Solution. Use the defining relation \(I = ABC\cdots K\), with all \(k\) letters, which gives resolution \(k\). Write down the full \(2^{k-1}\) design in the first \(k - 1\) factors and set the level of the last factor in each run equal to the product of the levels of the others. For \(k = 4\), \(D = ABC\): the runs are \((1)\), \(ad\), \(bd\), \(ab\), \(cd\), \(ac\), \(bc\), \(abcd\), the \(2^{4-1}_{\mathrm {IV}}\) design with \(I = ABCD\).

Solution. (a) The runs with an even number of letters from both \(\{a, b, c\}\) and \(\{b, c, d\}\) are \((1)\), \(bc\), \(abd\), \(acd\), the principal block of the confounding scheme in Section 9. On them \(ABC = BCD = -\), so the defining relation is \(I = -ABC = -BCD = AD\), including the generalised interaction \(ABC \cdot BCD = AD\). The aliases are \begin {gather*} A = D = -BC = -ABCD , \\ B = ABD = -AC = -CD , \\ C = ACD = -AB = -BD . \end {gather*} The word \(AD\) has only two letters: the design has resolution II, and the main effects \(A\) and \(D\) are aliased with each other.

(b) The four responses are \((1) = 19\), \(bc = 17\), \(abd = 18\), \(acd = 19\). Each estimate is a contrast divided by \(2\): \begin {gather*} \hat A = \tfrac 12(18 + 19 - 19 - 17) = 0.5 , \\ \hat B = \tfrac 12(17 + 18 - 19 - 19) = -1.5 , \\ \hat D = \tfrac 12(18 + 19 - 19 - 17) = 0.5 . \end {gather*} The estimates of \(A\) and \(D\) are identical, because the \(A\) and \(D\) columns coincide on these runs: the design cannot distinguish the two factors at all. Choosing two three-factor interactions whose product is a two-factor interaction is sound for blocking, where the confounded effects are lost to blocks, but disastrous for fractionation, where the generalised interaction becomes a word of the defining relation. For a quarter fraction of a \(2^4\) the best possible is resolution II in any case, which is why such small fractions are rarely used with four factors.

Solution. (a) The generalised interaction is \(ABCD \cdot CDE = ABE\), so \(I = ABCD = CDE = ABE\), of resolution III. The alias sets are

(b) The four quarter fractions are given by the signs of \(ABCD\) and \(CDE\):

The first is the principal fraction. (It does not contain \((1)\), because \(CDE\) has an odd number of letters and is \(-\) at \((1)\); the fraction containing \((1)\) is the second, \(I = ABCD = -CDE = -ABE\).)

(c) In the principal fraction the responses are \(ac = 34\), \(bc = 24\), \(ad = 29\), \(bd = 30\), \(e = 30\), \(abe = 24\), \(cde = 31\), \(abcde = 24\). Each estimate is the contrast divided by \(4\): \begin {align*} \hat A &= \tfrac 14\bigl [(34 + 29 + 24 + 24) - (24 + 30 + 30 + 31)\bigr ] = -4/4 = -1.0 , \\ \hat E &= \tfrac 14\bigl [(30 + 24 + 31 + 24) - (34 + 24 + 29 + 30)\bigr ] = -8/4 = -2.0 , \\ \widehat {BD} &= \tfrac 14\bigl [(34 + 30 + 30 + 31) - (24 + 29 + 24 + 24)\bigr ] \\ &= \tfrac 14(125 - 115) = 10/4 = 2.5 . \end {align*}

By the alias structure these estimate \(A + BE + BCD + ACDE\), \(E + AB + CD + ABCDE\) and \(BD + AC + ADE + BCE\); in particular the estimate of \(BD\) is equally an estimate of \(AC\).

Solution. The first fraction estimates \(A + BC\), the second \(A - BC\) (Theorem 10.1.2). So \(\frac 12(l_A + l_A')\) estimates \(A\) and \(\frac 12(l_A - l_A')\) estimates \(BC\), and similarly for the others. For three factors the alternate fraction is also the full fold-over (reversing all three signs maps \(a\) to \(bc\), and so on), and the two fractions together form the full \(2^3\) design, in which nothing is aliased.