Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 30 / 5

Code the two levels of each factor as $\pm 1$. For a factor subset $A$, its factorial contrast column is the character ${\chi }_A(x)={\prod }_{j\in A}{x}_j$. Products of columns correspond to symmetric differences of subsets, because $x_j^2=1$.

Let $G_1,\dots ,G_k$ be defining words whose incidence vectors are linearly independent over ${\mathbb{F}}_2$. Write $x_j=(-1{)}^{u_j}$, with $u_j\in {\mathbb{F}}_2$. Specifying signs ${\chi }_{G_i}(x)=s_i$ is then a rank-$k$ system of binary linear equations. Every one of its $2^k$ right-hand sides has $2^{n-k}$ solutions. Thus $k$ independent defining contrasts split the full $2^n$ design into $2^k$ equal fractions. Independence is necessary: dependent defining words do not yield $2^k$ distinct sets.

The defining contrast subgroup has $2^k$ words. On a selected fraction its word $G$ has a constant sign $s_G$, soHence the coefficients of $A$ and $AG$ cannot be estimated separately without assumptions or extra runs: this is aliasing in a fractional factorial design. Conversely, averaging a character over the affine binary solution space vanishes unless its word is in the defining subgroup, so these signed subgroup cosets give all aliases. Here a lowercase treatment label records the factors at their high levels, and $(1)$ puts all five at their low levels; signs must be retained when listing aliases.

On these eight runs, $W=M$ and $SCT=-1$. The two independent defining words therefore give the full signed defining relationMultiplying each desired factorial contrast by its four words gives all aliases. In the following equalities a minus sign means opposite contrast columns:Thus two main effects, $W$ and $M$, are inseparable even when every unwanted interaction vanishes. In addition, the important interaction $ST$ aliases with the main effect $C$. Design (a) is unsuitable. The seven-column design matrix for the intercept, the five main effects and $ST$ has rank five, confirming the two independent estimability failures.

For this fraction, $SCW=-1$ and $SCMT=1$, so their product is $WMT=-1$. The signed defining relation isThe complete factorial contrast alias classes areThe five main effects and $ST$ occupy six distinct alias classes, and none aliases with the intercept. Every undesired two-factor term in these classes is assumed negligible, and all terms of order at least three are also assumed negligible. Thus design (b) is suitable under the stated interaction assumptions. Its intercept and six requested columns are mutually orthogonal, with $X^{T}X=8I_7$, so each coded regression coefficient is ${\hat{\beta }}_A=8^{-1}{\sum }_{i=1}^8{\chi }_A(x_i)Y_i$. The corresponding high-minus-low main effect is $2{\hat{\beta }}_A$.

There is one residual degree of freedom, but no replicated setting and therefore no separate pure error estimate. Suitability here means estimability under the stated model; it does not protect estimates from active interactions that the assumptions exclude.