## Related questions with answers

Consider the $2^{6-2}$ design in table below. Suppose that after analyzing the original data, we find that factors C and E can be dropped. What type of $2^{k}$ design is left in the remaining variables?

$\scriptstyle\begin{array}{rrrrrrr} \text { Run } & A & B & C & D & E=A B C & F=B C D & (\times 10) \\ \hline 1 & - & - & - & - & - & - & 6 \\ 2 & + & - & - & - & + & - & 10 \\ 3 & - & + & - & - & + & + & 32 \\ 4 & + & + & - & - & - & + & 60 \\ 5 & - & - & + & - & + & + & 4 \\ 6 & + & - & + & - & - & + & 15 \\ 7 & - & + & + & - & - & - & 26 \\ 8 & + & + & + & - & + & - & 60 \\ 9 & - & - & - & + & - & + & 8 \\ 10 & + & - & - & + & + & + & 12 \\ 11 & - & + & - & + & + & - & 34 \\ 12 & + & + & - & + & - & - & 60 \\ 13 & - & - & + & + & + & - & 16 \\ 14 & + & - & + & + & - & - & 5 \\ 15 & - & + & + & + & - & + & 37 \\ 16 & + & + & + & + & + & + & 52 \\ \hline \end{array}$

Solution

VerifiedTable 14-28. shows a fractional $2^6$ experimental design.

If factors $C$ and $E$ are deleted that design is full $2^4$ design.

The table below describes the reasoning:

First six columns are copied from table 14-28

The 7th column shows the factor combinations associated with each row - the experiments that were preformed

The 8th column is derived from the 7th by removing factors $c$ and $e$ - these are the experiments that are preformed in the design, if $C$ and $E$ are irrelevant.

All the combinations of factors $A, B, D, F$ are found in the 8th column - so this is a full $2^4$ design.

$\begin{equation*} \begin{array}{| c | c c c c c c c c | }\hline & A & B & C & D & E & F & \text{experiment} & \begin{matrix}\text{experiment}\\ \text{without $C$ and $E$} \end{matrix}\\ \hline 1 & - & - & - & - & - & - & (1) & (1)\\ 2 & + & - & - & - & + & - & ae & a\\ 3 & - & + & - & - & + & + & bef & bf\\ 4 & + & + & - & - & - & + & abf & abf\\ 5 & - & - & + & - & + & + & cef & f\\ 6 & + & - & + & - & - & + & acf & af\\ 7 & - & + & + & - & - & - & bc & b\\ 8 & + & + & + & - & + & - & abce & ab\\ 9 & - & - & - & + & - & + & df & df\\ 10 & + & - & - & + & + & + & adef & adf\\ 11 & - & + & - & + & + & - & bde & bd\\ 12 & + & + & - & + & - & - & abd & abd\\ 13 & - & - & + & + & + & - & cde & d\\ 14 & + & - & + & + & - & - & acd & ad\\ 15 & - & + & + & + & - & + & bcdf & bdf\\ 16 & + & + & + & + & + & + & abcdef & abdf\\ \hline \end{array} \end{equation*}$

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