2ᵏ full factorial DOE

The language of designed experiments

• Factor: an input you set on purpose (mold temperature). Level: a setting of a factor (40 °C and 60 °C, coded − and +). Response: the output you measure.
• Run (treatment combination): one setting of every factor. A 2^k full factorial runs every combination of k factors at two levels, 2^k runs in all.
• Replicate: a complete independent repeat of a run, giving an estimate of pure error. A repeated measurement of the same part is not a replicate.
• Randomization protects against lurking variables such as tool wear or ambient temperature; blocking removes a known nuisance source such as material lot or day.
• The design is balanced and orthogonal: each factor is at each level equally often and the effects can be estimated independently of each other.

Effects and coefficients

The main effect of a factor is the average response at its high level minus the average at its low level. An interaction effect such as AB is computed the same way using the product of the coded columns; it measures how much the effect of A changes with the level of B. In coded units the regression coefficient is half the effect, and the constant is the grand mean. With r replicates, the standard error of every effect is 2s/√N, where s² is the pooled variance within runs and N = 2^k × r, and each effect gets a t test on 2^k(r − 1) degrees of freedom. With some replicates missing, effects come from the run means and the tests are approximate. An unreplicated design has no pure error; the tool then uses Lenth's pseudo standard error, which assumes most effects are inactive and needs k ≥ 3.

Reading the plots

• Main effects plot: a steep line is a large effect; a flat line, a small one. The direction tells you which level to use for the goal.
• Interaction plot: parallel lines mean no interaction. Lines that diverge or cross mean the best level of one factor depends on the other, and the main effects alone can mislead.
• Pareto of effects: bars past the reference line are statistically significant at the chosen α.

Common traps

• Two levels can only fit a straight line. Add center points to detect curvature before predicting between the levels.
• Always run confirmation trials at the chosen settings before changing the process.
• A full factorial has no aliasing. A fractional factorial saves runs by aliasing effects with each other, as below.

Fractional factorials: generators, aliases and resolution

A 2^k−p design runs 1/2^p of the full factorial. Write out a full factorial in k − p base factors, then set each extra factor from a generator: in a 2⁴⁻¹ with D = ABC, D is run at the sign of the A×B×C column. Multiplying the generator by D gives the defining relation I = ABCD (a letter times itself is I). Multiply any effect by the defining relation to find its aliases: A × ABCD = BCD, so the A column also estimates BCD, and AB × ABCD = CD, so AB and CD share a column. The experiment estimates the sum of the effects in each alias chain and cannot separate them.

The resolution is the length of the shortest word in the defining relation. Resolution III: main effects are aliased with two-factor interactions (screening many factors). Resolution IV: main effects are clear of two-factor interactions, but two-factor interactions are aliased with each other. Resolution V: main effects and two-factor interactions are clear of each other. Choose a fraction in section 2: the tool uses the standard minimum-aberration generators unless you type your own (a minus sign, as in D = −ABC, gives the other half), and lists the defining relation, the resolution and every alias chain. The effects table shows the chain behind each estimate, and the checks warn when a significant effect shares its chain with another plausible one. The button in section 2 loads a worked 2⁴⁻¹ example.

On the exam

CSSGB: expect DOE vocabulary questions, counting runs (2^k × replicates), computing a main or interaction effect from a small table, and reading main effects and interaction plots. The CQE asks you to apply the design principles, including confounding (VI.H.3), and to construct and analyze full factorials (VI.H.4) and two-level fractional factorials (VI.H.5). The CSSBB covers confounding and resolution (VII.A.2) and how confounding affects the use of fractional factorials (VII.A.5). Expect to find the aliases of an effect from a generator, state a design's resolution, and say why a resolution III design is risky when interactions are likely.

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