Significance testing: Each effect is a single coefficient, so hypotheses \( H_0: \beta = 0 \) are tested with \( t \)-tests (linear regression) or \( Z \)-tests (logistic regression). The orthogonality ensures tests are independent.

4.4 Normal Probability Plots of Effects

When \( n=1 \) (no replication), MSE cannot be estimated directly. Daniel’s method uses a half-normal or normal probability plot of the estimated effects:

• Under \( H_0 \) (all effects zero), effects are approximately normal with zero mean.
• Significant effects appear as outliers from the line fitted through the central bulk.
• Large effects that deviate from the line are flagged as potentially significant.

4.5 Center Points in 2^k Designs

Augment a \( 2^K \) factorial with \( n_c \) center point observations at \( x_1 = x_2 = \cdots = x_K = 0 \) to:

1. Provide an independent estimate of pure error.
2. Test for quadratic curvature (see Chapter 6).

where \( \bar{\hat{\eta}}_F \) = average fitted value at factorial conditions and \( \hat{\eta}_C \) = fitted value at center point. A \( t \)-test of \( H_0: \beta_{PQ} = 0 \) tests for overall quadratic curvature.

4.6 Worked Example: 2^4 Credit Card Factorial

A \( 2^4 \) factorial experiment (4 factors, 16 conditions, \( n=7500 \) each) tested factors: Annual Fee (\( x_1 \)), Account-Opening Fee (\( x_2 \)), Initial Interest Rate (\( x_3 \)), Long-term Interest Rate (\( x_4 \)) on credit card conversion rate (binary response → logistic regression). Selected output:

All main effects are significant. The \( x_1:x_2 \) and \( x_3:x_4 \) interactions are also significant. Interpretation: conversion rate is maximized when annual fee is lower, no account-opening fee, and both interest rates are low.

Chapter 5: Fractional Factorial Designs

5.1 Motivation: The 2^(k−p) Design

For large \( K \), \( 2^K \) conditions may be impractically many. A \( 2^{K-p} \) fractional factorial investigates \( K \) factors in only \( (1/2)^p \) as many conditions.

Principle of effect sparsity (justification): In practice, only a small fraction of effects are significant:

• ~40% of main effects are significant.
• ~10% of two-factor interactions are significant.
• ~5% of three-factor-and-higher interactions are significant.

Therefore, higher-order interaction columns in the model matrix can be “borrowed” to accommodate additional factors.

5.2 Aliasing and the Defining Relation

Aliasing: Associate the main effect of a new factor with an existing interaction column. This confounds the two effects — they cannot be separately estimated.

Every main effect is aliased with a two-factor interaction — a Resolution III design.

Resolution IV: main effects aliased with three-factor interactions (not two-factor interactions).

Every effect is aliased with 3 other effects (\( 2^p - 1 = 3 \)).

General rule: In a \( 2^{K-p} \) design, each estimated quantity jointly represents \( 2^p \) aliased effects.

5.3 Design Resolution

Higher resolution is preferred because main effects are not confounded with low-order interactions.

Projective property: A resolution \( R \) design projects onto a full factorial on any subset of \( R-1 \) factors. This means that if only \( R-1 \) factors are active, they form a full factorial sub-design without aliasing.

5.4 Minimum Aberration

When multiple generator choices yield the same resolution, compare designs by counting words of each length in the defining relation. The minimum aberration design minimizes the number of shortest words (length \( R \)), thereby minimizing the number of aliases at the lowest order.

Example — \( 2^{7-2}_\text{IV} \):

Design 3 has the fewest length-4 words (one), so it is the minimum aberration design.

5.5 Analysis of 2^(k−p) Designs

The analysis mirrors that of a full \( 2^K \) design, but with the aliasing structure in mind.

Steps:

1. Write out the model matrix for the \( 2^{K-p} \) design.
2. Fit regression (linear or logistic) with up to \( 2^{K-p} \) effects.
3. Interpret significant effects in light of the alias structure — under effect sparsity, attribute significance to the lower-order aliased effect.
4. Follow up with fold-over or additional runs to de-alias if needed.

Worked Example: \( 2^{8-4}_\text{IV} \) Wine Experiment

\( K=8 \) factors (Pinot clone, oak type, barrel age, yeast contact, stems, barrel toast, whole cluster, fermentation temp) investigated in only 16 conditions. Generators: \( E=BCD, F=ACD, G=ABC, H=ABD \). Defining relation: \( I = BCDE = ACDF = ABCG = ABDH = \cdots \)

Selected regression output (attributing 3FI estimates to aliased main effects E, F, G, H):

All main effects except H (fermentation temperature) are significant. Significant interactions after de-aliasing: DF, EG, FG interactions.

Chapter 6: Response Surface Methods

6.1 Overview of Response Optimization

Response Surface Methodology (RSM) is a sequential strategy for optimization:

• Phase 1: Factor screening (2-level designs identify important factors).
• Phase 2: Method of steepest ascent/descent (locate vicinity of optimum).
• Phase 3: Response surface design + second-order model (precisely locate optimum).
• Phase 4: Confirmation experiment.

Response surface models (Taylor approximations):

• First-order: \( \eta = \beta_0 + \sum_j \beta_j x_j \) (plane).
• First-order + interactions: adds \( \sum_{j<\ell} \beta_{j\ell} x_j x_\ell \) (twisted plane).
• Second-order: \( \eta = \beta_0 + \sum_j \beta_j x_j + \sum_{j<\ell}\beta_{j\ell} x_j x_\ell + \sum_j \beta_{jj} x_j^2 \) (quadratic surface — can have maxima/minima).

Only the second-order model can identify an optimum (maximum or minimum). It requires \( (K'+1)(K'+2)/2 \) parameters where \( K' \) is the number of quantitative factors.

6.2 Method of Steepest Ascent/Descent

Path of steepest ascent (maximize): \( \mathbf{x}' = \mathbf{x} + \lambda\mathbf{g} \)

Path of steepest descent (minimize): \( \mathbf{x}' = \mathbf{x} - \lambda\mathbf{g} \)

Step size \( \lambda = \Delta x_j / |\hat{\beta}_j| \) where factor \( j \) is chosen as the reference (e.g., the factor with most practical constraint), and \( \Delta x_j \) is the desired step in coded units for that factor.

Algorithm:

1. Start at origin \( \mathbf{x}_0 = \mathbf{0} \) (center of current \( 2^{K'} \) design).
2. Determine step size \( \lambda \).
3. Move along the path, collecting data at each step.
4. Continue until improvement ceases.
5. At best point, test for curvature.
   • No curvature → fit new first-order model; repeat.
   • Curvature detected → proceed to response surface design.

Checking for Curvature

Test \( H_0: \beta_{PQ} = 0 \) using a \( t \)-test in the model with linear predictor \( \beta_0 + \sum_j \beta_j x_j + \sum_{j<\ell}\beta_{j\ell}x_jx_\ell + \beta_{PQ} x_{PQ} \), where \( x_{PQ}=1 \) for factorial conditions and \( x_{PQ}=0 \) for center points.

Worked Example: Netflix Steepest Descent

Two factors: Preview Length (\( x_1 \), 90s to 120s) and Preview Size (\( x_2 \), 0.2 to 0.5 screen fraction). Goal: minimize average browsing time (users want to find content quickly). First \( 2^2 + \)CP experiment:

Curvature test: \( \hat{\beta}_{PQ} = -0.405 \), p-value = 0.064 — no significant curvature yet. First-order fit: \( \hat{\eta} = 21.722 + 0.448 x_1 - 0.539 x_2 \). Gradient: \( \mathbf{g} = (0.448, -0.539)^\top \). Steepest descent: decrease \( x_1 \) and increase \( x_2 \).

Step size with \( \Delta x_1 = 1/3 \) (steps of 5 seconds): \( \lambda = (1/3)/0.448 = 0.744 \). Path results:

Minimum at Step 6. Second \( 2^2 + \)CP in the new region (preview length 60–90s, size 0.6–0.8) yielded \( \hat{\beta}_{PQ} = 2.573 \), p-value \( < 10^{-15} \) — significant curvature detected. Proceed to response surface design.

6.3 Central Composite Design (CCD)

Choosing \( a \):

• Face-centered CCD (\( a=1 \)): reduces to a \( 3^{K'} \) design; only 3 levels per factor; cuboidal.
• Rotatable CCD (\( a = \sqrt[4]{2^{K'}} = 2^{K'/4} \)): equal prediction variance at all points equidistant from center; spherical. For \( K'=2 \): \( a = \sqrt{2} \approx 1.414 \).

CCD design matrix for \( K'=2 \):

Second-Order Model and Canonical Analysis

where \( \mathbf{b} = (\hat{\beta}_1, \ldots, \hat{\beta}_{K'})^\top \) and \( \mathbf{B} \) is the symmetric matrix with diagonal entries \( \hat{\beta}_{jj} \) and off-diagonal entries \( \hat{\beta}_{j\ell}/2 \).

• All \( \lambda_i < 0 \): stationary point is a maximum (response surface).
• All \( \lambda_i > 0 \): stationary point is a minimum.
• Mixed signs: stationary point is a saddle point.

Worked Example: Lyft Booking Rate (CCD)

Two factors: discount amount (\( x_1 \)) and discount duration (\( x_2 \)), with \( n=500 \) users per condition. CCD with \( a=1.4 \) (natural units: discount 15–85%, duration 1–8 days):

Second-order logistic regression output:

Stationary point (coded): \( x_{1s} = 0.007 \), \( x_{2s} = -0.973 \) → natural units: 50.2% discount, 2.1 days duration. Predicted booking rate: 0.792 (95% PI: 0.769–0.814). Both quadratic terms are significant and negative → the stationary point is a maximum.

6.4 Box-Behnken Designs

An alternative to CCD for \( K' \geq 3 \) factors. A Box-Behnken design uses 3 levels (−1, 0, +1) but never runs all factors at extreme levels simultaneously. Each condition lies on a sphere of radius \( \sqrt{2} \) for \( K'=3 \). The design is rotatable (or near-rotatable) and avoids extreme conditions (corners). For \( K'=3 \): 12 edge-midpoint conditions + center = 13 runs (vs. CCD’s \( 2^3 + 2(3) + 1 = 15 \)).

6.5 RSM with Qualitative Factors

When both quantitative and categorical factors are present, treat each combination of categorical factor levels as defining a separate response surface. Conduct RSM independently within each categorical combination, then compare the optima to identify the overall best operating condition.

Chapter 7: Split-Plot Designs and Variance Components

7.1 Split-Plot Structure

Motivating example: An experiment studies baking temperature (hard-to-change — oven must preheat) and baking time (easy-to-change). Whole plots are oven batches (set at a specific temperature); subplots are individual loaves within each batch tested at different times.

Design structure: With \( a \) whole-plot treatment levels applied to \( r \) whole plots, and \( b \) subplot treatment levels applied within each whole plot:

7.2 Split-Plot Model

The split-plot model with whole-plot factor A and subplot factor B in a RCBD arrangement:

where:

• \( \alpha_i \): effect of whole-plot factor level \( i \), \( i = 1,\ldots,a \).
• \( \delta_{ij} \sim \mathcal{N}(0, \sigma^2_\delta) \): whole-plot random error (whole-plot \( j \) within level \( i \)); \( j = 1,\ldots,r \).
• \( \beta_k \): effect of subplot factor level \( k \), \( k = 1,\ldots,b \).
• \( (\alpha\beta)_{ik} \): interaction between A and B.
• \( \varepsilon_{ijk} \sim \mathcal{N}(0, \sigma^2) \): subplot error (pure error within subplot).

Critical point: The two error terms \( \sigma^2_\delta \) and \( \sigma^2 \) are distinct. Whole-plot effects are tested against \( \sigma^2_\delta \) (whole-plot error MS), while subplot effects and interactions are tested against \( \sigma^2 \) (subplot error MS).

7.3 Split-Plot ANOVA Table

With \( r \) whole plots per whole-plot treatment (all nested in blocks \( b_0 \)):

where \( \phi_\alpha = \sum_i \alpha_i^2/(a-1) \), etc.

Hypothesis tests:

• \( H_0: \alpha_i = 0 \): \( F = \text{MS}_A / \text{MS}_{\text{WP error}} \sim F(a-1, (a-1)(r-1)) \)
• \( H_0: \beta_k = 0 \): \( F = \text{MS}_B / \text{MS}_{\text{SP error}} \sim F(b-1, a(r-1)(b-1)) \)
• \( H_0: (\alpha\beta)_{ik} = 0 \): \( F = \text{MS}_{AB} / \text{MS}_{\text{SP error}} \)

Common error: Using a single error term (as in a CRD) for both whole-plot and subplot effects leads to incorrect F-tests.

7.4 Variance Components Estimation

ANOVA Method of Moments

Set observed mean squares equal to their expected values and solve for variance components.

For a balanced one-way random-effects model with \( a \) groups and \( n \) observations per group:

Limitation: ANOVA estimates can be negative (since \( \hat{\sigma}^2_\alpha \) could be \( < 0 \) if \( \text{MS}_\text{between} < \text{MS}_\text{within} \)), which is not meaningful. Convention: truncate negative estimates to 0.

REML Estimation

Restricted Maximum Likelihood (REML) is preferred in practice. REML:

1. Accounts for uncertainty in fixed-effects estimates when estimating variance components.
2. Always yields non-negative variance component estimates (for standard models).
3. Is implemented in R via lme4::lmer().

For the split-plot model with whole-plot factor A (fixed), subplot factor B (fixed), and random whole-plot error, REML estimates both \( \sigma^2_\delta \) and \( \sigma^2 \) while properly accounting for fixed effects.

measures the proportion of total variance attributable to between-group differences. High ICC means observations within the same group are highly correlated.

7.5 Mixed Models for Split-Plot Designs

The split-plot model is a mixed model: fixed effects (treatment factors A and B, their interaction) plus random effects (whole-plot error).

where \( \mathbf{X}\boldsymbol{\beta} \) contains fixed treatment effects, \( \mathbf{Z}\mathbf{u} \) contains the random whole-plot effects (\( \mathbf{u} \sim \mathcal{N}(\mathbf{0}, \sigma^2_\delta \mathbf{I}) \)), and \( \boldsymbol{\varepsilon} \sim \mathcal{N}(\mathbf{0}, \sigma^2\mathbf{I}) \).

Marginal distribution: \( \mathbf{Y} \sim \mathcal{N}(\mathbf{X}\boldsymbol{\beta}, \sigma^2_\delta\mathbf{Z}\mathbf{Z}^\top + \sigma^2\mathbf{I}) \).

where \( \mathbf{V} = \sigma^2_\delta\mathbf{Z}\mathbf{Z}^\top + \sigma^2\mathbf{I} \).

7.6 Nested Designs

where \( \beta_{j(i)} \) is the effect of level \( j \) of B nested within level \( i \) of A.

ANOVA table for two-stage nested design (balanced, \( a \) levels of A, \( b \) levels of B within each A, \( n \) observations per B):

7.7 Repeated Measures Designs

Repeated measures arise when multiple measurements are taken on the same experimental unit over time or under different conditions. This is analogous to a split-plot design where:

• Whole-plot unit = subject (experimental unit).
• Within-subject factor = time/condition (equivalent to subplot factor).

where \( \rho \) is the correlation between any two measurements on the same subject. The Mauchly sphericity test checks whether this assumption holds; if violated, Greenhouse-Geisser or Huynh-Feldt corrections to the degrees of freedom are applied.

Appendix: Key Formulas and Summary Tables

A.1 Comparison of Design Structures

A.2 2^k and 2^(k−p) Design Summary

A.3 Response Surface Design Comparison

A.4 Critical Distributions Reference

A.5 Effect Hierarchy Principle

When analyzing factorial experiments, the following hierarchy guides interpretation:

1. Main effects are most likely to be important.
2. Two-factor interactions are next most likely.
3. Three-factor and higher interactions are progressively less likely to be significant.

This principle, combined with effect sparsity (few effects are truly active), justifies:

• Using \( 2^{K-p} \) fractional factorials for screening.
• Choosing design generators that alias main effects with high-order interactions.
• Treating significant high-order aliased effects as due to their lower-order alias.

A.6 Sequential Experimentation Strategy

The overall RSM workflow:

Each phase informs the next, making efficient use of experimental resources by not committing all runs to a single large experiment at the outset.

Chapter 8: Advanced Topics and Extensions

8.1 Analysis of Covariance (ANCOVA)

where \( \gamma \) is the common within-group regression coefficient (slope) for the covariate. The ANCOVA F-test for treatments adjusts for the covariate effect.

ANCOVA ANOVA table (one covariate, \( m \) treatments, \( N \) total observations):

Assumptions for ANCOVA:

1. Common slope \( \gamma \) across all treatment groups (homogeneity of regression slopes). Test with an interaction term \( \tau_j \times z \) in the model.
2. Covariate values \( z_{ij} \) are not influenced by the treatments (measured before treatment application, or known to be independent).
3. Standard regression assumptions for the errors.

Benefit: ANCOVA can substantially reduce \( \text{SSE} \), leading to a smaller MSE and more powerful treatment comparisons — especially when the covariate is highly correlated with the response.

8.2 Yates’ Algorithm for 2^k Designs

Yates’ algorithm provides a systematic, manual procedure for computing all \( 2^K \) effect estimates from a \( 2^K \) factorial experiment without matrix multiplication. It is especially useful for checking hand calculations.

Algorithm (illustrated for \( 2^3 \), standard order: (1), a, b, ab, c, ac, bc, abc):

1. List condition totals \( y_1, \ldots, y_{2^K} \) in standard (Yates) order.
2. Form column (1): upper half = sums of pairs, lower half = differences (later − earlier) of pairs.
3. Repeat \( K \) times total to generate column \( K \).
4. Divide column \( K \) by \( n \cdot 2^{K-1} \) (where \( n \) = reps per condition) to get effect estimates.

Worked Example (\( 2^3 \), \( n=1 \), condition totals in standard order):

Column 1 example: \( 6+8=14 \), \( 11+12=23 \)… upper half; \( 8-6=2 \), \( 12-11=1 \)… lower half.

The most influential factors are A (effect = 2.5) and C (effect = 2.0). The three-factor interaction ABC is negligible (effect = 0).

8.3 Plackett-Burman Designs

Plackett-Burman (PB) designs are a class of two-level screening designs that are Resolution III and allow studying \( N-1 \) factors in \( N \) runs, where \( N \) is a multiple of 4 but not necessarily a power of 2 (e.g., \( N = 12, 20, 24, 28, \ldots \)).

Comparison to \( 2^{K-p} \) designs:

• PB designs with \( N = 12 \) can accommodate up to 11 factors in 12 runs.
• The smallest \( 2^{K-p} \) design for 11 factors in 12+ runs is \( 2^{11-7} = 16 \) runs.
• PB designs are more economical for large \( K \), but they have complex, non-structured alias patterns — main effects are partially aliased with two-factor interactions, not fully confounded.

Subsequent rows are obtained by cyclic shifts; the final row is all \( -1 \).

Partial aliasing: Each main effect in a PB design is partially aliased with every two-factor interaction not involving it. The alias coefficient is \( \pm 1/3 \) for \( N=12 \). This means that even after identifying significant main effects, residual bias from interactions may exist, but the bias is spread evenly rather than concentrated.

Use case: PB designs are primarily used for initial factor screening when the number of factors is large (up to \( N-1 \)) and resources are very limited, accepting the limitation of complex aliasing.

8.4 Fold-Over of Fractional Factorial Designs

A fold-over is a follow-up experiment used to de-alias confounded effects identified in a \( 2^{K-p} \) screening experiment.

Principle: Run the complementary fraction (obtained by reversing the signs of all factor levels in the original fraction). The combined design has all main effects free of two-factor interaction aliases.

Complete fold-over of a Resolution III design: Appending the complementary fraction to a \( 2^{K-p}_\text{III} \) design yields a design where:

• All main effects are de-aliased from all two-factor interactions.
• The combined design is Resolution IV.
• Total runs: \( 2 \times 2^{K-p} \).

Partial fold-over: Reverse the signs of only one factor \( F_j \) in the original design. This de-aliases the main effect of \( F_j \) from all two-factor interactions involving \( F_j \). Useful when only one factor’s aliases are problematic.

Example: If the original \( 2^{5-2}_\text{III} \) design identified factors A, C, D as potentially significant, and the aliases are \( A = DE, C = BE, D = AE \), a fold-over on factor D would separate the \( D \) main effect from the \( AE \) interaction, clarifying whether the signal is due to D or AE.

8.5 Projection Properties of Fractional Factorials

Projection theorem: A \( 2^{K-p}_R \) design projects onto a full factorial design in any \( R-1 \) factors. Specifically:

• Resolution III design → full \( 2^2 \) in any 2 of the \( K \) factors.
• Resolution IV design → full \( 2^3 \) in any 3 of the \( K \) factors.
• Resolution V design → full \( 2^4 \) in any 4 of the \( K \) factors.

This property enables sequential learning: if the screening experiment identifies 2 (or 3) truly active factors among \( K \), the sub-design defined by those factors is already a full factorial — no further runs are needed to estimate their main effects and interactions.

Example: The \( 2^{4-1}_\text{IV} \) design (8 runs, 4 factors) projects onto a full \( 2^3 \) design in any 3 of the 4 factors. If factor D turns out to be inactive, the remaining 3-factor sub-design is a complete \( 2^3 \) factorial.

8.6 Desirability Functions for Multiple Responses

When optimizing several response variables simultaneously (multiple responses \( y_1, y_2, \ldots, y_r \)), the desirability function approach converts each response to an individual desirability \( d_i \in [0,1] \) and then combines them into an overall desirability \( D \).

Individual desirability for maximization (target \( T \), lower limit \( L \), upper limit \( U \)):

where \( s, t > 0 \) are shape parameters (s=t=1 gives linear ramps; s,t > 1 penalizes deviations more sharply).

where \( w_i > 0 \) are importance weights. The factor settings that maximize \( D \) represent the compromise optimum across all responses.

Procedure:

1. Fit separate second-order models for each response \( y_1, \ldots, y_r \).
2. Define individual desirability functions based on goals (maximize, minimize, or hit target).
3. Maximize \( D \) numerically over the factor space.
4. Evaluate the predicted values and desirabilities at the optimal settings.
5. Confirm with a confirmation run.

8.7 Robust Parameter Design

Developed by Genichi Taguchi (popularized in the West by Wu & Hamada), robust parameter design aims to find factor settings that minimize response variability (due to noise factors) while keeping the mean on target.

Key concepts:

• Control factors (\( x \)): factors the engineer can set and hold fixed in production.
• Noise factors (\( z \)): sources of variability that cannot be controlled during production (environmental conditions, material variation, operator differences) but can be controlled during experimentation.

Combined array (crossed) approach: Run a factorial design in the control factors crossed with a factorial design in the noise factors. For each combination of control settings, observe the response across noise conditions.

Signal-to-noise (SN) ratios (Taguchi’s approach): summarize location and dispersion jointly.

• Larger-is-better: \( \text{SN} = -10 \log_{10}\left(\frac{1}{n}\sum_i y_i^{-2}\right) \)
• Smaller-is-better: \( \text{SN} = -10 \log_{10}\left(\frac{1}{n}\sum_i y_i^{2}\right) \)
• Nominal-is-best: \( \text{SN} = 10 \log_{10}(\bar{y}^2/s^2) \)

Modern approach (response model method): Fit a model for the mean and a model for the log-variance as functions of control and noise factors. Then identify control factor settings that achieve the desired mean while minimizing variance.

The key insight: if a control factor \( x \) interacts with a noise factor \( z \), then changing \( x \) can reduce the effect of \( z \) on the response — making the product or process robust to noise.

8.8 Summary: Design Selection Guide

The following guide summarizes which design to choose based on the experimental situation:

Scenario 1: One factor, no nuisance factors, units are homogeneous → Use a CRD (one-way ANOVA).

Scenario 2: One factor, one known nuisance factor → Use an RCBD (two-way ANOVA without interaction).

Scenario 3: One factor, two known nuisance factors, same number of levels → Use a Latin square design (three-way ANOVA).

Scenario 4: One factor, one nuisance factor, but block size < number of treatments → Use a BIBD.

Scenario 5: Multiple factors (screening), all easy to change, budget for \( 2^K \) runs → Use a \( 2^K \) full factorial.

Scenario 6: Many factors (\( K \) large), budget limits runs to \( 2^{K-p} \) → Use a \( 2^{K-p} \) fractional factorial with maximum resolution and minimum aberration.

Scenario 7: Very large \( K \), extreme resource constraints → Use a Plackett-Burman design (Resolution III, non-regular).

Scenario 8: Active factors identified, seeking optimum location → Use method of steepest ascent/descent, then a central composite design for second-order modelling.

Scenario 9: Some factors hard-to-change (expensive to reset) → Use a split-plot design with appropriate two-error-term analysis.

Scenario 10: Multiple response variables to optimize simultaneously → Combine RSM with desirability functions.

8.9 Worked Example: Full 2^3 ANOVA Table

Consider a \( 2^3 \) factorial experiment studying factors A (temperature), B (pressure), C (catalyst) on yield (%), with \( n=2 \) replicates per condition. Condition totals:

Grand total \( T = 280 \), \( N = 16 \), correction factor \( \text{CF} = T^2/N = 280^2/16 = 4900 \).

Effect estimates (using contrast coefficients, divided by \( n \cdot 2^{K-1} = 2 \times 4 = 8 \)):

Each SS = \( n \cdot 2^{K-1} \cdot (\text{Effect}/2)^2 = 8 \cdot (\text{Effect}/2)^2 \) = \( 2 \cdot \text{Effect}^2 \):

ANOVA table (assuming \( \text{SSE} = 22 \) from replication error, df = 8):

At \( \alpha=0.05 \), \( F_{0.05, 1, 8} = 5.32 \). Factors A and C are significant. Factor B, and all interactions, are not significant at this level. The yield is maximized by setting temperature A high and catalyst C high, regardless of pressure B.

8.10 Worked Example: 2^(4-1) Alias Structure

Construct a \( 2^{4-1}_\text{IV} \) design with generator \( D = ABC \):

1. Write the \( 2^3 \) design in A, B, C.
2. Set \( D = ABC \) (product of all three factors’ signs).

Defining relation: \( I = ABCD \). Complete alias structure:

• Main effects: \( A = BCD \), \( B = ACD \), \( C = ABD \), \( D = ABC \)
• Two-factor interactions: \( AB = CD \), \( AC = BD \), \( AD = BC \)

This is Resolution IV: main effects are aliased only with three-factor interactions (assumed negligible by effect sparsity), but two-factor interactions are aliased with other two-factor interactions. To de-alias two-factor interactions, a fold-over or additional runs are needed.

Steepest Ascent Calculation Example

To minimize the response, use steepest descent. Choose step size anchored to factor D (harder to change): \( \Delta x_D = 0.5 \), so \( \lambda = 0.5/|{-2.4}| = 0.208 \).

Steps along the path of steepest descent (\( \mathbf{x}' = \mathbf{x} - \lambda\mathbf{g} \), with \( \mathbf{g} = (3.1, -2.4) \)):

Each step reduces the predicted response by approximately 4 units. Continue until improvement slows, then augment with center points to test for curvature.

Chapter 9: Inference in Linear Models — Review and Connections

9.1 The General Linear Model Framework

where:

• \( \mathbf{Y} \) is the \( N \times 1 \) vector of responses.
• \( \mathbf{X} \) is the \( N \times p \) model (design) matrix — full column rank assumed.
• \( \boldsymbol{\beta} \) is the \( p \times 1 \) vector of unknown parameters.
• \( \boldsymbol{\varepsilon} \) is the \( N \times 1 \) vector of random errors.

Residual sum of squares: \( \text{SSE} = \mathbf{Y}^\top(\mathbf{I} - \mathbf{H})\mathbf{Y} \) where \( \mathbf{H} = \mathbf{X}(\mathbf{X}^\top\mathbf{X})^{-1}\mathbf{X}^\top \) is the hat (projection) matrix. Unbiased estimator: \( \hat{\sigma}^2 = \text{SSE}/(N-p) = \text{MSE} \).

9.2 The Partial F-Test

The partial F-test compares a full model to a reduced model obtained by setting a subset of \( q \) parameters to zero.

This is the workhorse test for:

• Testing significance of a factor in ANOVA (\( q = \) number of levels \( - 1 \)).
• Testing significance of an interaction (\( q = \) number of interaction terms).
• Testing a group of regression coefficients simultaneously.

9.3 Model Diagnostics

After fitting any linear model, residual diagnostics verify the assumptions:

Residuals: \( \hat{\varepsilon}_i = y_i - \hat{y}_i = y_i - \mathbf{x}_i^\top\hat{\boldsymbol{\beta}} \).

Standardized residuals: \( r_i = \hat{\varepsilon}_i / (\hat{\sigma}\sqrt{1-h_{ii}}) \) where \( h_{ii} \) is the \( i \)-th diagonal of \( \mathbf{H} \).

Normality: Shapiro-Wilk test on residuals. Constant variance: Bartlett test (sensitive to normality) or Levene test (robust).

If normality fails: consider transformations (Box-Cox \( \lambda \)-family: \( y^{(\lambda)} = (y^\lambda - 1)/\lambda \) for \( \lambda \neq 0 \), \( \log y \) for \( \lambda = 0 \)).

9.4 Estimability and Aliasing in Matrix Terms

For a \( 2^{K-p} \) design with model matrix \( \mathbf{X} \) (columns = estimable effects), the relationship between the OLS estimate and the true effect is:

Here \( \mathbf{X}_1 \) = columns for estimable effects, \( \mathbf{X}_2 \) = columns for non-estimable (aliased) effects. For \( \pm 1 \) orthogonal designs, entries of \( \mathbf{A} \) are \( \pm 1/2^p \in \{0, \pm 1\} \), reflecting full confounding. When \( A_{ij} = 1 \), effect \( i \) is completely aliased with effect \( j \).

9.5 Confidence Intervals and Multiple Comparison Adjustments

These simultaneous methods maintain the familywise confidence level at \( 1-\alpha \) over all comparisons, ensuring that conclusions from the full set of intervals are jointly reliable.

9.6 Fixed vs. Random Effects

One-way random-effects ANOVA test (testing \( H_0: \sigma^2_\tau = 0 \)): Same F-statistic as fixed effects (\( F = \text{MSC}/\text{MSE} \)), but the interpretation differs — rejection means between-group variance exceeds within-group variance, not that specific group means differ.

Under \( H_0: \sigma^2_\tau = 0 \), both equal \( \sigma^2 \), giving \( F \sim F(m-1, N-m) \).

9.7 Power for Factorial Experiments

For a \( 2^K \) experiment with \( n \) replicates per condition (\( N = n \cdot 2^K \)), the test for a single effect (one \( \beta \)) is a \( t \)-test:

using the non-central \( t \)-distribution with non-centrality parameter \( \phi = |\beta|/(\sigma/\sqrt{n \cdot 2^{K-1}}) \).

(approximately, using normal approximation). Insight: For the same power and effect size, a \( 2^K \) design is more efficient than \( 2^K \) separate one-factor experiments because all \( N = n \cdot 2^K \) observations contribute to each main-effect estimate through orthogonality.