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I have more factors than runs — which design, and what will it fail to tell me?

Run a fraction of the full experiment and see exactly what it costs — the alias structure is on screen before a single run is booked.

Fractional Factorial (Screening Design) · improve · Black Belt

Use this when

  • More candidate factors than a full factorial's run budget allows
  • Screening many inputs down to the few worth a real experiment
  • Before agreeing to a design somebody else chose, to see what it confounds

Fractional factorial (screening design)

sandbox

2^(4-1): 4 factors in 8 runs — one half of the 16 the full design would need. Resolution IV.

Half of a 2^4. Main effects clear of two-factor interactions; the three two-factor pairs are aliased with each other.

Defining relation
I = ABCD
Resolution
IV
Runs saved
8

What each estimate is a sum of

This design never varied the aliased terms independently, so no arithmetic on its data can separate them. The chains below are fixed by the design, not by the results.

Every effect this design estimates, with the effects it is aliased with.
What this estimate ISLowest aliased orderValueVerdict
A + BCD319.000Noise-sized
B + ACD31.500Noise-sized
AB + CD2 — confounded with a two-factor interaction-1.000Noise-sized
C + ABD314.000Noise-sized
AC + BD2 — confounded with a two-factor interaction-18.500Noise-sized
BC + AD2 — confounded with a two-factor interaction19.000Noise-sized
ABC + D1 — confounded with a main effect16.500Noise-sized

Nothing stands out from the noise floor. That is a finding about the factors chosen and the size of this fraction rather than about the process — and with few contrasts and several large effects, the floor itself rises to meet them, so a null reading here is weak evidence either way. In this 2^(4-1) design every estimate is a SUM of its alias chain, not a single effect. It ran 8 of the 16 combinations, so the aliased terms were never varied independently and cannot be separated by any arithmetic on this data. Which member of a chain is doing the work is a question for the next experiment — a fold-over, or the complementary fraction. Naming one of them in a report is a choice made on prior belief, and it should be labelled as one.

Lenth PSE
24.750
Margin of error
93.162
Contrasts
7

This is a resolution IV design: main effects are clear of two-factor interactions, but the two-factor interactions are aliased with each other. A significant interaction estimate names a PAIR of candidates, and separating them needs more runs.

This design is unreplicated, so there is no estimate of experimental error — every degree of freedom went into effects. Significance here comes from Lenth's method, which builds a noise floor from the smaller contrasts themselves. It is a screening judgement, good at finding the effects worth a confirmation run; it is not an F test, and a borderline effect deserves the replicated follow-up rather than a verdict.

Run these in RANDOM order; enter them here in this order. Generated factors are set by the design, not chosen.
#ABCD
1−−−−
2+−−+
3−+−+
4++−−
5−−++
6+−+−
7−++−
8++++
How this is calculated
  • The design is built from a base 2k−p factorial in standard order, with each generated factor’s level set by its generator word. Generating the runs this way — rather than filtering the full design and keeping its order — is what makes the estimates match their labels.
  • The defining relation is the GROUP generated by the generator words: every product of a subset of them, so a quarter fraction has three words and not two. Forgetting those generalised interactions is the classic way to understate a design’s aliasing.
  • Resolution is the length of the shortest defining word. Resolution III aliases main effects with two-factor interactions; IV keeps main effects clear but aliases two-factor interactions with each other; V keeps both clear.
  • Every estimate is the sum of its alias chain, computed as the base design’s contrast. The value shown for “A” in a 2(4−1) is A + BCD, and the table says so rather than printing “A”.
  • Significance follows the full factorial rules: F tests when replicated, Lenth’s method when not — and with few contrasts and several large effects, Lenth’s floor rises to meet them, so a null reading in a small fraction is weak evidence either way.

Source: Montgomery, D.C., Design and Analysis of Experiments 8e, Ch. 8; Example 8.1 is the preset and the verification fixture. Box, Hunter & Hunter, Statistics for Experimenters 2e, Ch. 6.

How this is calculated

Two-level fractional factorial 2^(k-p). Runs are generated from a base 2^(k-p) factorial in standard order with each generated factor set by its generator word — not by filtering the full design, whose ordering would mislabel every effect while leaving the numbers plausible. The defining relation is the GROUP generated by the generator words, so a quarter fraction has 2^p - 1 = 3 words rather than 2; the generalised interactions alias as surely as the originals. Resolution is the length of the shortest defining word. Every estimate is the base design's contrast and is reported as the SUM of its alias chain, never under one member's name. Significance follows the full factorial: F tests when replicated, Lenth's method when not, with the caveat that a small fraction's noise floor rises to meet several large effects.

Source: Montgomery, D.C., Design and Analysis of Experiments 8e, Ch. 8 — two-level fractional factorials; Example 8.1 (the 2^(4-1) half fraction of the filtration experiment) is the preset and the verification fixture, checked both against its published estimates and against the exact identity that each half-fraction estimate equals the sum of its aliased pair from the full factorial. Box, Hunter & Hunter, Statistics for Experimenters 2e, Ch. 6.

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