23}} & \multicolumn{1}{c}{{\bf 123}} &
\multicolumn{1}{c|}{Block} \\ \hline
1&$-$&$-$&$-$&$+$& $+$&$+$&$-$& I\\
2&$-$&$-$&$+$&$+$& $-$&$-$&$+$& II\\
3&$-$&$+$&$-$&$-$& $+$&$-$&$+$& II\\
4&$-$&$+$&$+$&$-$& $-$&$+$&$-$& I\\
5&$+$&$-$&$-$&$-$& $-$&$+$&$+$& II\\
6&$+$&$-$&$+$&$-$& $+$&$-$&$-$& I\\
7&$+$&$+$&$-$&$+$& $-$&$-$&$-$& I\\
8&$+$&$+$&$+$&$+$& $+$&$+$&$+$& II\\ \hline
\end{tabular}
%\end{center}
\label{tab:2p3block2}
\end{table}

\es

%----------------------------------------------------------------
% 32--------------------------------------------------------------
\bs

\heading{A $2^3$ Design in 4 Blocks} \bi \item Similarly we can
use $\mathbf{B_1 = 12}$ and $\mathbf{B_2 = 13}$ to define two
independent blocking variables. The 4 blocks I, II, III and IV are
defined by $B_1 = \pm$ and $B_2 = \pm$ :

\begin{center}
\begin{tabular}{c|cc|}
\multicolumn{1}{c}{ } &
\multicolumn{2}{c}{${\bf B_1}$ } \\
\multicolumn{1}{c}{${\bf B_2}$ } & \multicolumn{1}{c}{$-$} &
\multicolumn{1}{c}{$+$} \\ \cline{2-3}
$-$&I&III\\
$+$&II&IV\\ \cline{2-3}
\end{tabular}
\end{center}

\vspace{0.3 in}

\item A $2^3$ design in 4 blocks is given in Table 9. Confounding
relationships: $B_1 = 12$, $B_2 = 13$, $B_1B_2 = 12 \times 13
=23$. Thus 12,13 and 23 are confounded with block effects and thus
sacrificed.

\ei \es

%----------------------------------------------------------------
% 33--------------------------------------------------------------
\bs

\heading{Arranging a $2^3$ Design in 4 Blocks}
\begin{table}[htbp]
\caption{Arranging a $2^3$ Design in Four Blocks of Size Two }
$ $\\
%\begin{center}
%\renewcommand{\baselinestretch}{1}
%\large\normalsize
\centering\small
\begin{tabular}{|r|rrrccrrc|} \hline
\multicolumn{1}{|c}{Run} & \multicolumn{1}{|c}{{\bf 1}} &
\multicolumn{1}{c}{{\bf 2}} & \multicolumn{1}{c}{{\bf 3}} &
\multicolumn{1}{c}{${\bf B_1(=12)}$} & \multicolumn{1}{c}{${\bf
B_2(=13)}$} & \multicolumn{1}{c}{{\bf 23}} &
\multicolumn{1}{c}{{\bf 123}} & \multicolumn{1}{c|}{block} \\
\hline
1&$-$&$-$&$-$&$+$& $+$&$+$&$-$& IV\\
2&$-$&$-$&$+$&$+$& $-$&$-$&$+$& III\\
3&$-$&$+$&$-$&$-$& $+$&$-$&$+$& II\\
4&$-$&$+$&$+$&$-$& $-$&$+$&$-$& I\\
5&$+$&$-$&$-$&$-$& $-$&$+$&$+$& I\\
6&$+$&$-$&$+$&$-$& $+$&$-$&$-$& II\\
7&$+$&$+$&$-$&$+$& $-$&$-$&$-$& III\\
8&$+$&$+$&$+$&$+$& $+$&$+$&$+$& IV\\ \hline
\end{tabular}

\label{tab:2p3block4}
\end{table}


\es

%----------------------------------------------------------------
% 34--------------------------------------------------------------
\bs

\heading{Minimum Aberration Blocking Scheme}

\bi

\item On page 38, \{I,12,13,23\} forms the \textbf{block defining
contrast subgroup} for the $2^3$ design in 4 blocks. For a more
complicated example ($2^5$ design in 8 blocks), see page 196 of
WH.

\item For any blocking scheme $b$, let $g_i (b)$ = number of
$i$-factor interactions that are confounded with block effects.
Must require $g_1 (b)=0$ (because no main effect should be
confounded with block effects). For any two blocking schemes $b_1$
and $b_2$, let $r$ = smallest $i$ such that $g_i (b_1) \neq
g_i(b_2)$. If $g_r (b_1) < g_r(b_2)$, $b_1$ is said to have {\em
less aberration} than scheme $b_2$. (This is justified by the
effect hierarchy principle). A blocking scheme has \emph{minimum
aberration} if no other blocking schemes have less aberration.

\item Minimum aberration blocking schemes are given in Table 4A.1
of WH.

\item Theory is developed under the assumption of no $block\times
treatment$  interactions. \ei

\es


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

 \bs
  \heading{Comments on Board}

 \vspace{3 in} \es




\end{document}
