Factorial DOE

Method

• Factors and coding: 2 to 10 factors A, B, C … K (Minitab skips I), each with a numeric low and high level. Coded units: low = −1, high = +1, the midpoint 0 (center points). The analysis codes each factor column from its own lowest and highest value; a row must have every factor at its low or high level (a corner point) or every factor at its midpoint (a center point).
• Designs: the full factorial 2^k, or a fraction 2^k−p: the first k − p factors in standard (Yates) order (A changes fastest), each added factor the product of the base factors in its generator. The generators below are minimum aberration designs (the highest resolution, then the fewest shortest words): an exhaustive search over every choice of generators in the reference script finds the same word length pattern for every entry. Unverified: whether each entry is the same set of words as Minitab’s catalogue could not be checked; an equivalent choice gives the same resolution and the same number of aliases, but the alias table may name other (equally short) words.
• Generator catalogue (fractions up to 128 runs; the full factorial is always offered):

  Fractional designs: generators (unverified against Minitab)

  Factors	Runs	Fraction	Resolution	Generators
  3	4	1/2	III	C = AB
  4	8	1/2	IV	D = ABC
  5	8	1/4	III	D = AB, E = AC
  5	16	1/2	V	E = ABCD
  6	8	1/8	III	D = AB, E = AC, F = BC
  6	16	1/4	IV	E = ABC, F = BCD
  6	32	1/2	VI	F = ABCDE
  7	8	1/16	III	D = AB, E = AC, F = BC, G = ABC
  7	16	1/8	IV	E = ABC, F = BCD, G = ACD
  7	32	1/4	IV	F = ABCD, G = ABDE
  7	64	1/2	VII	G = ABCDEF
  8	16	1/16	IV	E = BCD, F = ACD, G = ABC, H = ABD
  8	32	1/8	IV	F = ABC, G = ABD, H = BCDE
  8	64	1/4	V	G = ABCD, H = ABEF
  8	128	1/2	VIII	H = ABCDEFG
  9	16	1/32	III	E = ABC, F = BCD, G = ACD, H = ABD, J = ABCD
  9	32	1/16	IV	F = BCDE, G = ACDE, H = ABDE, J = ABCE
  9	64	1/8	IV	G = ABCD, H = ACEF, J = CDEF
  9	128	1/4	VI	H = ACDFG, J = BCEFG
  10	16	1/64	III	E = ABC, F = BCD, G = ACD, H = ABD, J = ABCD, K = AB
  10	32	1/32	IV	F = ABCD, G = ABCE, H = ABDE, J = ACDE, K = BCDE
  10	64	1/16	IV	G = BCDF, H = ACDF, J = ABDE, K = ABCE
  10	128	1/8	V	H = ABCG, J = BCDE, K = ACDF

• Defining relation, resolution and aliases: the defining relation is every word whose product is the same (+1 or −1) in all factorial runs (computed from the runs, so it also holds after a fold-over); the resolution is the length of its shortest word (III: main effects aliased with two-factor interactions; IV: main effects clear, two-factor interactions aliased with each other; V: both clear). The alias chain of a term is its product with every word of the relation; the alias table lists, as Minitab’s “Alias Structure”, each chain with its members up to the chosen order.
• Fold-over (fractions only): the runs are added again with the signs of every factor (or of one factor) reversed. Folding on all factors removes the words of odd length (a resolution III design becomes IV); folding on one factor frees that factor and its two-factor interactions. The folded runs are part of the same blocks (no extra block; FactoKit’s convention).
• Center points, replicates and blocks: center points are added to every block. Blocks hold whole replicates when the number of blocks divides the number of replicates (nothing is confounded), or each replicate is split into 2 or 4 blocks by confounding: the block generators are chosen so that the effects confounded with blocks (each read through its alias chain) are of the highest order possible — the sorted orders of the confounded chains are compared from the lowest up, ties go to the first in term order — and never a main effect. A full factorial in 2 blocks confounds the highest-order interaction (ABC…). Standard order runs through the blocks; within a block the factorial runs (replicate by replicate) come first, then the center points, as Minitab’s StdOrder.
• Randomization: the run order is shuffled within each block (blocks are run in order) by a Fisher–Yates shuffle driven by the Mulberry32 generator from the seed, so the same seed gives the same order in every browser and in the offline file. The worksheet (StdOrder, RunOrder, CenterPt with 1 = corner and 0 = center, Blocks, the factors in uncoded units and an empty response column) is listed in run order and saved as CSV or Excel.
• Model: the terms up to the chosen order (default 2: main effects and two-factor interactions) minus the terms unticked by hand, plus the blocks (effect coded: +1 in its block, −1 in the last block) and the center-point term (1 at the center points, 0 elsewhere), as Minitab’s defaults. Terms are taken in the order blocks, main effects, two-factor interactions …, center point; a term whose column adds nothing to the columns before it (Gram–Schmidt, tolerance 10⁻⁷) is left out and reported as aliased with the earlier term of its chain or as confounded with blocks — so only one term of each alias chain is estimated, the first in term order.
• Effects and coefficients: least squares by Householder QR (never by inverting XᵀX) on the coded columns. Effect = 2 × coefficient (the change of the mean response from the low to the high level). With error degrees of freedom: SE Coef = √(MSE · cⱼⱼ), T = Coef / SE Coef, two-sided P from the t distribution with the error degrees of freedom; VIF = cⱼⱼ · Σ (xᵢⱼ − x̄ⱼ)² (1 for an orthogonal design).
• Analysis of variance (adjusted sums of squares): Model = SST − SSE with p − 1 degrees of freedom; Blocks, each group (Linear, 2-Way Interactions …), each term and Curvature: the increase of SSE when they are left out, bᵀ C⁻¹ b over their coefficients (C = (XᵀX)⁻¹ restricted to them); F = MS / MSE. Curvature is the test of the center-point term: whether the mean at the center differs from the mean of the corners. Pure error is the error of a model with one mean per setting of the factors in the model (the center points one setting) plus the blocks when they are in the model; lack of fit = SSE − pure error, tested against it. S = √MSE, R-sq = 1 − SSE / SST, R-sq(adj) = 1 − MSE / (SST / (n − 1)), R-sq(pred) = 1 − PRESS / SST with PRESS = Σ (eᵢ / (1 − hᵢ))² (not defined when a leverage is 1, e.g. a model that fits every corner exactly).
• Lenth’s method (no degree of freedom left for error, e.g. an unreplicated design with every term in the model): s₀ = 1.5 · median |effect|, pseudo standard error PSE = 1.5 · median of the |effects| below 2.5 · s₀; t = effect / PSE with m / 3 degrees of freedom (m effects), margin of error ME = t(1 − α/2; m/3) · PSE and simultaneous margin SME = t(γ; m/3) · PSE with γ = (1 + (1 − α)^(1/m)) / 2 (Lenth 1989). An effect is significant when |effect| > ME, as Minitab marks effects with Lenth’s PSE.
• Effect plots: the Pareto chart shows |T| (with error degrees of freedom; reference line t(1 − α/2; error df)) or |effect| (Lenth; reference line ME), largest first. The normal plot shows the effects (or T) against the normal scores of the median ranks (i − 0.3) / (m + 0.4); the half-normal plot the absolute values against Φ⁻¹(0.5 + 0.5 · (i − 0.3) / (m + 0.4)). The line is where the effects would fall if none were real (slope 1 for T, PSE for effects); significant effects are labelled squares.
• Factorial plots: main effects plot = mean response of the corner points at each factor’s low and high level (data means; the overall mean dashed, the center points’ mean as a diamond); interaction plots = the means at the four combinations of two factors, one line per level of the second (dashed low, solid high), for the two-factor interactions in the model (the 12 largest). Parallel lines mean no interaction.
• Equation in uncoded units: each coded term ∏ (Xⱼ − Mⱼ) / Hⱼ (M the midpoint, H half the range) is expanded into products of the uncoded factors; the block terms are left out (the equation is the average over the blocks) and the center-point term is written as “Ct Pt” (1 at the center points).
• Response optimizer (one response, Derringer–Suich desirability with weight 1, as Minitab’s Response Optimizer): maximize: d = 0 below the lower limit, (y − L) / (T − L) up to the target, 1 above; minimize: 1 below the target, (U − y) / (U − T) up to the upper limit, 0 above; target: rising from L to T and falling from T to U. Empty limits take the lowest and highest response in the data. The prediction uses the model without blocks and without the center-point term. That model is linear in each factor, so its maximum and minimum over the cube (−1 … +1) are at corners, and maximize / minimize check every corner of the factors in the model (factors not in the model are shown at the center). A target between the lowest and highest corner prediction is reached on the line from the center of the design towards the corner on its side (the first point where the prediction equals the target; FactoKit’s rule, as the solution is not unique). The fit is given with its standard error and the confidence and prediction intervals at 1 − α. When the curvature test is significant the optimum may lie inside the region, which a response surface design would show.
• Checks: every design (runs, run order, defining relation, alias lines, block words) and every analysis number is compared with an independent Python script: designs built with numpy and itertools, the minimum aberration search, statsmodels OLS for the coefficients, SE, T, P and fitted values, refitted models for every adjusted sum of squares and for PRESS, statsmodels OLS of one mean per setting for the pure error, variance_inflation_factor, OLSInfluence for the standardized residuals, numpy and scipy for Lenth’s method, brute force over the corners and scipy brentq for the optimizer, get_prediction for the intervals. Published check: Montgomery, Design and Analysis of Experiments, Example 6.2 (filtration rate, 2⁴ unreplicated): the effects A 21.625, C 9.875, D 14.625, AC −18.125 and AD 16.625 and Lenth’s PSE 2.625 are reproduced (“Load unreplicated example”).
• Example: synthetic data, not from a real machine: shrinkage of an injection-moulded part in a 2⁵⁻¹ design (resolution V, E = ABCD) run on two days (two replicates as two blocks) with two center points per day. Mould temperature, holding pressure and their interaction are real, the other factors are not; there is no curvature.
• Files: CSV, TSV and TXT are read as if pasted (UTF-8, UTF-16, otherwise Windows-1257). XLSX: choose the sheet and the columns or a cell range; formula cells use the value Excel last saved, dates are kept as text. Files up to 20 MB are read in this browser only and never uploaded.
• XLSX files are read with SheetJS Community Edition 0.20.3, Apache License 2.0 (license), loaded only when an Excel file is opened or created.