## Related questions with answers

Construct a

$2^5$

design in four blocks. Select the appropriate effects to confound so that the highest possible interactions are confounded with blocks.

Solution

VerifiedOur task is to confound $2^k=2^5$ design in $2^p=2^2=4$ blocks. Hence, since $\boxed{p=2<5}$, we are starting by selecting 2 effects to be confounded such that no effect chosen is a generalized interaction of the others, and then the blocks can be constructed from 2 defining contrasts $L_1, L_2$ that are associated with these effects.

Further, since it is required to confound the highest possible interactions, $ABCDE$ should be the generalized interaction of the two effects from the start. These two effects can only be represented by a two-factor interaction and a three-factor interaction.

For example, let's two-factor interaction be $AC$ and three-factor interaction be $BDE$. Then, if $AC$ and $BDE$ are confounded with blocks, their generalized interaction is

$(AC)(BDE)=ABCDE\,.$

The design can be constructed by using the defining contrasts for $AC$ and $BDE$:

$L_1=x_1+x_3\,, \quad L_2=x_2+x_4+x_5\,,$

but here, this problem is solved using the following $\mathbf{R}$ code:

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