| Term | Definition | Example |
|---|
| Factor | A variable deliberately changed in the experiment to study its effect on the response. | Temperature, Pressure, Catalyst type, Speed. |
| Level | A specific value or setting of a factor. | Temperature at 150°C (low) and 180°C (high) = 2 levels. |
| Response (y) | The output variable measured as the result of the experiment. | Yield (%), Tensile strength (MPa), Defect count. |
| Treatment | A specific combination of factor levels applied to an experimental unit. | In a 2² design: (Temp=High, Pressure=Low) is one treatment. |
| Treatment combination | One complete set of factor level settings; one "run" of the experiment. | "ab" in Yates notation = Factor A high, Factor B high. |
| Experimental unit | The smallest entity to which a treatment is independently applied. Determines the error term. | One oven batch (not one cookie from the batch). |
| Observational unit | The entity on which a measurement is taken. Can be smaller than the experimental unit. | One cookie measured from an oven batch. Multiple cookies per batch = subsamples. |
| Replicate | An independent re-run of the entire treatment under the same conditions. | Running the (High Temp, Low Pressure) combination 3 separate times on 3 different days. |
| Repeat (subsample) | Re-measurement of the same experimental unit. NOT an independent run. | Taking 2 lab samples from the same reaction batch. These are NOT replicates! |
| Pseudoreplication | Treating subsamples or repeats as if they were independent replicates. Underestimates σ². | 5 cookies from 1 oven run treated as n=5 instead of n=1. Gives falsely small p-values. |
| Randomisation | Running treatments in random order to protect against lurking variables and time trends. | Using a random number generator to determine the sequence of 16 runs in a 2⁴ design. |
| Blocking | Grouping experimental units into homogeneous "blocks" to remove known nuisance variation. | Each day is a block. All treatments run within each day. Day-to-day variation removed from error. |
| Block | A group of homogeneous experimental units. Within-block variation is small. | One batch of raw material, one machine, one operator shift, one day. |
| Nuisance variable | A variable that affects the response but is NOT of primary interest. Managed by blocking or randomisation. | Ambient humidity, operator experience, raw material batch. |
| Lurking variable | An unknown variable that affects the response and may be confounded with treatment effects if not randomised. | Tool wear increasing over time — if runs aren't randomised, later treatments appear worse. |
| Main effect | The average change in response when a factor moves from its low to high level, averaged over all other factors. | Effect of Temperature = (avg response at 180°C) − (avg response at 150°C) = +12.5. |
| Interaction effect | When the effect of one factor depends on the level of another factor. Lines on interaction plot cross or diverge. | Temperature increases yield by +20 when Pressure is High, but only +5 when Pressure is Low → interaction = +15. |
| Contrast | A linear combination of treatment means (Σcᵢȳᵢ) where Σcᵢ=0. Tests a specific hypothesis. | Linear contrast for 3 levels: C = (−1)ȳ₁ + (0)ȳ₂ + (+1)ȳ₃ tests for a trend. |
| Orthogonality | Two columns/contrasts are orthogonal if their dot product = 0. Effects are estimated independently. | In a 2² design: A column = (−1,+1,−1,+1), B column = (−1,−1,+1,+1). Dot product = 1−1−1+1 = 0. |
| Coding | Transforming natural factor levels to coded values (−1, 0, +1) for analysis. | Temp: 150°C → −1, 165°C → 0, 180°C → +1. Formula: x = (X − X̄)/(Δ/2). |
| Effect sparsity | In most experiments, only a small fraction of possible effects are actually significant ("vital few"). | In a 2⁷ with 127 effects, typically only 5–10 are real; the rest are noise. |
| Effect heredity | An interaction is more likely to be active if its parent main effects are active. | If A and C are significant but B is not, then AC is more likely than BC. |
| Model hierarchy | If an interaction is in the model, both parent main effects must also be included. | If x₁x₂ is significant, keep both x₁ and x₂ in the model even if their p-values are >0.05. |
| Resolution (R) | Describes the severity of aliasing in a fractional factorial. Higher = better. | Res III: main↔2FI. Res IV: main clear, 2FI↔2FI. Res V: everything clear. |
| Defining relation | The identity equation (I = ...) that determines the alias structure of a fractional factorial. | 2⁴⁻¹ with D=ABC: defining relation is I = ABCD. Multiply any effect by ABCD to find its alias. |
| Alias (confounding) | Two effects that cannot be estimated separately because they produce the same contrast pattern. | In 2⁴⁻¹: AB is aliased with CD. The estimated "AB effect" is actually AB+CD combined. |
| Generator | The equation that defines how a column in a fractional factorial is created from other columns. | D = ABC means column D is the element-wise product of columns A, B, and C. |
| Fold-over | Adding a second fraction with all signs reversed to de-alias effects in a fractional factorial. | Original 2⁴⁻¹ Res IV (8 runs) + fold-over (8 runs) = 16 runs with Res V (all clear). |
| Projectivity | A Res R design has projectivity R−1: any R−1 factors form a full factorial. | Res IV (proj 3): pick any 3 of the k factors → you have a complete 2³. |
| Minimum aberration | Among designs of the same resolution, the one with fewest short words in the defining relation. | Two Res IV designs for 2⁶⁻²: one with word lengths (4,4,4), another with (4,4,6) → prefer the second. |
| Rotatability | A design where prediction variance depends only on distance from the centre, not direction. | CCD with α = 2^(k/4) is rotatable. The prediction variance is the same at (1,0) and (0,1). |
| Centre point | A run at the midpoint of all factors (all coded values = 0). Tests for curvature and provides pure error. | If Temp ranges 150–180°C and Time 60–90 min, the centre point is (165°C, 75 min). |
| Axial (star) point | A run where one factor is at ±α from the centre while all others are at 0. Part of CCD. | For k=2 CCD: axial points are (−α,0), (+α,0), (0,−α), (0,+α). α = 1.414 for rotatability. |
| Stationary point | The point on a response surface where the gradient is zero (∂ŷ/∂xᵢ = 0 for all i). May be a max, min, or saddle. | x* = −½B⁻¹b. If all eigenvalues of B are negative → maximum. |
| Saddle point | A stationary point that is a maximum in some directions and a minimum in others. | Eigenvalues of B: λ₁=−3.2, λ₂=+1.5. Mixed signs → saddle, NOT an optimum. |
| Lack of Fit (LOF) | A test for whether the fitted model systematically misses the true response. Uses replicated points. | F_LOF = MS_LOF/MS_PE. If significant → model is inadequate, need higher-order terms. |
| Pure error | Variability estimated from replicated runs at the same factor settings. Not affected by model misfit. | 5 centre points with responses 47, 49, 48, 50, 46 → s²_PE = var(these) = 2.5. |
| PRESS | Predicted Residual Error Sum of Squares. Leave-one-out cross-validation measure. | PRESS = Σ[eᵢ/(1−hᵢᵢ)]². Lower PRESS = better prediction. R²_pred = 1 − PRESS/SS_T. |
| Adequate Precision | Signal-to-noise ratio of the model's predicted values. Must be > 4. | AP = (max ŷ − min ŷ) / avg SE(ŷ) = 22.5. Excellent signal. |
| Variance component | The portion of total variance attributed to a specific random factor in a nested/random model. | σ²_Supplier = 4.7, σ²_Batch = 1.8, σ²_Error = 1.0. Supplier accounts for 63% of total. |
| Sphericity | Assumption in repeated measures that variances of all pairwise differences are equal. | Var(Y₁−Y₂) should equal Var(Y₁−Y₃) should equal Var(Y₂−Y₃). Mauchly's test checks this. |
| Carryover effect | In crossover designs, a residual effect from a previous treatment that persists into the next period. | Drug A's effect lasts into Period 2, contaminating Drug B's estimate. Requires adequate washout. |
| Signal-to-Noise ratio | Taguchi metric combining mean performance and variability into one number. Larger = better. | S/N_SB = −10 log₁₀[(1/n)Σyᵢ²] for smaller-is-better. S/N = −5.2 dB vs −12.1 dB → −5.2 is better. |
| Inner array × Outer array | Taguchi's crossed array: inner = control factors (OA), outer = noise factors (OA). Total = product of runs. | L₉ (inner, 4 control factors) × L₄ (outer, 2 noise factors) = 36 total runs. |
| Scheffé model | Regression model for mixtures with NO intercept. Σxᵢ = 1 makes β₀ redundant. | Linear: ŷ = β₁x₁ + β₂x₂ + β₃x₃. Quadratic adds βᵢⱼxᵢxⱼ blending terms. |
| Pseudo-component | Transformed mixture variable when components have lower bounds. Maps constrained region to full simplex. | If x₁ ≥ 0.2: x₁' = (x₁ − 0.2)/(1 − ΣLᵢ). The pseudo-component ranges 0–1. |
| Whole plot / Subplot | Split-plot terminology. Whole plot = batch at one HTC setting. Subplot = individual run within the batch. | Whole plot = oven at 200°C for 4 hours. Subplots = 3 different coatings tested in that oven run. |
| Hard-to-change (HTC) | A factor that is expensive, slow, or difficult to change between runs. | Oven temperature (2 hrs to stabilise), reactor pressure, machine calibration. |
| Easy-to-change (ETC) | A factor that can be changed quickly between runs within a whole plot. | Coating type, additive concentration, operator technique. |
| Desirability (dᵢ, D) | A 0-1 scale for each response. dᵢ=0 unacceptable, dᵢ=1 ideal. D = geometric mean of all dᵢ. | Yield: d₁=0.85 (good). Cost: d₂=0.60 (OK). D = √(0.85 × 0.60) = 0.714. |
| Box-Cox transformation | Power transformation y' = (y^λ − 1)/λ that stabilises variance and improves normality. | λ=0 → ln(y). λ=0.5 → √y. λ=−1 → 1/y. Choose λ from Box-Cox plot maximum. |
| VIF (Variance Inflation Factor) | Measures multicollinearity in non-orthogonal designs. VIF=1 ideal, >10 problematic. | D-optimal design: VIF(x₁)=1.2, VIF(x₁²)=3.8, VIF(x₁x₂)=1.0. All OK (<10). |
| EVOP | Evolutionary Operation. Running small experiments during production without disrupting quality. | Change temperature ±2°C from current (within spec). After 5 cycles, detect if +2°C improves yield. |
| Confirmation experiment | Runs at the predicted optimal settings to verify the model's prediction. | Model predicts yield=82% at (Temp=175, Time=90). Run 5 confirmation trials. Actual: 80, 83, 81, 82, 79. Mean=81 is within the PI → model validated. |
AI optimises blocking structures: ML clustering identifies natural groupings in experimental units. Adaptive randomisation algorithms ensure balanced designs even with constraints. Optimal design algorithms (D-optimal, I-optimal) generate custom designs when classical templates don't fit your constraints.
