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Consider the $2^{6}$ factorial design. Set up a design to be run in four blocks of 16 runs each. Show that a design that confounds three of the four-factor interactions with blocks is the best possible blocking arrangement.

Consider a

2^2

design in two blocks with AB confounded with blocks. Prove algebraically that

SS_AB = SS

Blocks*

The article “An Application of Fractional Factorial Designs" describes a $2^{5-1}$ (half-replicate of a $2^{5}$ design) involving the use of carbon dioxide $\left(\mathrm{CO}_{2}\right)$ at high pressure to extract oil from peanuts. The outcomes were the solubility of the peanut oil in the $CO_2$ (in mg oil/liter $CO_2),$ and the yield of peanut oil (in percent). The five factors were A: $CO_2$ pressure, B: $CO_2$ temperature, C: peanut moisture, D: $CO_2$ flow rate, and E: peanut particle size. The results are presented in the following table.

\begin{matrix} \text{Treatment} & \text{Solubility} & \text{Yield} & \text{Treatment} & \text{Solubility} & \text{Yield}\\ \text{e} & \text{29.2} & \text{63} & \text{d} & \text{22.4} & \text{23}\\ \text{a} & \text{23.0} & \text{21} & \text{ade} & \text{37.2} & \text{74}\\ \text{b} & \text{37.0} & \text{36} & \text{bde} & \text{31.3} & \text{80}\\ \text{abe} & \text{139.7} & \text{99} & \text{abd} & \text{48.6} & \text{33}\\ \text{c} & \text{23.3} & \text{24} & \text{cde} & \text{22.9} & \text{63}\\ \text{ace} & \text{38.3} & \text{66} & \text{acd} & \text{36.2} & \text{21}\\ \text{bce} & \text{42.6} & \text{71} & \text{bcd} & \text{33.6} & \text{44}\\ \text{abc} & \text{141.4} & \text{54} & \text{abcde} & \text{172.6} & \text{96}\\ \end{matrix}

a. Assuming third- and higher-order interactions to be negligible, compute estimates of the main effects and interactions for the solubility outcome. b. Plot the estimates on a normal probability plot. Does the plot show that some of the factors influence the solubility? If so, which ones? c. Assuming third- and higher-order interactions to be negligible, compute estimates of the main effects and interactions for the yield outcome. d. Plot the estimates on a normal probability plot. Does the plot show that some of the factors influence the yield? If so, which ones?

Question

Consider the
$2^6$
factorial design. Set up a design to be run in four blocks of 16 runs each. Show that a design that confounds three of the four-factor interactions with blocks is the best possible blocking arrangement.

Solution

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We have to consider $2^6$ factorial design. We need to set up design to be run in four blocks of 16 runs each. Also, we have to show that a design that confounds three of the four factor interactions with blocks is the best possible blocking arrangement.

We constructed the design by using defining contrast for $ABCD$ and $CDEF.$ To assign the treatment combinations to the four blocks, substitute them into the defining contrast, so