Home / DOE / Additional Information / Folding
Folding¶
Folding adds a second copy of a fractional factorial design with signs flipped, which breaks some of the aliasing the fraction created.
Setting it up¶
Fold design: appears on the replicates and blocks page of a 2-Level Factorial design, during the build or afterwards from QXL DOE New > Modify Design > Change Blocking/Folding. That page is headed Enter number of replicates and number of blocks on the builder and in Modify Design, and Enter number of replicates and blocks (optional) in the Design Wizard. Folding is offered for two-level factorials only, because a full factorial and a three-level design have nothing to unfold.
Two kinds:
| Fold | What it flips |
|---|---|
| Full fold | the sign of every factor |
| Single factor fold | the sign of one factor you name |
What it does to the design¶
The folded block is appended below the original runs, so folding always doubles the run count. An eight-run design becomes sixteen runs: the original eight, then eight more with the signs flipped.
The point of the flip is that a term and its alias do not both survive it. A full fold clears the aliasing between main effects and two-factor interactions; a single factor fold clears it for the factor you folded on.
A fold that would change nothing is refused¶
Quantum XL checks whether the fold actually breaks any aliasing, and declines it if not. Before the design is built it works out the defining relation, applies the fold to it, and asks whether the relation got shorter. If it did not, the design is not created and the message is Cannot fold this design.
The rule behind that check, stated in terms of the defining relation:
| Fold | Refused when |
|---|---|
| Single factor fold | no word in the defining relation contains that factor, so removing it breaks nothing |
| Full fold | no word in the defining relation has an odd number of letters, so nothing drops out |
The two cases you will meet in practice are a full fold of a resolution IV fraction, whose words are all even length, and a fold of a full factorial, which has no defining relation to shorten. Both are refused.
So folding never silently doubles a design for nothing. If the fold is accepted, it bought you something in the alias table; if it was not going to, you get the message instead of the runs.
A fold appends a second copy of the design below the first, so the run count doubles. The copy holds the same columns with some of them negated:
- Fold on all factors negates every factor column in the copy.
- Fold on one factor negates that factor's column in the copy and leaves every other column as it was.
What that does to the aliasing follows from how a word's column is formed. The column of a word of length \(L\) is the product of \(L\) factor columns, so the word's column in the copied half is multiplied by \(-1\) once for every one of its factors that the fold negates.
- Under a fold on all factors, all \(L\) of them are negated, so the word's column is multiplied by \((-1)^L\) and is negated exactly when \(L\) is odd. An odd-length word is then \(+1\) through the first half and \(-1\) through the second, which is not the identity column any more, so the word is no longer part of the defining relation. Even-length words are unchanged and stay.
- Under a fold on one factor, a word that contains that factor has exactly one of its factors negated and a word that does not contain it has none, so exactly the words containing the folded factor drop out and the rest stay.
Quantum XL applies that rule to the words of the defining relation before it builds anything, and asks whether any word drops out. If none does, the fold would double the run count and change no alias, so the design is not built and the message is Cannot fold this design.
Two consequences of the doubling are worth stating plainly. The fold is applied before replicates and centre points are added, so a folded and replicated design carries both multipliers. And the list of interactions the model can hold is not derived from the shortened relation: once the columns are doubled, Quantum XL rebuilds the alias table from the folded columns themselves and takes the model terms from that, so a term the fold has freed appears because the new columns no longer match, not because a word was struck off a list.
Full details: Aliasing and Resolution.