Fractional Factorials and Screening

Main idea

With many suspect factors, trying every combination gets expensive fast: seven factors need 128 runs. A fractional factorial runs a cleverly chosen subset instead. The price is that some effects get mixed up with others, so you mix them only with complicated interactions that are almost always negligible, and a handful of runs finds the few factors that matter.

A half fraction from a generator

2^{4-1}: \quad {\htmlData{sym=0}{D = ABC}} \implies {\htmlData{sym=1}{I}} = ABCD

What the symbols mean

$D = ABC$
Generator
The extra factor's column is set equal to a product of the others' columns.
$I$
Identity column
All + signs. The defining relation lists every product of columns that equals it.

Aliases

A = A\cdot I = A\cdot ABCD = {\htmlData{sym=0}{A^2}}BCD = BCD, \qquad {\htmlData{sym=1}{\ell_A}} = A + BCD

What the symbols mean

$A^2$
A column times itself
Every entry is (±1)² = 1, so A·A = I. That's how aliases are found.
$\ell_A$
Contrast estimate
What the fraction actually estimates: the sum of an effect and all its aliases.

Runs in a fraction

{\htmlData{sym=2}{N}} = 2^{{\htmlData{sym=0}{k}}-{\htmlData{sym=1}{p}}}, \qquad \text{fraction} = \frac{1}{2^{\htmlData{sym=1}{p}}}

What the symbols mean

$k$
Factors
How many factors are studied.
$p$
Generators
How many factors are assigned to interaction columns; each one halves the runs.
$N$
Runs
The size of the design.

Explanation

A fractional factorial is simply a full factorial with some of the runs deliberately left out, chosen so you can still estimate what matters. The catch is aliasing: with fewer runs, some sign columns become identical, so their effects can't be told apart and you see only their sum. The trick is to choose a generator like $D = ABC$ so that each main effect is mixed only with complicated interactions that are probably near zero. It sounds slippery, but it's the same averages-and-subtraction from the last lesson, with careful bookkeeping about what's mixed with what.

The scene. An injection-molding line makes warped housings, and the team has a list of suspects: melt temperature, injection speed, holding pressure, cooling time and more. Experience says only one or two really matter (effect sparsity), but nobody knows which. All $2^4 = 16$ combinations of four factors is affordable; seven factors would take 128 runs, weeks of molding time. Fractional factorials buy the answer at a fraction of the cost.

Building a half fraction. Write the full $2^3$ sign table for A, B and C, eight runs, and set the fourth factor's column equal to the ABC column. That generator, $D = ABC$ , makes a $2^{4-1}$ : four factors in eight runs. Multiplying both sides by D, and using $D \cdot D = I$ (the all-+ column), gives the defining relation $I = ABCD$ .

Aliasing: the price. In these eight runs the column for A is identical to the column for BCD: multiply $I = ABCD$ by A and use $A^2 = I$ . Effects with identical columns have identical contrasts, so the data estimate only their sum, $\ell_A = A + BCD$ . In the same way $B = ACD$ , $C = ABD$ , $D = ABC$ , $AB = CD$ , $AC = BD$ and $AD = BC$ . The fraction is cheap because it stops trying to separate effects you'd bet are zero.

Resolution. The length of the shortest word in the defining relation sets what is safe to estimate:

Resolution III (shortest word has 3 letters, like $I = ABC$ ): main effects are aliased with two-factor interactions. Pure screening.
Resolution IV ( $I = ABCD$ ): main effects are clear of two-factor interactions, but two-factor interactions are aliased with each other.
Resolution V ( $I = ABCDE$ ): main effects and two-factor interactions are all clear, if higher-order interactions are negligible.

With two or more generators, every product of the generator words is also in the defining relation, and that's where the shortest word often hides.

Screening in practice. Estimate every contrast, then plot them on a normal probability plot. Inert contrasts are averages of noise, so they behave like a normal sample and fall on a straight line through about zero; real effects fall off it. Then untangle aliases with effect heredity: if A and C are active and B and D aren't, a large $AC + BD$ contrast is almost surely AC.

When it's worth it. Screening pays when there are many candidate factors (five or more), runs are expensive and few factors are expected to matter. It's a first step, not the last. Follow up with a foldover, the fraction with the signs of one or more factors reversed, which separates aliased effects: averaging the two estimates isolates the effect, and half their difference isolates its alias. Then study the few active factors with a full 2ᵏ factorial.

A common mistake. It's tempting to read the contrast $\ell_{AC}$ as "the AC interaction." In this design it's AC plus BD, and the data alone can't say which. Deciding takes engineering judgment or more runs. Related slips: choosing a generator that aliases two interactions you suspect are real, forgetting the products of generators when finding the resolution, and fractioning a design that was cheap enough to run in full.

See it

Aliasing in a 2³⁻¹ half fraction, where A shares its column with BC and the true A effect is 4. The fraction can only report A + BC, so as the BC interaction swings from −4 to +4, the reported 'A effect' swings from 0, hiding a real effect completely, to 8, doubling it. C comes out right only because its alias, AB, happens to be zero. A foldover or the full 2³ would separate them.

A normal probability plot of the seven contrasts (µm) from the eight-run molding experiment. Contrasts that are pure noise behave like a normal sample and line up on a straight line near zero. Three land far off it, and those are the real effects: melt temperature, holding pressure and the AC + BD chain, which effect heredity reads as AC since A and C are both active.

Worked examples

Example 1

Four molding factors are screened in 8 runs: A melt temperature, B injection speed, C holding pressure and D cooling time, with the generator $D = ABC$ . In standard order of A, B and C the runs are $(1)$ , $ad$ , $bd$ , $ab$ , $cd$ , $ac$ , $bc$ , $abcd$ , with warpage $213, 238, 206, 241, 132, 224, 127, 219$ µm. Find the defining relation, the alias structure and the A contrast.

1
Multiplying $D = ABC$ by $D$ gives $I = ABCD$ : resolution IV.

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

A $2^{5-1}$ design for five factors uses the generator $E = ABCD$ . What are its defining relation and resolution?

Problem 2+5 XP

In the $2^{4-1}$ design with $I = ABCD$ , which effect is aliased with the AB interaction?

Problem 3+5 XP

A four-run screening of a spray-coating line uses the half fraction $C = AB$ (so $I = ABC$ ): A is spray pressure, B gun distance and C line speed. The runs and coating thicknesses are $a = 42$ , $b = 31$ , $c = 36$ and $abc = 47$ µm. Compute the contrast for A, in µm. What does it actually estimate?

Problem 4+5 XP

To untangle that alias, the team runs the other half fraction ( $I = -ABC$ ) and gets $\ell'_A = 7$ µm, which estimates $A - BC$ . Estimate A and BC separately, in µm. Enter A, then BC.

Problem 5+5 XP

An unreplicated 16-run screening experiment gives 15 effect estimates. On a normal probability plot, 12 of them fall along a straight line through about 0, and 3 sit far above the line at the upper right. What do you conclude?

Problem 6+5 XP

A screening study has 6 candidate factors, and each run ties up a test furnace for 4 hours. How many furnace hours does a $2^{6-2}$ design save compared with the full $2^6$ factorial?

Problem 7+5 XP

A $2^{5-2}$ design (8 runs) uses the generators $D = AB$ and $E = AC$ . What are its complete defining relation and its resolution?

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

Return to the molding experiment (generator $D = ABC$ ; warpage in µm: $(1) = 213$ , $ad = 238$ , $bd = 206$ , $ab = 241$ , $cd = 132$ , $ac = 224$ , $bc = 127$ , $abcd = 219$ ). The A contrast is 61, and the B, D, $AB + CD$ and $AD + BC$ contrasts are all within $\pm 3.5$ . Compute the C contrast and the $AC + BD$ contrast, decide what the second one is, and predict the warpage at the best settings of A and C using a model with A, C and AC. Enter the C contrast, the $AC + BD$ contrast and the predicted warpage.

Where it's used

Screening designs are how engineers start on a process with many knobs: tuning a new injection mold, setting up a plasma etch recipe, choosing wave-soldering parameters or finding which of a dozen formulation ingredients drives a coating's adhesion. Resolution III and IV fractions, and their cousins the Plackett–Burman designs, routinely cut weeks of testing to days. The standard follow-up, a foldover and then a full factorial on the active factors, is built into how industrial experiments are planned.

2ᵏ Factorial Designs X̄ and R Control Charts

Fractional Factorials and Screening

Main idea

A half fraction from a generator

2^{4-1}: \quad {\htmlData{sym=0}{D = ABC}} \implies {\htmlData{sym=1}{I}} = ABCD

What the symbols mean

$D = ABC$
Generator
The extra factor's column is set equal to a product of the others' columns.
$I$
Identity column
All + signs. The defining relation lists every product of columns that equals it.

Aliases

A = A\cdot I = A\cdot ABCD = {\htmlData{sym=0}{A^2}}BCD = BCD, \qquad {\htmlData{sym=1}{\ell_A}} = A + BCD

What the symbols mean

$A^2$
A column times itself
Every entry is (±1)² = 1, so A·A = I. That's how aliases are found.
$\ell_A$
Contrast estimate
What the fraction actually estimates: the sum of an effect and all its aliases.

Runs in a fraction

{\htmlData{sym=2}{N}} = 2^{{\htmlData{sym=0}{k}}-{\htmlData{sym=1}{p}}}, \qquad \text{fraction} = \frac{1}{2^{\htmlData{sym=1}{p}}}

What the symbols mean

$k$
Factors
How many factors are studied.
$p$
Generators
How many factors are assigned to interaction columns; each one halves the runs.
$N$
Runs
The size of the design.

Explanation

Resolution. The length of the shortest word in the defining relation sets what is safe to estimate:

Resolution III (shortest word has 3 letters, like $I = ABC$ ): main effects are aliased with two-factor interactions. Pure screening.
Resolution IV ( $I = ABCD$ ): main effects are clear of two-factor interactions, but two-factor interactions are aliased with each other.
Resolution V ( $I = ABCDE$ ): main effects and two-factor interactions are all clear, if higher-order interactions are negligible.

With two or more generators, every product of the generator words is also in the defining relation, and that's where the shortest word often hides.

See it

Worked examples

Example 1

1
Multiplying $D = ABC$ by $D$ gives $I = ABCD$ : resolution IV.

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

A $2^{5-1}$ design for five factors uses the generator $E = ABCD$ . What are its defining relation and resolution?

Problem 2+5 XP

In the $2^{4-1}$ design with $I = ABCD$ , which effect is aliased with the AB interaction?

Problem 3+5 XP

Problem 4+5 XP

To untangle that alias, the team runs the other half fraction ( $I = -ABC$ ) and gets $\ell'_A = 7$ µm, which estimates $A - BC$ . Estimate A and BC separately, in µm. Enter A, then BC.

Problem 5+5 XP

Problem 6+5 XP

A screening study has 6 candidate factors, and each run ties up a test furnace for 4 hours. How many furnace hours does a $2^{6-2}$ design save compared with the full $2^6$ factorial?

Problem 7+5 XP

A $2^{5-2}$ design (8 runs) uses the generators $D = AB$ and $E = AC$ . What are its complete defining relation and its resolution?

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.