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Disclaimers ctor on the response depends on the level of the other factor(s).} One-factor-at-a-time experiments (where each factor is investigated separately by keeping all the remaining factors constant) do not reveal the interaction effects between the factors. Further, in one-factor-at-a-time experiments full randomization is not possible.

To illustrate factorial experiments consider an experiment where the response is investigated for two factors, $A$ and $B$. Assume that the response is studied at two levels of factor $A$ with $A_{% \text{low}}$ representing the lower level of $A$ and $A_{\text{high}}$ representing the higher level of $A$. Similarly, let $B_{\text{low}}$\ and $% B_{\text{high}}$ represent the two levels of factor $B$ that are being investigated in this experiment. Since there are two factors with two levels, a total of $2\times 2=4$ combinations exist ($A_{\text{low}}$-$B_{% \text{low}}$, $\ A_{\text{low}}$-$B_{\text{high}}$, $A_{\text{high}}$-$B_{% \text{low}}$, $\ A_{\text{high}}$-$B_{\text{high}}$). Thus, four runs are required for each replicate if a factorial experiment is to be carried out in this case. Assume that the response values for each of these four possible combinations are obtained as shown in Table 6.3.\FRAME{dtbpFU}{% 2.2762in}{1.2246in}{0pt}{\Qcb{Table 6.3: Two-factor factorial experiment.}}{% }{Figure}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "T";width 2.2762in;height 1.2246in;depth 0pt;original-width 3.2119in;original-height 1.7097in;cropleft "0";croptop "1";cropright "1";cropbottom "0";tempfilename 'Ch6TwoFactExTable1.wmf';tempfile-properties "XNPR";}}

\subsection{Investigating Factor Effects}

The effect of factor $A$ on the response can be obtained by taking the difference between the average response when $A$ is high and the average response when $A$ is low. The change in the response due to a change in the level of a factor is called the \emph{main effect} of the factor% \index{Main effect}. The main effect of $A$ as per the response values in Table 6.3 is:% \begin{eqnarray*} A &=&Average% \text{ }response\text{ }at\text{ }A_{\text{high}}-Average\text{ }response% \text{ }at\text{ }A_{\text{low}} \\