Aliasing on Purpose
11 min read
An experiment with \(d\) two-level factors sets each factor high or low, runs the configuration, and gets back one number: the yield of a reaction, the accuracy of a model. There are \(2^d\) possible runs, and each one costs something real. Run them all and every effect can be separated: what each factor does on its own, and how factors reinforce or cancel one another. Run fewer and effects merge, leaving only sums.
Fractional factorial design is the craft of choosing which effects merge. Classically the merging is damage, managed by pushing it onto interactions small enough to ignore. But when the number we want is itself a sum of interactions, there is no damage at all: the right fraction merges exactly the effects we meant to add, and computes their sum with no error and no assumption.
Two Factors by Hand
The smallest experiment that can interact has \(d = 2\) factors: say a reaction whose yield depends on temperature and a catalyst, each set high or low, so the four runs give four yields \(f(+,+),\ f(+,-),\ f(-,+),\ f(-,-)\), temperature’s level written first.
What does temperature do? With the catalyst high, its effect is half the change in yield when it flips, \(\tfrac{1}{2}\big(f(+,+) - f(-,+)\big)\), and with the catalyst low it is \(\tfrac{1}{2}\big(f(+,-) - f(-,-)\big)\). If the two numbers agree, temperature does the same thing no matter the catalyst. If they disagree, the factors interact, and there is no single “effect of temperature”, only an effect at each setting of the catalyst.
The standard bookkeeping splits the pair into an average and a swing. The main effect \(\beta_1\) of temperature is the average of the two numbers, and the interaction \(\beta_{12}\) is half their difference, how much temperature’s effect moves when the catalyst flips. Writing a run as \(x = (x_1, x_2)\) with levels \(x_j = \pm 1\) and expanding, both are signed averages of all four yields:
\[\beta_1 = \frac{1}{4} \sum_{\text{runs}} f(x)\, x_1, \qquad \beta_{12} = \frac{1}{4} \sum_{\text{runs}} f(x)\, x_1 x_2.\]
The main effect signs each yield by temperature’s level; the interaction signs it by whether the two levels agree. The catalyst’s main effect \(\beta_2\) and the grand mean \(\beta_\emptyset\), the plain average, complete the set: four runs, four numbers.
The recipe is the same at every \(d\). A run is \(x = (x_1, \dots, x_d)\), and each set \(T\) of factors owns one effect, the average of the responses signed by the product of \(T\)’s levels:
\[\beta_T = \frac{1}{2^d} \sum_{\text{runs}} f(x)\, \chi_T(x), \qquad \chi_T(x) = \prod_{j \in T} x_j,\]
where the sign pattern \(\chi_T\) is called the parity of \(T\), with \(\chi_\emptyset \equiv 1\). Singletons give main effects, larger sets give \(|T|\)-factor interactions: \(2^d\) runs, \(2^d\) effects. And the effects are not just summaries; they carry the whole response.
Claim (the effects basis): every response decomposes as
\[f(x) = \sum_{T} \beta_T\, \chi_T(x),\]
and the signed averages above are the only coefficients that work.
Proof
At \(d = 2\) the display reads \(f(x) = \beta_\emptyset + \beta_1 x_1 + \beta_2 x_2 + \beta_{12} x_1 x_2\), four equations that can be checked by hand. In general, two facts do the work. Parities multiply into parities, \(\chi_T\, \chi_U = \chi_{T \,\triangle\, U}\), where \(T \triangle U\) is the symmetric difference, since a factor in both squares away. And every nonempty parity sums to zero over the runs: pick a factor in it and pair each run with the run that flips that factor’s level, so the paired terms cancel. Together the two facts make distinct parities orthogonal,
\[\sum_{\text{runs}} \chi_T(x)\, \chi_U(x) = \begin{cases} 2^d & \text{if } T = U, \\ 0 & \text{otherwise,} \end{cases}\]
so the \(2^d\) parity columns are a basis for the \(2^d\)-dimensional space of responses, and multiplying \(f = \sum_T \beta_T\, \chi_T\) by \(\chi_U\) and summing over all runs isolates \(2^d \beta_U\): the coefficients are the signed averages, and only they.
The proof uses only that every combination of levels appears exactly once, so it holds for a full factorial on any collection of \(\pm 1\) variables, however few. We’ll use that again, on an experiment much smaller than the cube.
Two Runs
The first experiment anyone actually runs is two runs, not \(2^d\): set every factor high, flip factor 1 low, and take half the difference of the two responses. In symbols the runs are \(x^{+} = (+1, \dots, +1)\) and \(x^{-}\), the same but with \(x_1 = -1\). Substituting the expansion from the claim into the difference:
\[\begin{align} f(x^{+}) - f(x^{-}) &= \sum_{T} \beta_T \big( \chi_T(x^{+}) - \chi_T(x^{-}) \big) \\ &= \sum_{T \ni 1} 2\, \beta_T, \end{align}\]
since a parity missing factor 1 agrees at the two runs and drops out, while a parity containing it contributes \(1 - (-1) = 2\). So half the difference is a clean sum:
\[\frac{f(x^{+}) - f(x^{-})}{2} = \sum_{T \ni 1} \beta_T.\]
This is the effect of factor 1 with everything else on: the main effect plus every interaction involving factor 1, in one number. At \(d = 2\) it is the first quantity we computed by hand, the effect of temperature at catalyst-high, and the sum it equals is \(\beta_1 + \beta_{12}\): average plus swing. Two runs can’t tell the effects in the sum apart, but the sum itself arrives exact.
There is a name for what just happened. Restricted to the two runs kept, every parity containing factor 1 became the same column of signs, \((+, -)\), which is \(x_1\)’s own, and every parity missing factor 1 became the constant column \((+, +)\). Parities that agree on every run kept are aliased: no arithmetic on the responses can split their coefficients apart, and the experiment determines only the sum within each alias group. Here there are two groups, the effects with factor 1 and the effects without, and the two responses determine both sums.
In the figure, each corner of the cube at \(d = 3\) is one run, labeled by the set of factors it sets high, and the two runs kept are the teal edge. Shrinking an experiment does not blur its effects into noise; it merges them into groups, and the group sums come out exact. Whether that is damage depends on the question. If we wanted \(\beta_1\) alone, two runs are hopeless. If we wanted factor 1’s contribution with everything else on, the group sum is the answer, and it arrived without error.
What two runs never offer is a choice: every effect containing factor 1 lands in one heap. Keeping more runs buys control over which effects merge.
Half the Runs
The smallest experiment with a real choice is \(d = 3\) with a budget of four runs. Keep the four runs whose levels satisfy one defining relation:
\[x_1 x_2 x_3 = +1.\]
Four of the eight corners of the cube obey it, the teal tetrahedron in the figure below. Multiply both sides by \(x_1\), using that levels square to one:
\[x_2 x_3 = x_1 \quad \text{on every run kept.}\]
The parity of \(\{2,3\}\) restricts to \(x_1\)’s column, so \(\beta_{23}\) aliases \(\beta_1\).
Multiplying the relation by \(x_2\) or \(x_3\) instead aliases \(\beta_{13}\) with \(\beta_2\) and \(\beta_{12}\) with \(\beta_3\), and the relation itself restricts \(\chi_{123}\) to the constant column, where it joins the mean. Eight effects merge into four groups of two, each effect paired with its complement:
\[\{\beta_\emptyset, \beta_{123}\}, \qquad \{\beta_1, \beta_{23}\}, \qquad \{\beta_2, \beta_{13}\}, \qquad \{\beta_3, \beta_{12}\}.\]
To read off a group’s sum, run the signed average on the fraction. For the group of \(\beta_1\), sign the four responses by \(x_1\)’s column:
\[\begin{align} \frac{1}{4} \sum_{\text{runs kept}} f(x)\, x_1 &= \sum_{T} \beta_T \cdot \frac{1}{4} \sum_{\text{runs kept}} \chi_T(x)\, x_1 \\ &= \beta_1 + \beta_{23}, \end{align}\]
where the first line substitutes the expansion and swaps the two sums. The second line is the orthogonality from the proof, applied to the fraction itself: the defining relation forces \(x_3 = x_1 x_2\), so the four kept runs are a full factorial on factors 1 and 2, and the four columns \(1, x_1, x_2, x_1 x_2\) are its parities, orthogonal over the kept runs. Every parity of the cube restricts to one of those four columns, so the inner average is one for the two parities restricting to \(x_1\)’s column, \(\chi_1\) and \(\chi_{23}\), and zero for the six restricting elsewhere. The same read-off against \(x_2\), \(x_3\), or the constant column returns the other three group sums. Four runs, four sums, each one exact: half the runs is half the information, with a precise ledger of which half.
As a check, the complementary fraction, the four gray corners obeying \(x_1 x_2 x_3 = -1\), satisfies \(x_2 x_3 = -x_1\), so its read-off returns \(\beta_1 - \beta_{23}\). Average the two fractions’ reads and \(\beta_1\) emerges, difference them and \(\beta_{23}\) does: the two halves together are the full factorial again, as they must be.
Can you see which fraction would alias \(\beta_1\) with \(\beta_{123}\) instead of \(\beta_{23}\)?
Choosing the Collisions
The defining relation is the dial. Keep the runs with \(x_2 x_3 = +1\): factors 2 and 3 now rise and fall together. On every run kept \(\chi_{23} = 1\), so multiplying any parity by it changes nothing, and each effect \(T\) merges with \(T \triangle \{2,3\}\):
\[\{\beta_\emptyset, \beta_{23}\}, \qquad \{\beta_1, \beta_{123}\}, \qquad \{\beta_2, \beta_{3}\}, \qquad \{\beta_{12}, \beta_{13}\}.\]
The group holding factor 1’s main effect is now \(\{\beta_1, \beta_{123}\}\), its contamination pushed up to order three. The price moved elsewhere. \(\beta_2\) and \(\beta_3\) share a group, so this fraction can never tell factors 2 and 3 apart, which is fair, since it never moves them separately. In general a fraction with one defining relation pairs each effect \(T\) with \(T \triangle D\), where \(D\) is the set of factors in the relation, its defining word: choosing the word chooses the collisions, casualties included.
Halving again stacks words. Keep only the runs with \(x_2 = +1\) and \(x_3 = +1\), and the two words act at once: each effect merges with \(T \triangle \{2\}\), \(T \triangle \{3\}\), and \(T \triangle \{2, 3\}\), groups of four. The group of \(\beta_1\) is \(\{\beta_1, \beta_{12}, \beta_{13}, \beta_{123}\}\), every effect containing factor 1: this is the two-run experiment again, revealed as the most aggressive fraction, its folded-in interactions just one large alias group. The accounting holds in general: \(k\) words, none a product of the others, cut the \(2^d\) runs to \(2^{d-k}\) and merge the effects into \(2^{d-k}\) groups of \(2^k\), and the fraction determines exactly the group sums, nothing less and nothing more.
The Two Contracts
A fraction hands back a short list of exact sums, and there are two ways to want it.
The classical contract assumes. A screening experiment with many factors and a small budget chooses its words long, so that main effects share groups only with interactions, never with each other, and with interactions as deep as possible; this craft dates to the crop and factory experiments of the 1940s, and the standard treatment is Wu & Hamada’s Experiments [WH, ’11]. The resolution of a fraction is the length of its shortest word, products of words included, and higher resolution pushes every group’s contamination deeper: our tetrahedron, one word of length three, is resolution III, its main effects polluted already at order two. If interactions beyond the second order are negligible, a resolution-V fraction returns every main effect and every two-factor interaction on its own. Either way the sums are read as single effects, and the reading is honest exactly when the assumed-away terms really vanish.
The modern contract sums. Sometimes the estimand is a sum of interactions, and then no assumption is needed at all: the fraction computes the estimand. The two-run experiment is the everyday case, since the effect of a factor in a fixed environment is a sum of interactions, and everyone already computes it from a fraction. The Shapley value, the standard attribution score in cooperative game theory and machine learning, is a sharper case: it weights every interaction by its order alone, so it only ever needs same-order sums of the interactions containing a factor. Zhou, Mee, Hamers & Zheng [ZMHZ, JASA ’25] construct fractions that merge effects only within an order, and read off every factor’s exact Shapley value from at most \(4d^2 - 4\) of the \(2^d\) runs whenever no five factors interact.
An experiment too small to separate effects still determines a full slate of sums, and determines them exactly. The craft is arranging for those to be the sums you wanted.