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Two Level Factorial Designs¶

QXL DOE New > Create Design > Modeling > Create 2-Level Factorial Design

A two-level factorial runs every factor at two settings. Full or fractional, 2 to 15 factors, 4 to 128 runs.

What this design gives you¶

Each factor is run at a low and a high setting only, so the model can hold main effects and interactions but no quadratic term: with two points on a factor there is no third point to bend a curve through. Adding center points gives a check on curvature without making it estimable per factor.

A fractional design costs fewer runs and pays for it in aliasing, which the dialog reports as the design's resolution before you commit to it.

Choosing the design¶

The first page is headed Select 2-level factorial design. and holds a grid. The number of factors runs along the top, 2 to 15, and the number of runs down the side, 4, 8, 16, 32, 64 and 128. Each cell that holds a design shows its resolution as a Roman numeral, and hovering a cell reports the factors, the runs and the resolution.

Under the grid two lines describe the design you have selected. For the 7-factor, 16-run cell they read:

1/8 Fractional Factorial Design, 7 Factors in 16 Runs.

Resolution: IV. Main effects are aliased with three-ways, and two-ways are aliased with two-ways.

The first line names the fraction as either Full Factorial Design or 1/n Fractional Factorial Design, where n is a power of two. It is not a word like half or quarter: this grid reports the fraction numerically, and n can reach 2048 in the widest cells.

The second line is the word Resolution, the resolution as a Roman numeral, and then a fixed description. These are the six the grid can produce, verbatim. A resolution II description also exists in the software, but no cell in this grid ever reaches it:

Resolution	Description
Full	There is no aliasing.
III	Main effects are aliased with two-ways.
IV	Main effects are aliased with three-ways, and two-ways are aliased with two-ways.
V	Main effects are aliased with four-ways, and two-ways are aliased with three-ways.
VI	Main effects are aliased with five-ways, and two-ways are aliased with four-ways.
VII	Main effects are aliased with six-ways, and two-ways are aliased with five-ways.
VIII	Main effects are aliased with seven-ways, and two-ways are aliased with six-ways.

Note that from resolution IV up, each description has a second clause about the two-way interactions, and that clause is the one that tells you whether the interactions can be separated from each other.

A full factorial reports the word Full where the Roman numeral would be, not a number, and its description is There is no aliasing.

The rest of the pages¶

Enter Factor Names and Coding takes the factor names and their low and high settings. See Factor Names and Coding.

Define outputs (responses) takes the responses. See Define outputs.

Enter number of replicates and number of blocks is the last page, and for this design type it carries four things rather than two: replicates, blocking, Fold design: and Number of center points per block:. Two-level factorials are the only designs that set center points and folding on this page. See Define Replicates and Blocks.

Aliasing and blocking¶

The generators that build a fractional design are fixed per combination of factors and runs. The whole aliasing report is written to the design sheet, in a block below the run data headed Aliasing information. Inside that block, in order, are Resolution and its line, then Blocked Resolution and its line if the design is blocked, then Identity, which is the defining relation printed as const=ABD=..., and then Alias Table, whose lines name the terms that cannot be told apart. A design with nothing aliased prints All terms are free from aliasing. instead of those headings.

The sheet's Resolution line is a shorter sentence than the one in the dialog. It gives the main-effect aliasing only, as Resolution IV. Main effects are aliased with three-ways., and leaves out the second clause about the two-way interactions that the Select design grid shows. So the part that tells you whether the two-ways can be separated from each other is on screen while you choose the design, and is not carried onto the sheet. The table above is the full wording.

The regression sheet carries no alias table. Aliasing appears there inside a term's coded name instead, so a term reads A=BC.

Blocking does not add a table of its own. With b block generators there are 2^b - 1 blocking lines, each naming the terms the block effect is confounded with, and they are written as the first lines of the Alias Table, above the factor and interaction lines. Those terms are left out of the factor and interaction lines, so a term confounded with a block is listed once, among the blocking lines. What blocking does add is the Blocked Resolution heading and its line. Blocking on replicates instead skips the whole mechanism and produces no blocking lines. See Blocking.

The Blocked Resolution line reads Resolution n. Blocks are aliased with terms., so a full factorial with no aliasing of its own can still have a blocked resolution.

How those lines are formed¶

A two-level design codes every factor as a column of \(+1\) and \(-1\), one entry per run. A full factorial on \(k\) factors needs \(2^k\) runs, because that is how many distinct combinations of \(+1\) and \(-1\) there are. A fractional design has fewer runs than that, so some factors cannot be given a column of their own.

Quantum XL builds a fractional design in two steps. The factors that do get their own column, \(\log_2 F\) of them for a design of \(F\) runs, are laid out as a full factorial: their columns carry every combination of \(+1\) and \(-1\) exactly once, which fills all \(F\) rows. Each of the remaining

\[ g = k - \log_2 F \]

factors is then given the entrywise product of two or more of those columns. The equation naming that product is the factor's generator. Written the way the source writes it, a generator reads \(D = AB\), which means

\[ x_D = x_A \odot x_B \]

where \(\odot\) is entrywise multiplication: run by run, \(x_D\) is the product of the entries of \(x_A\) and \(x_B\) in that run. Two entries of \(\pm 1\) multiply to \(\pm 1\), so the new column is a legal two-level column.

The consequence is immediate, and it is the source of every alias in the design. The column belonging to factor D and the column belonging to the interaction AB are not merely similar, they are the same list of numbers. Nothing computed from the design can separate them.

The generators are looked up, not calculated. A reader expecting a formula that produces them will not find one. Quantum XL carries a fixed table with one entry per combination of factor count and run count: the code selects first on the number of factors, from 2 to 15, then on the number of runs, from 4 to 128, and each case assigns literal lists of factor positions. There are 49 such combinations, 214 stored design generators among them, and a further 338 stored block generators for the blocked cases. Nothing about a generator is derived while a design is being built. It is read out of that table.

A resolution number is stored beside each entry of that table, and a blocked resolution beside each block count within it. Those stored numbers are what the design description at the top of the design sheet reports: the stored resolution when the design is not blocked, the stored blocked resolution when it is. The Roman numeral in the design selection grid is stored too, but in a second table of its own, written into the dialog code and indexed by the grid's row and column rather than by anything the generators carry. Neither of those is the number in the sheet's aliasing block. That one is computed from the design's own columns after the design exists, and the section below says how.

Every generator can be rewritten as a statement about the identity column \(I\), the column whose every entry is \(+1\). Multiply both sides of \(D = AB\) by \(x_D\):

\[ x_D \odot x_D = x_A \odot x_B \odot x_D \]

The left side is \(I\), because every entry of \(x_D\) is \(+1\) or \(-1\) and both square to \(+1\). So

\[ I = ABD \]

and the generator has become a word: a product of factor letters whose column is the identity column. Quantum XL forms one such word from each generator by taking the generator's own letters and adding the letter of the factor that generator defines.

The words multiply, and their products are words as well. If \(I = ABD\) and \(I = ACE\) then

\[ I = I \odot I = ABD \odot ACE = AABCDE = BCDE \]

because \(x_A \odot x_A = I\) removes the repeated letter. Multiplication of two words is therefore performed by writing their letters together and then deleting every letter that appears an even number of times, keeping one copy of every letter that appears an odd number of times.

The defining relation is the closure of the generator words under that multiplication. With \(g\) generators, Quantum XL forms the \(g\) words first, then multiplies out every subset of two or more of them, which gives

\[ 2^g - 1 \]

words in all. The relation is then sorted, shortest word first. The identity itself is not one of the listed words. It is the thing they are all equal to.

The design sheet prints the relation under the heading Identity, as a single line that begins with the label const and lists the words separated by equals signs, for example const=ABD=ACE=BCDE. The printed line is not read back from the generator table. It is found by scanning the finished design: every product of factor columns up to the size limit described under the alias table is formed, and any product whose column holds the same value in every run is collected onto that line. A design in which no product is constant prints no Identity line at all.

The length of a word \(W\), written \(\lvert W \rvert\), is the number of letters it holds after the cancelling described above. Because a letter that appears twice is removed outright, a reduced word never repeats a letter, so its length is a count of distinct factors: \(ABD\) has length 3 and \(BCDE\) has length 4. Quantum XL stores a word as a sorted list of factor positions and takes the length from the size of that list, so the length is read after every cancellation has already happened, never before.

Word length is what connects the defining relation to the aliases. Take a word \(W\) with \(\lvert W \rvert = L\) and multiply it by one of its own letters, say \(A\):

\[ A = A \odot I = A \odot W \]

and \(A \odot W\) has \(A\) cancelled out of it, so it is an interaction of the other \(L - 1\) letters of \(W\). A main effect is therefore aliased with an interaction of \(L - 1\) factors for every word of length \(L\) that contains it. The same arithmetic runs from any term, not only from a main effect: multiplying any term by any word gives the term it is aliased with, and that term is written with the letters the two do not have in common, every shared letter having cancelled.

So the shortest word in the defining relation fixes the closest pairing the design contains, and the longer words describe pairings that are further apart.

The resolution reported on the design sheet is

\[ R = 1 + \min \{\, n : \text{a main effect is aliased with an } n \text{-factor interaction} \,\} \]

that is, one more than the number of factors in the smallest interaction that any main effect is aliased with. Quantum XL does not read this off the words. It pairs terms first, by comparing columns in the way the alias table section describes, and records for each pair it finds the two term sizes involved. It then looks for the smallest \(n\) for which the pair (main effect, \(n\)-factor interaction) was recorded, and reports \(n + 1\).

The two routes give the same number. A word of length \(L\) that contains a main effect makes that main effect aliased with an interaction of \(L - 1\) factors, and no shorter pairing can exist, so

\[ R = \min_{W} \lvert W \rvert \]

the length of the shortest word in the defining relation. Reading resolution as a shortest word length and reading it as one more than the closest aliased pair are two descriptions of the same design.

\(R\) is written as a Roman numeral wherever it is displayed. Three cases sit outside the ordinary numbering:

No aliasing at all. No product of columns matches any other and none of them is constant, so no pair is ever recorded. A full factorial with no blocking is this case. The sheet still writes the Aliasing information title, and prints All terms are free from aliasing. under it instead of the resolution, identity and alias table headings.
Aliasing that does not touch a main effect. If no pair involves a main effect but some pair involves two interactions, no Roman numeral is produced. The line names the smallest such pair directly, as Some m-ways are aliased with n-ways.
Resolution II. The arithmetic allows \(n = 1\), one main effect aliased with another, which gives \(R = 2\).

The search has a ceiling, and that ceiling decides the largest resolution that can be reported. Products are formed up to 7 factors when the design has 15 factors or fewer, and up to 3 factors when it has more than 15, which the larger Taguchi arrays reach. A main effect aliased with a 7-factor interaction is the deepest pairing the search can see, so resolution VIII is the largest number the design sheet can report.

A line of the alias table is a list of terms that share one column of the design, printed with equals signs between them, for example A=BD=CE. Every term on the line has the identical list of \(+1\) and \(-1\) values down the runs, so the design carries one column of information for all of them and no fit can tell them apart.

The lines are built by comparison, not by algebra. Quantum XL forms every main effect and every interaction up to the size ceiling given above, turns each one into a column by multiplying its factors' columns entrywise, and then compares columns:

Two terms go on the same line when their columns agree entry by entry, within a tolerance of \(10^{-7}\) on each entry.
They also go on the same line when the columns agree after one of them is multiplied by \(-1\). The term is then printed with a leading minus sign, as in A=-BD, which says the two columns carry the same information with opposite signs.

Two kinds of term never reach the table. A term whose column is constant down every run belongs to the defining relation and is collected onto the Identity line instead. A term whose column is constant within every block is confounded with a block and is collected onto a blocking line instead. Each term is placed once and once only, so a term on the Identity line or on a blocking line does not appear again among the alias table lines.

A line can hold a single term. That is what the table prints for a term the comparison matched with nothing, and it means that term is aliased with nothing.

Within a line the terms are ordered by length first, shortest leading, and by factor position among terms of equal length. The lines are then sorted against each other by comparing their leading terms the same way. Terms are written with one letter per factor, A for the first factor and onward, and in a design of more than 26 factors as F1, F2 and onward instead.

The regression sheet carries no alias table. It prints the alias inside the term's own name instead: the term reads A=BC, and where the line holds more than two terms the name ends in =....

Full details: Aliasing and Resolution.

Folding¶

An accepted fold appends a copy of the design below the original, doubling the run count. Fold on all factors flips the sign of every factor in that copy; Fold on factor 'name' flips only that one factor's column and leaves the rest of the copy identical. Either way the run count doubles. A fold that would not shorten the defining relation is refused with Cannot fold this design. See Folding.