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\bs
  \heading{Unit 5: Fractional Factorial Experiments at Two Levels}

\bc Source : Chapter~5 (sections 5.1 - 5.5, part of section 5.6).
\ec

\vspace{.1 in}

\bi
 \item Leaf Spring Experiment (Section 5.1)
 \item Effect aliasing, resolution, minimum aberration criteria (Section 5.2).
 \item Analysis of Fractional Factorials (Section 5.3).
 \item Techniques for resolving ambiguities in aliased effects (Section 5.4).
 \item Choice of designs, use of design tables (Section 5.5).
 \item Blocking in $2^{k-p}$ designs (Section 5.6).
\ei
 \es




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

\bs
  \heading{Leaf Spring Experiment}
\bi
 \item $y$ = free height of spring, target = 8.0 inches. \\
 \textbf{Goal} : get $y$ as close to 8.0 as possible (nominal-the-best problem).

 \item Five factors at two levels, use a 16-run design with three
 replicates for each run. It is a $2^{5-1}$ design, 1/2 fraction
 of the $2^5$ design.

 \begin{table}[htbp]
\caption{Factors and Levels, Leaf Spring Experiment}
$ $\\
\centering
%\renewcommand{\baselinestretch}{1}
%\large\normalsize
\begin{tabular}{|ll|ll|} \hline
\multicolumn{2}{|c}{} &
\multicolumn{2}{|c|}{Level} \\
\multicolumn{2}{|c}{Factor} & \multicolumn{1}{|c}{$-$} &
\multicolumn{1}{c|}{$+$} \\ \hline
$B.$ & high heat temperature ($^{\circ}$F)& 1840& 1880\\
$C.$ & heating time (seconds)& 23 & 25\\
$D.$ & transfer time (seconds)& 10 & 12\\
$E.$ & hold down time (seconds)& 2 & 3\\
$Q.$ & quench oil temperature  ($^{\circ}$F)& 130-150& 150-170\\
\hline
\end{tabular}
%\renewcommand{\baselinestretch}{1.5}
%\large\normalsize
\label{tab:springfac}
\end{table}

 \ei

 \es

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

\bs
  \heading{Leaf Spring Experiment: Design Matrix and Data}
\begin{table}[htbp]
\caption{Design Matrix and Free Height Data, Leaf Spring
Experiment}
$ $\\
%\renewcommand{\baselinestretch}{1}
%\large\normalsize
%\small
\centering\tiny
\begin{tabular}{|rrrrr|rrr|rrr|} \hline
\multicolumn{5}{|c|}{Factor} & \multicolumn{3}{c|}{} &
\multicolumn{3}{c|}{} \\
\multicolumn{1}{|c}{$B$} & \multicolumn{1}{c}{$C$} &
\multicolumn{1}{c}{$D$} & \multicolumn{1}{c}{$E$} &
\multicolumn{1}{c|}{$Q$} & \multicolumn{3}{c|}{Free Height}&
\multicolumn{1}{c}{$\bar{y}_i$}& \multicolumn{1}{c}{$ s^{2}_{i}$}&
\multicolumn{1}{c|}{$\ln s^{2}_{i}$} \\ \hline
$-$&$+$&$+$&$-$&$-$&7.78&7.78&7.81&7.7900&0.0003&-8.1117\\
+  &$+$&$+$&+&$-$&8.15&8.18&7.88&8.0700&0.0273&-3.6009\\
$-$&$-$&$+$&+&$-$&7.50&7.56&7.50&7.5200&0.0012&-6.7254\\
+  &$-$&$+$&$-$&$-$&7.59&7.56&7.75&7.6333&0.0104&-4.5627\\
$-$&$+$&$-$&+&$-$&7.94&8.00&7.88&7.9400&0.0036&-5.6268\\
+  &$+$&$-$&$-$&$-$&7.69&8.09&8.06&7.9467&0.0496&-3.0031\\
$-$&$-$&$-$&$-$&$-$&7.56&7.62&7.44&7.5400&0.0084&-4.7795\\
+  &$-$&$-$&$+$&$-$&7.56&7.81&7.69&7.6867&0.0156&-4.1583\\
$-$&$+$&$+$&$-$&+&7.50&7.25&7.12&7.2900&0.0373&-3.2888\\
+  &$+$&$+$&+&+&7.88&7.88&7.44&7.7333&0.0645&-2.7406\\
$-$&$-$&$+$&+&+&7.50&7.56&7.50&7.5200&0.0012&-6.7254\\
+  &$-$&$+$&$-$&+&7.63&7.75&7.56&7.6467&0.0092&-4.6849\\
$-$&$+$&$-$&+&+&7.32&7.44&7.44&7.4000&0.0048&-5.3391\\
+  &$+$&$-$&$-$&+&7.56&7.69&7.62&7.6233&0.0042&-5.4648\\
$-$&$-$&$-$&$-$&+&7.18&7.18&7.25&7.2033&0.0016&-6.4171\\
+  &$-$&$-$&+&+&7.81&7.50&7.59&7.6333&0.0254&-3.6717\\ \hline
\end{tabular}
\label{tab:springdes}
\end{table}
\tiny


 \es

 %---------------------------------------------------------------------
%  4 -----------------------------------------------------------------
\bs
  \heading{Why Use Fractional Factorial Designs?}

\bi \item If a $2^5$ design is used for the experiment, its 31
degrees of freedom would be allocated as follows:

\begin{table}[htbp]
%\caption{Number of Effects in a $2^5$ Design}
\centering
\begin{tabular}{|cccccc|} \hline
\multicolumn{1}{|c}{} & \multicolumn{1}{c}{Main} &
\multicolumn{4}{c|}{Interactions} \\
\multicolumn{1}{|c}{ } & \multicolumn{1}{c}{Effects} &
\multicolumn{1}{c}{2-Factor} & \multicolumn{1}{c}{3-Factor} &
\multicolumn{1}{c}{4-Factor} & \multicolumn{1}{c|}{5-Factor} \\
\hline
\# & 5 & 10 & 10 & 5 & 1\\
\hline
\end{tabular}
\large\normalsize \label{tab:estfull}
\end{table}

\item Using effect hierarchy principle, one would argue that
4fi's, 5fi and even 3fi's are not likely to be important. There
are 10+5+1 = 16 such effects, half of the total runs! Using a
$2^5$ design can be wasteful (unless 32 runs cost about the same
as 16 runs.)

\item Use of a FF design instead of full factorial design is
usually done for economic reasons. Since there is \textbf{no free
lunch} , what \textbf{price to pay}? See next.

\ei \es
%---------------------------------------------------------------------
%  5 -----------------------------------------------------------------
\bs
 \heading{Effect Aliasing and Defining Relation}

 \bi
  \item In the design matrix, col $E$ = col $B \times$ col $C \times$ col
  $D$. That means,
\begin{eqnarray*}
\bar{y}(E+)-\bar{y}(E-)=\bar{y}(BCD+)-\bar{y}(BCD-).
\end{eqnarray*}
Therefore the design is not capable of distinguishing $E$ from
$BCD$. The main effect $E$ is \textbf{aliased} with the
interaction $BCD$. Notationally,
\begin{eqnarray} E = BCD \;\; \mbox{ or } \;\; {\bf I} = BCDE , \nonumber \end{eqnarray}
${\bf I}$ = column of $+$'s is the identity element in the group
of multiplications. (Notice the mathematical similarity between
aliasing and confounding. What is the difference?)

\item ${\bf I} = BCDE$ is the \textbf{defining relation} for the
$2^{5-1}$ design. It implies all the 15 effect aliasing relations
: \footnotesize
\begin{eqnarray*}
\begin{array}{l}
B=CDE, \;\; C=BDE,\;\; D=BCE,\;\; E=BCD, \\
BC = DE,\;\; BD=CE,\;\; BE=CD, \\
Q=BCDEQ,\;\; BQ=CDEQ,\;\; CQ=BDEQ,\;\; DQ=BCEQ,  \\
EQ=BCDQ,\;\; BCQ = DEQ,\;\; BDQ=CEQ,\;\; BEQ=CDQ.
\end{array}
\label{LeafAlias}
\end{eqnarray*}
\footnotesize

\ei \es

%---------------------------------------------------------------------
%  6 -----------------------------------------------------------------
\bs
 \heading{Clear Effects}

 \bi
  \item A main effect or two-factor interaction (2fi) is called
  \textbf{clear} if it is not aliased with any other m.e.'s or 2fi's and
  \textbf{strongly clear} if it is not aliased with any other m.e.'s, 2fi's
  or 3fi's. Therefore a clear effect is \emph{estimable} under the
  assumption of negligible 3-factor and higher interactions and a
  strongly clear effect is \emph{estimable} under the weaker
  assumption of negligible 4-factor and higher interactions.
  \item In the $2^{5-1}$ design with ${\bf I} = BCDE$, which
  effects are clear and strongly clear?\\
  Ans: $B$, $C$, $D$, $E$ are clear, $Q$, $BQ$, $CQ$, $DQ$, $EQ$
  are strongly clear.
  \item Consider the alternative plan $2^{5-1}$ design with ${\bf I} =
  BCDEQ$. (It is said to have resolution V because the length of
  the defining word is 5 while the previous plan has resolution
  IV.) It can be verified that all five main effects are strongly clear and
  all 10 2fi's are clear. (\emph{Do the derivations}). This is a very
  good plan because \emph{each} of the 15 degrees of freedom is either clear
  or strongly clear.


\ei \es

%---------------------------------------------------------------------
%  7 -----------------------------------------------------------------
\bs
 \heading{Defining Contrast Subgroup for $2^{k-p}$ Designs}

 \bi
  \item A $2^{k-p}$ design has $k$ factors, $2^{k-p}$ runs, and it
  is a $2^{-p}$th fraction of the $2^k$ design. The fraction is
  defined by $p$ \emph{independent} defining words. The group formed by
  these $p$ words is called the \textbf{defining contrast subgroup}. It has
  $2^p-1$ words plus the identity element $\bf I$.

  \item \textbf{Resolution} = shortest wordlength among the $2^p-1$ words.

  \item Example: A $2^{6-2}$ design with ${\bf 5=12}$ and ${\bf 6=134}$. The two independent defining
  words are ${\bf I=125}$ and ${\bf I= 1346}$. Then ${\bf I = \bf 125 \times 1346 =
23456}$. The defining contrast subgroup = $\{\bf{ I, 125, 1346,
23456}\}$. The design has resolution III.
 \ei \es

%---------------------------------------------------------------------
%  8 -----------------------------------------------------------------
\bs
 \heading{Deriving Aliasing Relations for the $2^{6-2}$ design}
 \bi
  \item For the same $2^{6-2}$ design, the defining contrast
  subgroup is
  \begin{eqnarray*}
{\bf I = 125 = 1346 = 23456}. \label{eqn:defingrelation}
\end{eqnarray*}
All the 15 degrees of freedom (each is a coset in group theory)
are identified. \tiny
\begin{equation}
{\bf
\begin{array}{lllllll}
\bf I&=& \bf 125 &=& \bf 1346 &=& \bf 23456, \\
\bf 1 &=& \bf 25 &=& \bf 346 &=& \bf 123456, \\
\bf 2 &=& \bf 15 &=& \bf 12346 &=& \bf 3456, \\
\bf 3 &=& \bf 1235 &=& \bf 146 &=& \bf 2456,  \\
\bf 4 &=& \bf 1245 &=& \bf 136 &=& \bf 2356,  \\
\bf 5 &=& \bf 12 &=& \bf 13456 &=& \bf 2346, \\
\bf 6 &=& \bf 1256 &=& \bf 134 &=& \bf 2345, \\
\bf 13 &=& \bf 235 &=& \bf 46 &=& \bf 12456, \\
\bf 14 &=& \bf 245 &=& \bf 36 &=& \bf 12356, \\
\bf 16 &=& \bf 256 &=& \bf 34 &=& \bf 12345, \\
\bf 23 &=& \bf 135 &=& \bf 1246 &=& \bf 456, \\
\bf 24 &=& \bf 145 &=& \bf 1236 &=& \bf 356, \\
\bf 26 &=& \bf 156 &=& \bf 1234 &=& \bf 345, \\
\bf 35 &=& \bf 123 &=& \bf 1456 &=& \bf 246, \\
\bf 45 &=& \bf 124 &=& \bf 1356 &=& \bf 236, \\
\bf 56 &=& \bf 126 &=& \bf 1345 &=& \bf 234.
\end{array}
} \label{eqn:2^6-2Aliases}
\end{equation}
\tiny\normalsize
 \item It has the clear effects: \textbf{3, 4, 6, 23,
24, 26, 35, 45, 56}. It has resolution III.
 \ei \es

 %---------------------------------------------------------------------
%  9 -----------------------------------------------------------------
\bs
 \heading{WordLength Pattern and Resolution}
 \bi
  \item Define $A_i$ = number of defining words of length $i$.
$W = (A_3, A_4, A_5, \ldots)$ is called the \textbf{wordlength
pattern}. In this design, $W$ = (1, 1, 1, 0). It is required that
$A_2$ = 0. (Why? No main effect is allowed to be aliased with
another main effect.)

\item {\bf Resolution} = smallest $r$ such that $A_r \geq 1$.

\item {\bf Maximum resolution criterion}: For fixed $k$ and $p$,
choose a $2^{k-p}$ design with maximum resolution.

\item {\bf Rules for Resolution IV and V Designs:}
\begin{equation}
\label{eqn:ResolutionRules}
\begin{array}{ll}
{\bf (i)} & {\it In \; any\;  resolution\;  IV\;  design, \; the \;main \;effects \;are \;clear}. \\
{\bf (ii)} & {\it In \;any \;resolution \;V \;design,\;
the \;main \;effects \;are \;strongly} \\
&{\it  clear \; and \;the \; \mbox{\it two-factor} \;interactions \; are \;clear}. \\
{\bf (iii)} & {\it Among\; the\; resolution\; IV\; designs\; with\; given\; k\; and\; p,}\\
&{\it those\; with\; the\; largest\; number\; of\; clear\; \mbox{\it two-factor} } \\
&{\it interactions\;are\; the\; best. }
\end{array}
\end{equation}
 \ei \es

 %---------------------------------------------------------------------
%  10 -----------------------------------------------------------------
\bs
 \heading{A Projective Rationale for Resolution}
 \bi
  \item For a resolution $R$ design, its projection onto any $R$-1
  factors is a full factorial in the $R$-1 factors. This would
  allow \emph{effects of all orders among the $R$-1 factors to be
  estimable}. (\textbf{Caveat}: it makes the assumption that other factors are
  inert.)

\begin{figure}[h]
\centerline{\psfig{figure=ch5_projection.eps,width=3in,height=2in}}
\caption{$2^{3- 1}$ Designs Using $\mathbf{I} = \pm \mathbf{123}$
and Their Projections to $2^{2}$~Designs.\label{fig4.1}}
\end{figure}

 \ei \es

%---------------------------------------------------------------------
%  11 -----------------------------------------------------------------
\bs
 \heading{Minimum Aberration Criterion}
 \bi
  \item Motivating example: consider the two $2^{7-2}$ designs:
  \begin{eqnarray*}
\begin{array}{rlllllll}
d_1: & {\bf I} & = & {\bf 4567} & = & {\bf 12346} & = & {\bf 12357}, \\
d_2: & {\bf I} & = & {\bf 1236} & = & {\bf 1457} & = & {\bf
234567}.
\end{array}
\label{d1d2}
\end{eqnarray*}
Both have resolution IV, but \[ W(d_{1}) = ( 0, 1, 2, 0, 0 )
\mbox{ and } W(d_{2}) = ( 0, 2, 0, 1, 0) . \] Which one is better?
Intuitively one would argue that $d_1$ is better because $A_4(d_1)
= 1 < A_4(d_2) = 2$. (Why? Effect hierarchy principle.)

\item For any two $2^{k-p}$ designs $d_1$ and $d_2$, let $r$ be
the smallest integer such that $A_r(d_1) \neq A_r(d_2)$. Then
$d_{1}$ is said to have {\em less aberration} than $d_{2}$ if
$A_r(d_1) < A_r(d_2)$. If there is no design with less aberration
than $d_{1}$, then $d_{1}$ has {\em minimum aberration}. \item
Throughout the book, this is the \emph{major criterion used for
selecting fractional factorial designs}. Its theory is covered in
the Mukherjee-Wu (2006) book.

\ei \es

%---------------------------------------------------------------------
%  12 -----------------------------------------------------------------
\bs
 \heading{Analysis for Location Effects}
 \bi
  \item \emph{Same} strategy as in full factorial experiments \emph{except} for the
  \textbf{interpretation and handling of aliased effects}.

\item For the location effects (based on $\bar{y}_i$ values), the
factorial effects are given in Table 3 and the corresponding
half-normal plot in Figure 2. Visually one may judge that $Q, B,
C, CQ$ and possibly $E, BQ$ are significant. One can apply the
studentized maximum modulus test (see section 4.14, not covered in
class) to confirm that $Q, B, C, CQ$ are significant at 0.05 level
(see pp. 219 and 221).

\item The $B \times Q$ and $C \times Q$ plots (Figure 3) show that
they are synergystic.

\item For illustration, we use the model
\begin{equation}
\begin{array}{ll}
\hat{y}  = & 7.6360 +0.1106 x_B +0.0519 x_E +0.0881 x_C -0.1298 x_Q  \\
  & +0.0423 x_B x_Q -0.0827 x_C x_Q
\end{array}
\label{eqn:springlocationmodel}
\end{equation}

\ei \es

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

\bs
 \heading{Factorial Effects, Leaf Spring Experiment}

\begin{table}[htbp]
\caption{Factorial Effects, Leaf Spring Experiment}
$ $\\
%\renewcommand{\baselinestretch}{1}
%\large\normalsize
\centering\tiny
\begin{tabular}{|lrr|} \hline
\multicolumn{1}{|c}{Effect} & \multicolumn{1}{c}{$\bar{y}$} &
\multicolumn{1}{c|}{$\ln{s^2}$} \\ \hline
$B$&0.221&1.891\\
$C$&0.176&0.569\\
$D$&0.029&-0.247\\
$E$&0.104&0.216\\
$Q$&-0.260&0.280\\
$BQ$&0.085&-0.589\\
$CQ$&-0.165&0.598\\
$DQ$&0.054&1.111\\
$EQ$&0.027&0.129\\
$BC$&0.017&-0.002\\
$BD$&0.020&0.425\\
$CD$&-0.035&0.670\\
$BCQ$&0.010&-1.089\\
$BDQ$&-0.040&-0.432\\
$BEQ$&-0.047&0.854\\ \hline
\end{tabular}
\label{tab:springeff}
\end{table}
\tiny


  \es


%---------------------------------------------------------------------
%  14 -----------------------------------------------------------------
\bs
 \heading{Half-normal Plot of Location Effects, Leaf Spring Experiment}

\begin{figure}[htbp]
\centerline{\psfig{figure=picture/springhnpm.ps,height=2.3in,width=2.7in,angle=0}}
%\vspace{2.5in}
\caption{\footnotesize {Half-Normal Plot of Location Effects, Leaf
Spring Experiment}}
\end{figure}
\es


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

\bs
 \heading{Interaction Plots}
\begin{figure}[htbp]
\centerline{\psfig{figure=picture/springbqcq.ps,height=2.8in,width=2.7in,angle=0}}
%\vspace{2.5in}
\caption{\footnotesize {$B \times Q$ and $C \times Q$ interaction
plots, Leaf Spring Experiment}}
\end{figure}
\es


%---------------------------------------------------------------------
%  16 -----------------------------------------------------------------
\bs
 \heading{Analysis for Dispersion Effects}

 \vspace{.5 in}

\bi \item For the dispersion effects (based on $z_i = \ln s_i^2$
values), the half-normal plot is given in Figure 4. Visually only
effect $B$ stands out. This is confirmed by applying the
studentized maximum modulus test (see pp.163-164 of WH, 2000). For
illustration, we will include $B, DQ, BCQ$ in the following model,

\begin{eqnarray}
\ln{\hat{\sigma}^2}= -4.9313 + 0.9455 x_B +  0.5556 x_D x_Q -
0.5445 x_B x_C  x_Q . \label{eqn:springdispersionmodel}
\end{eqnarray}

\ei \es

%---------------------------------------------------------------------
%  17 -----------------------------------------------------------------
\bs
 \heading{Half-normal Plot of Dispersion Effects, Leaf Spring Experiment}

\begin{figure}[htbp]
\centerline{\psfig{figure=picture/springhnps2.ps,height=2.3in,width=2.7in,angle=0}}
%\vspace{2.5in}
\caption{\footnotesize {Half-Normal Plot of Dispersion Effects,
Leaf Spring Experiment}}
\end{figure}

\es

%---------------------------------------------------------------------
%  18 -----------------------------------------------------------------
\bs
 \heading{Two-Step Procedure for Optimization}

\bi
 \item Step 1: To minimize $s^2$ (or $\ln s^2$) based on eq. (4),
 choose $B = -$. Based on the $D \times Q$ plot (Figure 5), choose
 the combination with the lowest value, $D = +$, $Q = -$. With
 $B = -$ and $Q = -$, choose $C = +$ to attain the minimum in the $B \times C \times Q$
 interaction  plot (Figure 6). Another confirmation: they lead to
 $x_B = -, x_Dx_Q = -$ and $x_Bx_Cx_Q = +$ in the model (4),
 which make each of the last three terms negative.

 \item Step 2: With $BCDQ$ = $( -, +, +, -)$,
\begin{eqnarray*}
\hat{y}& = &7.6360 +0.1106 (-1) +0.0519 x_{E} +0.0881 (+1) -0.1298 (-1)  \nonumber \\
&& +0.0423 (-1) (-1) -0.0827 (+1) (-1) \nonumber \\
&=& 7.8683 + 0.0519 x_{E}. \label{eqn:SpringLocationFitted}
\end{eqnarray*}
By solving $\hat{y}= 8.0$, $x_E$ = 2.54.

\textbf{Warning}: This is way outside the experimental range for
factor $E$. Such a value may not make physical sense and the
predicted variance value for this setting may be too optimistic
and not substantiated.


\ei \es

%---------------------------------------------------------------------
%  19 -----------------------------------------------------------------
\bs
 \heading{Interaction Plots for Dispersion Effects}

\begin{center}
\begin{tabular}{cc}
\psfig{figure=picture/springdq.ps,height=2.5in,width=2.in,angle=0}
&
\psfig{figure=picture/springbcqdeq.ps,height=2.5in,width=2.in,angle=0} \\
Figure~5 : $D \times Q$ Interaction Plot &
Figure~6 : $B \times C \times Q$ Interaction Plot \\
\end{tabular}
\end{center}

\es
%---------------------------------------------------------------------
%  20 -----------------------------------------------------------------
\bs
 \heading{Techniques for Resolving Ambiguities in Aliased Effects}
\vspace{0.2 in}
 \bi
 \item Among the three factorial effects that feature in model
 (4), $B$ is clear and $DQ$ is strongly clear.
 \item However, the term $x_B x_C x_Q$ is aliased with $x_D
 x_E x_Q$ (See bottom of page 5). The following three techniques
 can be used to resolve the ambiguities.
 \item \emph{Subject matter knowledge} may suggest some effects in the
 alias set are not likely to be significant (or does not have a
 good physical interpretation).
 \item Or use \emph{effect hierarchy principle} to \emph{assume away} some
 higher order effects.
 \item Or use a \textbf{follow-up experiment} to \textbf{de-alias} these effects.
 Two methods are given in section 5.4 of WH.
\ei

\es

%---------------------------------------------------------------------
%  24 -----------------------------------------------------------------
\bs
 \heading{Fold-over Technique}

\bi

 \item Suppose the original experiment is based on a
$2^{7-4}_{III}$ design with generators
\begin{eqnarray*}
d_{1}: \;\; {\bf 4=12, \;\; 5=13,\;\;  6=23,\;\;  7=123} .
\label{eqn:d1}
\end{eqnarray*}
None of its main effects are clear.

 \item To de-alias them, we can choose another 8 runs (no. 9-16 in Table 4) with \textbf{reversed} signs for each of the 7 factors. This follow-up design $d_2$ has the
 generators
\begin{eqnarray*}
d_{2}: {\bf 4=-12, 5=-13, 6=-23, 7=123.} \label{eqn:d2}
\end{eqnarray*}
With the extra degrees of freedom, we can introduce a new factor
\textbf{8} (or a blocking variable) for run number 1-8, and
\textbf{-8} for run number 9-16. See Table 4.

\item The combined design $d_1+d_2$ is a $2^{8-4}_{IV}$ design and
thus all main effects are clear. (Its defining contrast subgroup
is on p.227 of WH).

\ei


 \es

%---------------------------------------------------------------------
%  25 -----------------------------------------------------------------
\bs
 \heading{Augmented Design Matrix Using Fold-over Technique}
\begin{table}[htbp]
\caption{Augmented Design Matrix Using Fold-Over Technique}
$ $\\
%\renewcommand{\baselinestretch}{1}
%\large\normalsize
\centering \tiny
\begin{tabular}{|cccccccc|c|} \hline
\multicolumn{1}{|c}{} & \multicolumn{7}{c|}{$d_1$} &
\multicolumn{1}{c|}{} \\
\multicolumn{1}{|c}{Run} & \multicolumn{1}{c}{\bf 1} &
\multicolumn{1}{c}{\bf 2} & \multicolumn{1}{c}{\bf 3} &
\multicolumn{1}{c}{\bf 4=12} & \multicolumn{1}{c}{\bf 5=13} &
\multicolumn{1}{c}{\bf 6=23} & \multicolumn{1}{c|}{\bf 7=123} &
\multicolumn{1}{c|}{\bf 8} \\ \hline
1&$-$&$-$&$-$&$+$&$+$&$+$&$-$&$+$\\
2&$-$&$-$&$+$&$+$&$-$&$-$&$+$&$+$\\
3&$-$&$+$&$-$&$-$&$+$&$-$&$+$&$+$\\
4&$-$&$+$&$+$&$-$&$-$&$+$&$-$&$+$\\
5&$+$&$-$&$-$&$-$&$-$&$+$&$+$&$+$\\
6&$+$&$-$&$+$&$-$&$+$&$-$&$-$&$+$\\
7&$+$&$+$&$-$&$+$&$-$&$-$&$-$&$+$\\
8&$+$&$+$&$+$&$+$&$+$&$+$&$+$&$+$\\ \hline \multicolumn{1}{|c}{} &
\multicolumn{7}{c|}{$d_2$}&
\multicolumn{1}{c|}{} \\
\multicolumn{1}{|c}{Run} & \multicolumn{1}{c}{\bf -1} &
\multicolumn{1}{c}{\bf -2} & \multicolumn{1}{c}{\bf -3} &
\multicolumn{1}{c}{\bf -4} & \multicolumn{1}{c}{\bf -5} &
\multicolumn{1}{c}{\bf -6} & \multicolumn{1}{c|}{\bf -7} &
\multicolumn{1}{c|}{\bf -8} \\ \hline
9&$+$&$+$&$+$&$-$&$-$&$-$&$+$&$-$\\
10&$+$&$+$&$-$&$-$&$+$&$+$&$-$&$-$\\
11&$+$&$-$&$+$&$+$&$-$&$+$&$-$&$-$\\
12&$+$&$-$&$-$&$+$&$+$&$-$&$+$&$-$\\
13&$-$&$+$&$+$&$+$&$+$&$-$&$-$&$-$\\
14&$-$&$+$&$-$&$+$&$-$&$+$&$+$&$-$\\
15&$-$&$-$&$+$&$-$&$+$&$+$&$+$&$-$\\
16&$-$&$-$&$-$&$-$&$-$&$-$&$-$&$-$\\ \hline
\end{tabular}
\label{tab:FoldoverDesign}
\end{table}
\tiny

\es

%---------------------------------------------------------------------
%  26 -----------------------------------------------------------------
\bs
 \heading{Fold-over Technique: Version Two}

\bi

 \item Suppose one factor, say \textbf{5}, is very important. We want
 to de-alias \textbf{5} and all 2fi's involving \textbf{5}.
 \item Choose, instead, the following $2^{7-4}_{III}$ design
 \begin{eqnarray*}
d_{3}:  {\bf 4=12, 5=-13, 6=23, 7=123}.
\end{eqnarray*}

Then the combined design $d_1+d_3$ is a $2^{7-3}_{III}$ design
with the generators
\begin{eqnarray}
d': {\bf 4=12, 6=23, 7=123}. \label{eqn:dprimeDesign}
\end{eqnarray}

Since \textbf{5} does not appear in (5), \textbf{5} is strongly
clear and all 2fi's involving \textbf{5} are clear. However, other
main effects are not clear (see Table 5.7 of WH for $d_1+d_3$).

\item Choice between $d_2$ and $d_3$ depends on the priority given
to the effects (class discussions).

 \ei

 \es


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

\heading{Critique of Fold-over Technique}

\vspace{0.3 in}

\begin{itemize}
\item Fold-over technique is not an efficient technique. It
requires doubling of the run size and can only de-alias a
\textit{specific} set of effects. In practice, after analyzing the
first experiment, a set of effects will emerge and need to be
de-aliased. It will usually require much \textit{fewer} runs to
de-alias a few effects.

\item A more efficient technique that does not have these
deficiencies is the optimum design approach given in Section
5.4.2.

\end{itemize}


\es

%---------------------------------------------------------------------
%  21 -----------------------------------------------------------------
\bs
 \heading{Optimal Design Approach for Follow-Up Experiments}

\begin{itemize}

\item This approach add runs according to a particular optimal
design criterion. The $D$ and $D_{s}$ criteria shall be discussed.

\item Optimal design criteria depend on the assumed model. In
general, the model should contain:

1. All effects and their aliases (except those judged unimportant
a priori or by the effect hierarchy principle) identified as
significant in the initial experiment.

2. A block variable that accounts for differences in the average
value of the response over different time periods.

3. An intercept.

\item In the leaf spring experiment, we specify the model:

$E(z)=\beta_0+\beta_{bl}x_{bl}+\beta_Bx_B+\beta_{DQ}x_Dx_Q+\beta_{BCQ}x_Bx_Cx_Q+\beta_{DEQ}x_Dx_Ex_Q,$\\
where $z=ln(s^2)$, and $\beta_{bl}$ is the block effect.

\end{itemize}

\es

%---------------------------------------------------------------------
%  22 -----------------------------------------------------------------
\bs
 \heading{$D$-Criterion}

\begin{itemize}

\item In Table 5, the columns $B$, $C$, $D$, $E$, and $Q$ comprise
the design matrix while the columns $B$, $block$, $BCQ$, $DEQ$,
$DQ$ comprise the model matrix. Two runs are to be added to the
original 16-run experiment. There are $2^{10}=1024$ possible
choices of factor settings for the follow up runs (runs 17 and 18)
since each factor can take on either the + or - level in each run.

\item For each of the 1024 choices of settings for $B$, $C$, $D$,
$E$, $Q$ for runs 17 and 18, denote the corresponding model matrix
by $X_d, d=1,...,1024$. We may choose the factor settings $d^*$
that maximizes the $D$-criterion, i.e.
$$ {max_d}|X^T_dX_d|=|X^T_{d^*}X_{d^*}|.$$
Maximizing the $D$ criterion minimizes the volume of the
confidence ellipsoid for all model parameters $\beta$.

\item 64 choices of $d$ attain the maximum D value of 4,194,304.
Two are:

$d_1: (B,C,D,E,Q)=(+++-+)$ and $(++-+-)$

$d_2: (B,C,D,E,Q)=(+++-+)$ and $(-++++)$

\end{itemize}

 \es


%---------------------------------------------------------------------
%  23 -----------------------------------------------------------------
\bs
 \heading{Augmented Model Matrix and Design Matrix}

\begin{table}[htbp]
\caption{Augmented Design Matrix and Model Matrix, Leaf Spring
Experiment}
$ $\\
\centering \tiny
%\renewcommand{\baselinestretch}{1}
%\large\normalsize
%\small
\begin{tabular}{|r|ccccc|cccc|} \hline
\multicolumn{1}{|c|}{Run} & \multicolumn{1}{c}{$B$} &
\multicolumn{1}{c}{$C$} & \multicolumn{1}{c}{$D$} &
\multicolumn{1}{c}{$E$} & \multicolumn{1}{c|}{$Q$} &
\multicolumn{1}{c}{Block} & \multicolumn{1}{c}{$BCQ$} &
\multicolumn{1}{c}{$DEQ$} & \multicolumn{1}{c|}{$DQ$} \\ \hline
1 &$-$&$+$&$+$&$-$&$-$&$-$&$+$&$+$&$-$\\
2 &$+$&$+$&$+$&$+$&$-$&$-$&$-$&$-$&$-$\\
3 &$-$&$-$&$+$&$+$&$-$&$-$&$-$&$-$&$-$\\
4 &$+$&$-$&$+$&$-$&$-$&$-$&$+$&$+$&$-$\\
5 &$-$&$+$&$-$&$+$&$-$&$-$&$+$&$+$&$+$\\
6 &$+$&$+$&$-$&$-$&$-$&$-$&$-$&$-$&$+$\\
7 &$-$&$-$&$-$&$-$&$-$&$-$&$-$&$-$&$+$\\
8 &$+$&$-$&$-$&$+$&$-$&$-$&$+$&$+$&$+$\\
9 &$-$&$+$&$+$&$-$&$+$&$-$&$-$&$-$&$+$\\
10&$+$&$+$&$+$&$+$&$+$&$-$&$+$&$+$&$+$\\
11&$-$&$-$&$+$&$+$&$+$&$-$&$+$&$+$&$+$\\
12&$+$&$-$&$+$&$-$&$+$&$-$&$-$&$-$&$+$\\
13&$-$&$+$&$-$&$+$&$+$&$-$&$-$&$-$&$-$\\
14&$+$&$+$&$-$&$-$&$+$&$-$&$+$&$+$&$-$\\
15&$-$&$-$&$-$&$-$&$+$&$-$&$+$&$+$&$-$\\
16&$+$&$-$&$-$&$+$&$+$&$-$&$-$&$-$&$-$\\ \hline
17&$$&$$&$$&$$&$$&$+$&$$&$$&$$\\
18&$$&$$&$$&$$&$$&$+$&$$&$$&$$\\ \hline
\end{tabular}
\label{tab:augdes}
\end{table}
%\setlength{\tabcolsep}{5pt}
%\renewcommand{\baselinestretch}{1}
\tiny

 \es

%---------------------------------------------------------------------
%  24 -----------------------------------------------------------------
\bs
 \heading{$D_s$-Criterion}

\begin{itemize}

\item We may primarily be interested in the estimation of $BCQ$
and $DEQ$. Let $|X^T_dX_d|$ be partitioned as

\begin{eqnarray*}
\left(
\begin{array} {rr}
X^T_1X_1 & X^T_1X_2 \\
X^T_2X_1 & X^T_2X_2 \\
\end{array}
 \right) ,
\end{eqnarray*}

where $X_d=[X_1, X_2]$, with $X_2$ corresponding to the variables
$BCQ$ and $DEQ$. Then the lower right $2$x$2$ submatrix of
$(X^T_dX_d)^{-1}$ can be shown to be
$(X^T_2X_2-X^T_2X_1(X^T_1X_1)^{-1}X^T_1X_2)^{-1}$ and the optimal
choice of $d$ is one that satisfies
$${max_d}|(X^T_2X_2-X^T_2X_1(X^T_1X_1)^{-1}X^T_1X_2)|.$$

This is the $D_s$-optimal criterion, where $s$ denotes subset.

\item For $d_1$, the lower right $2$x$2$ submatrix of $X^T_dX_d$
have criterion value $128$. On the other hand, the $D_s$-criterion
value for $d_2$ is 113.78. $d_1$ is $D_s$-optimal, not $d_2$.

\end{itemize}
 \es

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

\heading{Use of Design Tables}

\begin{itemize}

\item Tables are given in Appendix 5A. Minimum aberration (MA)
designs are given in the tables. If two designs are given for same
$k$ and $p$, the first is an MA design and the second is better in
having a larger number of clear effects. Two tables are given on
next pages.

\item In Table 7, the first $2^{9-4}$ design has MA and 8 clear
2fi's. The second $2^{9-4}$ design is the second best according to
the MA criterion but has 15 clear 2fi's. Details on p. 234 of WH.
Using Rule (iii) in eq.(2) on page 9, the second design is better
because both have resolution IV.


\item It is not uncommon to find a design with slightly worse
aberration but more clear effects. Thus \textbf{the number of
clear effects} should be used as a \textit{supplementary
criterion} to the MA criterion.

\end{itemize}

\es

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

\bs

  \heading{Table 6: 16-Run $2^{k-p}$ FFD
$(k-p=4)$}


\begin{center}

($k$ is the number of factors and F\&R is the fraction and
resolution.)


\centering\scriptsize


\begin{tabular}{|l|l|p{2.5in}|p{1.3in}|} \hline


 $k$ & F\&R & \multicolumn{1}{l|}{Design Generators}
 & \multicolumn{1}{l|}{Clear Effects} \\ \hline

$5$ & $2^{5-1}_{V}$ & $5=1234$ & all five main effects, all $10$
2fi's
\\ \hline

$6$ & $2^{6-2}_{IV}$ & $5=123$, $ 6=124$ & all six main effects \\
\hline

$6^{*}$ & $2^{6-2}_{III}$ & $5=12$, $ 6=134$ & $3$, $ 4$, $ 6$, $
23$,
      $ 24$, $26$, $ 35$, $ 45$, $ 56 $ \\ \hline

$7$ & $2^{7-3}_{IV}$ & $5=123$, $ 6=124$, $ 7=134$ & all seven
main effects \\ \hline

$8$ & $2^{8-4}_{IV}$ & $5=123$, $ 6=124$, $ 7=134$, $ 8=234$ & all
 eight main effects \\ \hline

$9$ & $2^{9-5}_{III}$ & $5=123$, $ 6=124$, $ 7=134$, $ 8=234$,
$9=1234$ & none\\ \hline

$10$ & $2^{10-6}_{III}$ & $5=123$, $ 6=124$, $ 7=134$, $ 8=234$,
$9=1234$, $ t_{0}=34$ & none \\ \hline

$11$ & $2^{11-7}_{III}$ & $5=123$, $ 6=124$, $ 7=134$, $ 8=234$,
 $9=1234$, $ t_{0}=34$, $ t_{1}=24$ & none \\ \hline

$12$ & $2^{12-8}_{III}$ & $5=123$, $ 6=124$, $ 7=134$, $ 8=234$,
   $9=1234$, $ t_{0}=34$, $ t_{1}=24, t_{2}=14$ & none \\ \hline

$13$ & $2^{13-9}_{III}$ & $5=123$, $ 6=124$, $ 7=134$, $ 8=234$,
 $9=1234$, $ t_{0}=34$, $ t_{1}=24, t_{2}=14$, $t_{3}=23$ & none \\ \hline

$14$ & $2^{14-10}_{III}$ & $5=123$, $ 6=124$, $ 7=134$, $ 8=234$,
 $9=1234$, $ t_{0}=34$, $ t_{1}=24, t_{2}=14$, $t_{3}=23$, $ t_{4}=13$
 & none \\ \hline

$15$ & $2^{15-11}_{III}$ & $5=123$, $ 6=124$, $ 7=134$, $ 8=234$,
 $9=1234$, $ t_{0}=34$, $ t_{1}=24, t_{2}=14$, $t_{3}=23$, $
 t_{4}=13$, $ t_{5}=12$ & none \\ \hline
\end{tabular}

\small
\end{center}


\es


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


 \bs
  \heading{Table 7: 32 Run $2^{k-p}$ FFD ($k-p=5$, $6 \le k \le 11$)}



\begin{center}

 ($k$ is the number of factors and F\&R is the fraction and resolution.)


%\vspace{0.2in}


\centering\scriptsize


\begin{tabular}{|l|l|p{2.in}|p{1.6in}|}

\hline

$k$& F\&R & \multicolumn{1}{l|}{Design Generators}
 & \multicolumn{1}{l|}{Clear Effects} \\ \hline

$6$ & $2^{6-1}_{VI}$& $6=12345$
    & all six main effects, all $15$ $2$fi's \\

\hline

$7$ & $2^{7-2}_{IV}$& $6=123$, $7=1245$
    & all seven main effects, $14$, $15$, $17$, $24$,
   $25$, $27$, $34$, $35$, $37$, $45$, $46$,
   $47$, $56$, $57$, $67$\\

\hline

$8$ & $2^{8-3}_{IV}$& $6=123$, $7=124$, $8=1345$
    & all eight main effects, $15$, $18$, $25$, $28$,
$35$, $38$, $45$, $48$, $56$, $57$, $58$, $68$, $78$\\


\hline


$9$ & $2^{9-4}_{IV}$& $6=123$, $7=124$, $8=125$, $9=1345$
    & all nine main effects, $19$, $29$, $39$, $49$,
       $59$, $69$, $79$, $89$\\


\hline


$9$ & $2^{9-4}_{IV}$& $6=123$, $7=124$, $8=134$, $9=2345$
    & all nine main effects, $15$, $19$, $25$, $29$, $35$,
     $39$, $45$, $49$, $56$, $57$, $58$, $59$, $69$, $79$, $89$\\


\hline

$10$ & $2^{10-5}_{IV}$& $6=123$, $7=124$, $8=125$, $9=1345$,
$t_{0}=2345$
    & all $10$ main effects \\


\hline

$10$& $2^{10-5}_{III}$& $6=12$, $7=134$, $8=135$, $9=145$,
$t_{0}=345$
    & $3$, $4$, $5$, $7$, $8$, $9$, $t_{0}$, $23$,
   $24$, $25$, $27$, $28$, $29$, $2t_{0}$, $36$, $46$,
   $56$, $67$, $68$, $69$, $6t_{0}$\\


\hline

$11$ & $2^{11-6}_{IV}$& $6=123$, $7=124$, $8=134$,
 $9=125$, $t_{0}=135$, $t_{1}=145$
   & all $11$ main effects  \\


\hline

$11$& $2^{11-6}_{III}$& $6=12$, $7=13$, $8=234$, $9=235$,
$t_{0}=245$, $t_{1}=1345$
    & $4$, $5$, $8$, $9$, $t_{0}$, $t_{1}$, $14$, $15$,
        $18$, $19$, $1t_{0}$, $1t_{1}$\\


\hline

\end{tabular}
\scriptsize
\end{center}

 \es


%---------------------------------------------------------------------
%  31 -----------------------------------------------------------------
\bs
 \heading{Choice of Fractions and Avoidance of Specific
Combinations}

\vspace{0.2 in}

\begin{itemize}

\item A $2^{k-p}$ design has $2^p$ choices. In general, use
randomization to choose one of them. For example, the $2^{7-3}$
design has 8 choices $4 =\pm 12, 5 = \pm 13, 6 = \pm 23.$ Randomly
choose the signs.

\item If specific combinations (e.g., $(+++)$ for high pressure,
high temperature, high concentration) are deemed undesirable or
even disastrous, they can be avoided by choosing a fraction that
does not contain them. Example on p.237 of WH.


\end{itemize}

\es


%---------------------------------------------------------------------
%  32 -----------------------------------------------------------------
\bs
 \heading{Blocking in FF Designs}

Example: Arrange the $2^{6-2}$ design in four ($=2^2$) blocks with
\scriptsize
\begin{eqnarray*}
{\bf I=1235=1246=3456}.
\end{eqnarray*}
\normalsize Suppose we choose \scriptsize
\begin{eqnarray*}
{\bf B_1=134}, {\bf B_2=234} \; , {\bf B_1B_2 = 12}.
\end{eqnarray*}
\normalsize Then \scriptsize
\begin{eqnarray*}
{\bf B_1=134=245=236=156}, \\
{\bf B_2= 234=145=136=256}, \\
{\bf B_1B_2  = 12 = 35 = 46= 123456};
\end{eqnarray*} \normalsize
{\small i.e., these effects are confounded with block effects and
cannot be used for estimation. Among the remaining 12 degrees of
freedom, six are main effects and the rest are} \scriptsize
\[
\begin{array}{llllllllllllll}
\bf 13 & = & \bf 25 & = & \bf 2346 & = & \bf 1456, \\
\bf 14 & = & \bf 26 & = & \bf 2345 & = & \bf 1356, \\
\bf 15 & = & \bf 23 & = & \bf 2456 & = & \bf 1346, \\
\bf 16 & = & \bf 24 & = & \bf 2356 & = & \bf 1345, \\
\bf 34 & = & \bf 56 & = & \bf 1245 & = & \bf 1236, \\
\bf 36 & = & \bf 45 & = & \bf 1256 & = & \bf 1234 .
\end{array}
\]
\scriptsize

 \es

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

\bs
 \heading{Use of Design Tables for Blocking}

\vspace{0.4 in}

\begin{itemize}

\item Among the 15 degrees of freedom for the blocked design on
page 33, 3 are allocated for block effects and 6 are for clear
main effects (see Table 8). The remaining 6 degrees of freedom are
six pairs of aliased two-factor interactions.

\item For the same $2^{6-2}$ design, if we use the block
generators $B_1=13, B_2=14$, there are a total of 9 clear effects
(see Table 8): $3,4,6,23,24,26,35,45,56$.

Thus, the total number of clear effects for this blocked design is
3 more than the total number of clear effects for the blocked
design on page 33. However, only the main effects $3,4,6$ are
clear.

\end{itemize}

\vspace{0.2 in}

\es

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

\bs


 \tiny
\begin{table}

\heading{Table 8: Sixteen-Run $2^{k - p}$ Fractional Factorial
Designs in
$2^{q}$ Blocks}{%
\tabcolsep=4pt\begin{tabular}{@{}cccp{8pc}p{6pc}p{8pc}@{}}
\hline  &  &  & Design  & Block  &  \\
       $k$ & $p$ & $q$ &  Generators &  Generators & Clear Effects \\
\hline
5 &1 &1 &$5 = 1234$ &$B_{1} = 12$ &all five main effects, all 2fi's except 12 \\
5 &1 &2 &$5 = 1234$ &$B_{1} = 12$, \par $B_{2} = 13$ &all five main effects, 14, 15, 24, 25, 34, 35, 45 \\
5 &1 &3 &$5 = 123$ &$B_{1} = 14$, \par $B_{2} = 24$, \par $B_{3} = 34$ &all five main effects \\
6 &2 &1 &$5 = 123, 6 = 124$ &$B_{1} = 134$ &all six main effects \\
6 &2 &1 &$5 = 12, 6 = 134$ &$B_{1} = 13$ &3, 4, 6, 23, 24, 26, 35, 45, 56 \\
6 &2 &2 &$5 = 123, 6 = 124$ &$B_{1} = 134$, \par $B_{2} = 234$ &all six main effects \\
6 &2 &2 &$5 = 12, 6 = 134$ &$B_{1} = 13$, \par $B_{2} = 14$ &3, 4, 6, 23, 24, 26, 35, 45, 56 \\
6 &2 &3 &$5 = 123, 6 = 124$ &$B_{1} = 13$, \par $B_{2} = 23$, \par $B_{3} = 14$ &all six main effects \\
\hline
\end{tabular}}
{}{}
\end{table}

\es

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

\bs


 \tiny
\begin{table}

\heading{Table 8: Sixteen-Run $2^{k - p}$ Fractional Factorial
Designs in
$2^{q}$ Blocks (Cont.)}{%
\tabcolsep=4pt\begin{tabular}{@{}cccp{8pc}p{6pc}p{8pc}@{}}
\hline  &  &  & Design  & Block  &  \\
       $k$ & $p$ & $q$ &  Generators &  Generators & Clear Effects \\
\hline
7 &3 &1 &$5 = 123, 6 = 124$, \par $7 = 134$ &$B_{1} = 234$ &all seven main effects \\
7 &3 &2 &$5 = 123, 6 = 124$, \par $7 = 134$ &$B_{1} = 12$, \par $B_{2} = 13$ &all seven main effects \\
7 &3 &3 &$5 = 123, 6 = 124$, \par $7 = 134$ &$B_{1} = 12$, \par $B_{2} = 13$, \par $B_{3} = 14$ &all seven main effects \\
8 &4 &1 &$5 = 123, 6 = 124$, \par $7 = 134, 8 = 234$ &$B_{1} = 12$ &all eight main effects \\
8 &4 &2 &$5 = 123, 6 = 124$, \par $7 = 134, 8 = 234$ &$B_{1} = 12$, \par $B_{2} = 13$ &all eight main effects \\
8 &4 &3 &$5 = 123, 6 = 124$, \par $7 = 134, 8 = 234$ &$B_{1} = 12$, \par $B_{2} = 13$, \par $B_{3} = 14$ &all eight main effects \\
9 &5 &1 &$5 = 12, 6 = 13$, \par $7 = 14$, $8 = 234$, \par $9 = 1234$ &$B_{1} = 23$ &none \\
9 &5 &2 &$5 = 12, 6 = 13$, \par $7 = 14, 8 = 234$, \par $9 = 1234$ &$B_{1} = 23$, \par $B_{2} = 24$ &none \\
\hline
\end{tabular}}
{}{}
\end{table}

\es

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

\bs
 \heading{Use of Design Tables for Blocking}

\vspace{0.4 in}

\begin{itemize}

\item More FF designs in blocks are given in Appendix 5B of WH.
You only need to learn how to use the tables and interpret the
results. Theory or criterion used in choosing designs are not
required.

\end{itemize}

\vspace{0.2 in}

\es

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

 \bs
  \heading{Comments on Board}

 \vspace{3 in} \es


\end{document}
