Aliased designs: identify effects a reduced test cannot separate
A smaller experiment can estimate a combined contrast while being unable to separate the effects inside it. Before reporting that a workflow component helped, inspect the design columns and their alias relationships. More repetitions of the same combinations can make a mixed contrast more precise without revealing which component generated it.
The useful output is the actual run matrix, coding dictionary, alias table and a feasible follow-up design. This is a design-identification problem. It is different from correcting multiple p-values and from deciding whether correlated observational predictors produce unstable coefficients. Here the dependence is deliberately built into the combinations that were tested.
Define three factors and the missing combinations
Consider three hypothetical research-workflow options: A changes the document format, B adds a source-location checklist, and C changes the review display. Each has a low setting coded minus one and a high setting coded plus one. No setting is assumed better before measuring an outcome.
A full two-level experiment has eight combinations. To save runs, suppose the team tests only four and chooses C by multiplying A and B. The actual matrix is:
| Run | A | B | C = A×B |
|---|---|---|---|
| 1 | −1 | −1 | +1 |
| 2 | +1 | −1 | −1 |
| 3 | −1 | +1 | −1 |
| 4 | +1 | +1 | +1 |
The NIST discussion of confounding and aliasing explains the consequence of this generator. Locate its column-multiplication example and the section describing the complete alias pattern. The following numerical response model is invented for this article, rather than taken from NIST’s experimental results.
Derive the alias table instead of trusting coefficient labels
Because C equals A×B, multiplying both sides by C gives the defining relation I = A×B×C, where I is the all-ones column. Squaring any minus-one/plus-one column gives I. Multiplying the defining relation by each factor therefore yields:
| Column in the selected fraction | Indistinguishable column |
|---|---|
| A | B×C |
| B | A×C |
| C | A×B |
| I | A×B×C |
A fitted model may print a row called C, but those four observations cannot distinguish C’s coefficient from the A×B coefficient. Under a full mean representation, the quantity in that row combines them. The name chosen by software does not add combinations that were never observed.
This is not evidence that the interaction is actually large. It is evidence that the selected design cannot tell. Assuming the interaction is negligible can be a screening assumption, but it must be visible. A small standard error for the mixed contrast does not verify that assumption.
Reproduce a numerical mixture
Assume the fictional true mean response is twenty plus A minus B plus four C plus two A×B. The response measures completion time in minutes for a fixed verification task, assuming comparable accuracy and task scope; lower means faster. It is not an ordinal quality grade with invented distances. We use coefficients under the stated minus-one/plus-one coding.
In the initial fraction, replace A×B with C. The observable mean becomes twenty plus A minus B plus six C. Substituting the four rows gives responses 26, 16, 12 and 26.
| Initial fraction | C high mean | C low mean | High-minus-low contrast |
|---|---|---|---|
| C = A×B | (26 + 26) / 2 = 26 | (16 + 12) / 2 = 14 | 12 |
The contrast of twelve corresponds to twice the mixed coded coefficient of six. It does not establish that C’s standalone coefficient is six or that its standalone high-minus-low effect is twelve. In the assumed full model, the C coefficient is four and the interaction coefficient is two; both contribute to the first contrast.
Many other pairs adding to six would fit this initial fraction equally well. The data alone cannot choose between them. Reporting only the fitted C row would replace an unidentified decomposition with an unsupported component claim.
Add a complementary fraction
Now collect the other four combinations by setting C = −A×B. For this three-factor example, that set is also obtained by reversing all signs of the initial rows and reordering them. The NIST mirror-image foldover discussion describes augmenting appropriate fractional designs with reversed-sign runs to break main-effect and two-factor alias links.
Under the same hypothetical mean model, the complementary runs are:
| Run | A | B | C = −A×B | Mean response |
|---|---|---|---|---|
| 5 | −1 | −1 | −1 | 18 |
| 6 | +1 | −1 | +1 | 24 |
| 7 | −1 | +1 | +1 | 20 |
| 8 | +1 | +1 | −1 | 18 |
Here C-high averages twenty-two and C-low averages eighteen. The high-minus-low contrast is four, or twice a coded coefficient of two. This fraction estimates the C coefficient minus the A×B coefficient because their columns now have opposite signs.
Call the first mixed coded coefficient six and the second two. Their half-sum is four, recovering the assumed C coefficient; their half-difference is two, recovering A×B. Combining the fractions supplies all eight combinations, so those two columns are no longer identical. In a real experiment, observed values fluctuate and need appropriate uncertainty estimates.
The coding distinction matters: the recovered coefficient four corresponds to a high-minus-low C effect of eight in this representation. Do not mix a coefficient from one fraction with a raw contrast from another. Preserve the table and arithmetic so a reviewer can find a factor-of-two error.
If run six fails and only the other three complementary outcomes remain, the simple balanced calculation changes. The observed C-high mean becomes twenty rather than twenty-two, giving a high-minus-low difference of two. Mechanically applying the earlier half-sum formula would produce a C coefficient of 3.5, not the assumed four. The factor balance has changed. Retain the failed run and examine the observed design matrix and chosen model; some lower-order contrasts may remain estimable, but the original balanced shortcut is no longer justified.
Separate replication from identification
Repeating the original four combinations ten times supplies more information about their response distributions. It can estimate repeatability and improve precision for the combined coefficient. It still leaves C identical to A×B on every observed row, so it cannot separate their contributions.
Conversely, one run at each of all eight combinations permits a complete decomposition of eight cell means, but a model containing every factorial term uses all eight means. It has no within-cell pure-error estimate. Identification of contrasts and estimation of their uncertainty are separate requirements; a full run table does not remove the need for thoughtful replication.
Preserve run order, operator, document set and software version. Follow-up runs collected later require attention to learning, changed tasks and factor-specific version effects. A common time-block offset can be modeled where appropriate, but a changed response to the factors is not repaired just by adding a block label. Randomization and suitable repeated sessions should follow the intended comparison.
Test another boundary: infeasible combinations
Sometimes C = A×B is not a cost-saving choice but a genuine constraint: a display option may exist only with particular input formats and checklist states. The complementary combinations may then be impossible or represent a different product.
Do not promise that a foldover can solve that physical restriction. Define a comparison on the feasible combinations, redesign the factors or collect another kind of evidence. A component-specific causal claim may remain unavailable. The correct response to an impossible cell is not to fill it with a prediction and treat the prediction as a performed experiment.
Likewise, sparsity of effects is an assumption about which terms are small, not a measurement that the aliased terms equal zero. If subject knowledge makes it reasonable for initial screening, record the rationale and what follow-up evidence would overturn it. Keep the mixed estimate visible until the decomposition is supported.
Deliver the design audit
| Item | Evidence to retain | Passing check |
|---|---|---|
| Factor coding | Exact low/high definitions | Same settings across fractions |
| Actual runs | Per-run settings and outcomes | No predicted cell labeled observed |
| Generator | Column equality and sign | Reproduces the executed matrix |
| Aliases | Main and interaction columns | Every component claim is identifiable or conditional |
| Follow-up | Feasible complementary combinations | Breaks the specific alias of interest |
| Precision | Repetitions, units, blocks | Error calculation matches data collection |
Reopen the component conclusion when omitted combinations become feasible, a workflow version changes or evidence suggests a previously neglected interaction. Save both the screening assumption and the augmented design. The defensible report explains which mixed contrast was originally identified and what new observations, rather than a new coefficient label, allowed its parts to be separated.
FAQ
Can more repeated runs remove the alias?
Not if they repeat exactly the same fraction. They can improve precision without changing the column relationship.
Does aliasing prove there is an interaction?
No. It establishes that the chosen design cannot separate specified effects. The actual interaction still needs evidence.