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\begin{cslide}
\Stitle{Confidence Intervals for Predicted Response}

Each response variable has some {\em true} value, which we will never
know.

From $r$ replicas of an experiment (holding factor levels
constant), we can {\em estimate} response variable's true value by
statistical inference.

How far is our estimate from the true value?  

Suppose our data satisfies three conditions:

\begin{description}

\item[Independence of error components:]
%
each experiment outcome is in no way related to the outcome of any
other experiment

\item[Normal distribution of errors ($e_{ij}$) with zero mean]

\item[Homogeneity of error variance:]
%
the variance of the errors $e_{i.}$ for all $i$ must be equal

\end{description}

<P>Then $t$-distribution yields confidence interval for some
significance level ($\alpha$).

<P>
<UL>
<LI>
<A HREF="a491-4/a491-4.html">Computing confidence intervals...
</A>
</LI>
</UL>
</BODY>

\end{cslide}

\begin{cslide}
\Stitle{Computing confidence intervals}

\subsection{Review of CI}

Recall CI formula from Ch 13.2 in Jain for $n$ samples:

\[
q \pm ts/\sqrt(n)
\]

For $2^kr$ factorial design, $n=2^2r$.  But what is sample standard
deviation {\em s}?

\subsection{Computing Sample Standard Deviation}

Sample variance of errors ($e_{ij}$'s) is:
\[
s_e^2 = \frac{SSE}{2^2(r-1)}
\]
where $SSE = \sum_{i,j}e^2_{ij}$.

Denominator is degrees of freedom.  It's less than number of
experiments performed (i.e., $2^2r$) because errors for all replicas
for each combination of factor levels sums to zero (i.e., $\sum_j
e_{ij} = 0$).

Let {\em q} denote any effect or interaction:
\[
q \in \{ q_0, q_A , q_B,q_{AB} \}
\]

Therefore the desired sample standard deviation of $q$ is
\[
s_q = \frac{s_e}{\sqrt{2^2r}}
\]

\subsection{Computing CI for Effects}

CI for an effect $q$ is
\[
q \pm t_{[1- \alpha/2; 2^2(r-1)]}\sqrt{s_q}
\]

\end{cslide}


\begin{cslide}
\Stitle{Example}

Continuing the memory-cache study, the sample standard
deviation of errors is:
\begin{eqnarray*}
s_{e} & = & \sqrt{  SSE / 2^{2}(r-1)} \\
      & = & \sqrt{102/8} \\
      & = & 3.57
\end{eqnarray*}

The sample standard deviation of each effect or interaction
$q$ is:
\[
s_{q} = s_{e} / \sqrt{(2^{2}r)} = 1.03
\]

The t-value at $2^{2}(3-1) = 8$ degrees of freedom and $\alpha=0.1$ is
$t_{[0.95;8]} =1.86$.  Thus CI for each effect is:

\[
        q \pm (1.86)(1.03)
\]

Therefore:

\begin{tabular}{l c c}
Effect or      &   Sample & Confidence \\
interaction    & Mean   &  Interval    \\ \hline
$q_{0}$         &          41.0  &     [39.08, 42.91] \\
$q_{A}$        &          21.5   &    [19.58, 23.41]    \\                 
$q_{B}$        &            9.5   &    [  7.58, 11.41] \\
$q_{AB}$       &          5.0    &    [  3.08,   6.91]
\end{tabular}

\end{cslide}

\begin{cslide}
\Stitle{How To Interpret Confidence Interval}

Note that all CI's exclude zero; therefore {\em all} are effects and
interactions significant!

Suppose we choose $\alpha=0.1$ and perform many $2^{2}r$ factorial
experiments.  Then for 90\% of them the {\em true} mean value of each
effect or interaction lies in the confidence interval calculated for
that experiment.

Thus in 5 percent of the experiments, the true mean will {\em not}
lie in the confidence interval!

\end{cslide}

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