2ᵏ Factorial Designs

Main idea

Try every combination of your factors, each at just two levels, low and high. Then every effect, and every interaction between factors, is a simple comparison: the average result where it's "high" minus the average where it's "low," and every one of those comparisons gets to use all the runs.

An effect is a contrast

\text{effect} = {\htmlData{sym=0}{\bar y_{+}}} - \bar y_{-} = \frac{\text{contrast}}{{\htmlData{sym=1}{n}}\,2^{{\htmlData{sym=2}{k}}-1}}

What the symbols mean

$\bar y_{+}$
Average at +
Mean of all runs where this effect's sign column is +.
$n$
Replicates
Runs at each combination of levels.
$k$
Number of factors
A full design has 2ᵏ combinations.

Coded regression model

\hat y = \bar y + \frac{A}{2}{\htmlData{sym=0}{x_A}} + \frac{B}{2}x_B + \frac{{\htmlData{sym=1}{AB}}}{2}{\htmlData{sym=0}{x_A}}x_B, \qquad x = \pm 1

What the symbols mean

$x_A$
Coded level
−1 at the low setting and +1 at the high: x = (setting − center)/(half the range).
$AB$
Interaction effect
Half the difference between A's effect at high B and at low B.

Precision of an effect

SE(\text{effect}) = \frac{2{\htmlData{sym=0}{s}}}{\sqrt {\htmlData{sym=1}{N}}}, \qquad SS_{\text{effect}} = \frac{\htmlData{sym=1}{{N}}}{4}(\text{effect})^2

What the symbols mean

$s$
Error SD
From replicates, or from pooling negligible high-order effects.
$N$
Total runs
n·2ᵏ.

Explanation

A two-level factorial is simply an experiment that tries every combination of a few settings, each at a low and a high value, like feed slow or fast with speed low or high. The name says it: each factor gets two levels, and the runs cover every combination. The math is mostly one move: an effect is the average response at the "+" runs minus the average at the "−" runs. An interaction is the same move on a column made by multiplying two factors' signs, and it measures whether one knob's effect depends on where the other knob is set. Averages and subtraction, that's it.

The scene. A machinist wants smoother aluminum parts and has two knobs: feed rate (A) and spindle speed (B). Changing one knob at a time (feed up at low speed, then speed up at low feed) never visits high feed with high speed, so it can't see whether the knobs interact. A $2^2$ factorial runs all four corners: $(1)$ with both low, $a$ with A high, $b$ with B high and $ab$ with both high. The lowercase letters name the factors that are high; listed this way the runs are in standard order.

Main effects. The effect of A is the average response at high A minus the average at low A. With cell means:

A = \frac{a + ab}{2} - \frac{(1) + b}{2}

Each half averages over both levels of B, so the A effect is measured at both speeds, for free. That's the hidden bargain of factorials: every run helps estimate every effect.

Interactions. A's effect at high B is $ab - b$ ; at low B it's $a - (1)$ . The interaction is half their difference:

AB = \frac{(ab - b) - (a - (1))}{2} = \frac{(1) + ab}{2} - \frac{a + b}{2}

the "same-sign" corners minus the "opposite-sign" corners. On an interaction plot (mean response against A, one line for each level of B) parallel lines mean no interaction; lines that converge, diverge or cross mean the effect of one factor depends on the other.

Sign tables. Write a column of $-1$ s and $+1$ s for each factor in standard order; an interaction's column is the product of its factors' columns. Every effect is the sum of $y$ where its column is $+$ , minus the sum where it's $-$ , divided by half the number of runs. A $2^3$ has 8 runs and 7 effects: A, B, C, AB, AC, BC and ABC.

Effects are regression coefficients, doubled. With coded levels $x = \pm1$ , any two columns have as many matching signs as mismatched ones, so the columns are orthogonal and $X^TX = N\,I$ . Least squares from multiple regression then gives each coefficient on its own: $b = \sum xy/N$ , the contrast over $N$ . An effect is the change from $x = -1$ to $x = +1$ , two units, so $b = \text{effect}/2$ . Orthogonality is also why dropping one term changes no other coefficient: the confounding trouble of plant records is designed away.

Is an effect real? Each effect is a difference of two averages of $N/2$ runs, so $\operatorname{Var}(\text{effect}) = \sigma^2/(N/2) + \sigma^2/(N/2) = 4\sigma^2/N$ and $SE = 2s/\sqrt N$ . With replicates, $s^2$ is the pooled variance within cells; compare $t = \text{effect}/SE$ with the $t$ critical value on the error degrees of freedom, as in testing against a spec. In an unreplicated design, pool the small high-order interactions as error, or use a probability plot of the effects (see fractional factorials).

A common mistake. When an interaction is large, it's tempting to report the main effect of A on its own. But the A effect is then an average of two different slopes, one at each level of B, and it describes neither. Read the interaction plot first. Two smaller slips: dividing a contrast by the wrong count (use half the runs, $N/2$ ), and actually running in standard order. Standard order is for bookkeeping; the runs themselves go in random order.

See it

An interaction plot of surface roughness against feed rate, with one line per spindle speed. When the interaction AB is 0 the lines are parallel, so feed has the same effect at both speeds. The milling data sit at AB = −0.6, where the lines visibly fan: feed adds 2.0 µm at low speed but only 0.8 µm at high speed. The main effect, 1.4 µm, is just the average of those two slopes.

The 2³ reactor experiment as a cube: A (temperature) runs left to right, B (catalyst) bottom to top and C (stirring) front to back, with each corner's yield. Stepping through the seven sign columns highlights the four + corners for each effect, which is (sum of the highlighted corners − sum of the others)/4. Notice how A, B and AB stand out at 8, 4 and 3.1, while C, AC, BC and ABC all sit within ±0.4, which is noise.

Worked examples

Example 1

Surface roughness Ra (µm) of milled aluminum, two replicates per corner: $(1)$ : $1.9, 2.1$ ; $a$ (high feed): $3.8, 4.2$ ; $b$ (high speed): $1.5, 1.7$ ; $ab$ : $2.5, 2.3$ . Estimate the effects and test them, with $t = 2.776$ for 4 degrees of freedom.

1
Cell means: $(1) = 2.0$ , $a = 4.0$ , $b = 1.6$ , $ab = 2.4$ .

Practice

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

Problem 1+5 XP

A $2^2$ experiment on a solder reflow oven varies peak temperature (A) and conveyor speed (B) and measures the void area under a chip, in percent. The cell means are $(1) = 12$ , $a = 8$ , $b = 15$ and $ab = 7$ . Find the main effect of peak temperature.

Problem 2+5 XP

On an interaction plot of mean yield against catalyst level, the line for low temperature slopes up and the line for high temperature slopes down; they cross. What's the right conclusion?

Problem 3+5 XP

A $2^3$ experiment on a batch reactor (A temperature, B catalyst concentration, C stirring rate) gives yields (%) in standard order: $(1) = 55.6$ , $a = 59.9$ , $b = 56.5$ , $ab = 67.4$ , $c = 55.5$ , $ac = 61.0$ , $bc = 56.4$ , $abc = 67.7$ . Estimate the AB interaction.

Problem 4+5 XP

In a $2^3$ design, which four runs have a $+$ sign in the AC column?

Problem 5+5 XP

For the milling experiment ( $\bar y = 2.5$ µm, effects $A = 1.4$ , $B = -1.0$ , $AB = -0.6$ ), write the coded model and predict the roughness with feed halfway between its center and high levels ( $x_A = 0.5$ ) and high spindle speed ( $x_B = 1$ ), in µm.

Problem 6+5 XP

A $2^3$ design is run twice, 16 runs in all. The pooled within-cell SD is $s = 1.2$ and the C effect is estimated as 1.5. Compute the $t$ statistic for C and compare it with $t = 2.306$ (8 degrees of freedom).

Problem 7+5 XP

The reactor experiment is unreplicated. Treat AC, BC and ABC (effects $0.4$ , $-0.2$ and $-0.2$ ) as noise and pool their sums of squares into an error term. Compute $F$ for the C effect, $0.3$ , and compare it with $F_{0.05,1,3} = 10.13$ .

Challenge problem+25 XP

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

For the reactor experiment, (a) find the effect of temperature (A) at low catalyst and at high catalyst, and (b) using the coded model with only the A, B and AB terms, predict the yield at the best corner. Enter the two A effects, then the predicted yield in percent.

Where it's used

Two-level factorials are the workhorse of industrial experimentation: tuning injection-molding and machining parameters, optimizing chemical and semiconductor recipes, calibrating engine maps, and choosing settings for welding and 3D printing. Engineers read the interaction plots to find settings that work together, and the coded model becomes the transfer function a process controller uses. The same designs underpin robust design, where factors are chosen to make a product insensitive to noise it can't control.

Randomization, Replication and Blocking Fractional Factorials and Screening

2ᵏ Factorial Designs

Main idea

An effect is a contrast

\text{effect} = {\htmlData{sym=0}{\bar y_{+}}} - \bar y_{-} = \frac{\text{contrast}}{{\htmlData{sym=1}{n}}\,2^{{\htmlData{sym=2}{k}}-1}}

What the symbols mean

$\bar y_{+}$
Average at +
Mean of all runs where this effect's sign column is +.
$n$
Replicates
Runs at each combination of levels.
$k$
Number of factors
A full design has 2ᵏ combinations.

Coded regression model

\hat y = \bar y + \frac{A}{2}{\htmlData{sym=0}{x_A}} + \frac{B}{2}x_B + \frac{{\htmlData{sym=1}{AB}}}{2}{\htmlData{sym=0}{x_A}}x_B, \qquad x = \pm 1

What the symbols mean

$x_A$
Coded level
−1 at the low setting and +1 at the high: x = (setting − center)/(half the range).
$AB$
Interaction effect
Half the difference between A's effect at high B and at low B.

Precision of an effect

SE(\text{effect}) = \frac{2{\htmlData{sym=0}{s}}}{\sqrt {\htmlData{sym=1}{N}}}, \qquad SS_{\text{effect}} = \frac{\htmlData{sym=1}{{N}}}{4}(\text{effect})^2

What the symbols mean

$s$
Error SD
From replicates, or from pooling negligible high-order effects.
$N$
Total runs
n·2ᵏ.

Explanation

Main effects. The effect of A is the average response at high A minus the average at low A. With cell means:

A = \frac{a + ab}{2} - \frac{(1) + b}{2}

Each half averages over both levels of B, so the A effect is measured at both speeds, for free. That's the hidden bargain of factorials: every run helps estimate every effect.

Interactions. A's effect at high B is $ab - b$ ; at low B it's $a - (1)$ . The interaction is half their difference:

AB = \frac{(ab - b) - (a - (1))}{2} = \frac{(1) + ab}{2} - \frac{a + b}{2}

See it

Worked examples

Example 1

1
Cell means: $(1) = 2.0$ , $a = 4.0$ , $b = 1.6$ , $ab = 2.4$ .

Practice

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

Problem 1+5 XP

Problem 2+5 XP

On an interaction plot of mean yield against catalyst level, the line for low temperature slopes up and the line for high temperature slopes down; they cross. What's the right conclusion?

Problem 3+5 XP

Problem 4+5 XP

In a $2^3$ design, which four runs have a $+$ sign in the AC column?

Problem 5+5 XP

Problem 6+5 XP

Problem 7+5 XP

Challenge problem+25 XP

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