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    DOE / screening

    Fractional factorial design

    A screening experiment rarely needs the full factorial. Halving the runs aliases one effect with another, and the defining relation says exactly which. This builds the design, blocks it on a term you can afford to lose, and shows which estimates are clean and which are folded together.

    Fraction2^3Blocks1Alias groups7
    Runs
    8
    16 in the full factorial
    Resolution
    R4
    I = ABCD
    Runs per block
    —
    No blocking
    Largest effect
    A 4.00
    True effect 4.00
    Design matrix
    Coded levels in standard order, with the response each run produces.
    8 runs
    Design matrix
    RunABCDBlockResponse
    1−−−−—50.3
    2+−−+—50.8
    3−+−+—44.3
    4++−−—51.8
    5−−++—51.8
    6+−+−—52.3
    7−++−—45.8
    8++++—53.3

    Standard order. Randomise the run order before executing the experiment.

    Inspect the model
    Choose a quick result view, a scaled comparison, or the complete diagnostic set.
    Estimated against true effects
    Where the bars differ, an alias has folded another term into the estimate.
    Estimated against true effects-3.02-1.380.2561.893.535.17AABBCCDDABABCDCDACACBCBCADADEstimateTrue effect
    Effect magnitude
    Ranked by size — the order a normal plot would put them in.
    Effect magnitude0.0000.9441.892.833.784.72AAABABCDCDBBCCACACBCBCDDADADBDBD|Effect|
    Canonical equations
    The exact relationships implemented by this engine.

    I = ABCD · Every alias follows from these words.

    e = (2/n) Σ xᵢ yᵢ · Difference between the high and low level means.

    effect · I = effect · word · Multiplying by a defining word gives the alias.

    |e| vs Φ⁻¹(0.5 + 0.5·p) · The visual test for which effects are real.

    R = min |word| · R4 means no main effect shares a word with a two-factor interaction.

    engineering methodology

    A clear path from inputs to results.

    Screening experiments rarely need the full factorial. Halving the runs costs you an alias: one effect is estimated as the sum of two. The defining relation says exactly which pairs are folded together, and blocking lets you absorb a nuisance shift in the day or the batch — as long as you spend a term you can afford to lose.

    Step-by-step methodWorked exampleAssumptions explained

    How to use

    1. Pick the fraction: full, half, quarter or eighth.
    2. Turn blocking on to split the runs across two days or batches.
    3. Set the effects the process actually has, including a hidden interaction.
    4. Read the design matrix, the defining relation and the alias groups.
    5. Compare the estimated effects with the truth to see what the design can separate.

    The formula

    I=ABCD  ;  e=2n∑ixiyi  ;  R=min⁡∣w∣I = ABCD \;;\; e = \frac{2}{n}\sum_i x_i y_i \;;\; R = \min |w|

    Reference: Box, Hunter & Hunter, Statistics for Experimenters, ch. 12 (fractional factorials, resolution and blocking).

    Worked example

    2⁴⁻¹ with D = ABC, eight runs

    Inputs
    • Four factors, one generator D = ABC, so eight runs instead of sixteen.
    • Effects: A 4.0, B −2.5, C 1.5, AB 2.0, and a hidden CD of 1.5.
    Calculation path
    1. 1

      The defining relation is I = ABCD, so the design is resolution IV.

    2. 2

      A is aliased with BCD and B with ACD; main effects stay clear of two-factor interactions.

    3. 3

      AB is aliased with CD. The estimate for AB is 2.0 + 1.5 = 3.5.

    4. 4

      The same column estimates CD at 3.5 — two numbers, one estimate.

    5. 5

      Blocking on AB would split the runs 4 and 4, but it would also give up AB and CD.

    Result

    8 runs · resolution IV · AB = CD = 3.5 from a true 2.0 and 1.5.

    Interpreting the result

    Resolution IV is the usual compromise for screening: main effects are estimated cleanly, but two-factor interactions come in pairs. If the follow-up experiment shows that one of a pair matters, the other has to be separated by folding the design over — running the complementary fraction. Blocking spends a term, so block on the highest-order interaction you are willing to lose.

    Assumptions & limitations

    • Two-level factors, orthogonal ±1 coding, no centre points.
    • Responses are generated from the effect model you set, so the fit shows what the design can recover rather than a measurement.
    • No noise: every run is the exact model value.
    • Standard run order; randomise before executing.

    Frequently asked questions

    What does resolution IV mean in practice?

    Main effects are clear of two-factor interactions, but two-factor interactions are aliased with each other. You can screen factors safely and cannot interpret an interaction without a follow-up.

    How many runs do I need?

    Enough to estimate the effects you care about with some degrees of freedom for error. A half fraction of five factors gives 16 runs; an eighth gives 8 and costs resolution.

    Why block on a high-order interaction?

    Because you are unlikely to need it. Confounding blocks with a main effect would destroy the very thing the experiment is for.

    Can I add centre points?

    Yes, and they test for curvature. This page covers the two-level screening stage; the response-surface page handles the curved follow-up.

    Methodology last reviewed 2026-10-21. Results are informational and not financial or engineering advice — see our disclaimer.