## Related questions with answers

Montgomery (2001) describes a $2^{4-1}$ fractional factorial design used to study four factors in a chemical process. The factors are A = temperature, B = pressure, C = concentration, and D = stirring rate, and the response is filtration rate. The design and the data are as follows:

$\scriptstyle\begin{array}{cccccc} & & & & &\text{Treatment} & \text{Filtration} \\ \text { Run } & A & B & C & D=A B C & \text { Combination } & \text { Rate } \\ \hline 1 & - & - & - & - & (1) & 45 \\ 2 & + & - & - & + & a d & 100 \\ 3 & - & + & - & + &b d & 45 \\ 4 & + & + & - & - & a b & 65 \\ 5 & - & - & + &+ & c d & 75 \\ 6 & + & - & + & - & a c & 60 \\ 7 & - & + & + & - &b c & 80 \\ 8 & + & + & + & - &a b c d & 96 \\ \hline \end{array}$

(a) Write down the alias relationships. (b) Estimate the factor effects. Which factor effects appear large? (c) Project this design into a full factorial in the three apparently important factors and provide a practical interpretation of the results.

Solution

Verified#### a) Aliases

This design is defined by $D=ABC$.

If that equality is true, using the calculation with effects we have:

$\begin{gather*} D^2=ABCD\\ \boxed{I=ABCD} \end{gather*}$

So, as is standard, the identity element ($I$) is the highest order interaction.

This can be verified if we find the column in the given table for $ABCD$ - all elements would be "+"

The aliased effects are obtained using multiplication by $I$:

$\begin{gather*} I=ABCD\\ A=A\cdot I =A\cdot ABCD=BCD\\ \text{likewise}\\ B=B\cdot I =B\cdot ABCD=ACD\\ C=C\cdot I =C\cdot ABCD=ABD\\[2ex] AB=AB\cdot I =AB\cdot ABCD=CD\\ \text{likewise}\\ AC=AC\cdot I =AC\cdot ABCD=BD\\ AD=AD\cdot I =AD\cdot ABCD=BC \end{gather*}$

All effects and interactions are on the left or the right in the previous formulas, and they are aliased with the one on the other end. q

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