7.5 Advanced Statistical Design & Analysis

7.5 Advanced Statistical Design & Analysis

Design of Experiments (DOE) is a structured statistical methodology used to manipulate input process factors ($X$) systematically and evaluate their effect on key response variables ($Y$). While full factorial designs evaluate all possible factor combinations, resource constraints in complex manufacturing often demand advanced DOE techniques.

This section covers Fractional Factorial Designs ($2^{k-p}$), Analysis of Variance (ANOVA) for DOE, and Response Surface Methodology (RSM) using Central Composite and Box-Behnken designs.

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1. Fractional Factorial Designs ($2^{k-p}$)

When the number of factors $k$ increases, the number of treatment combinations in a full $2^k$ factorial design grows exponentially (e.g., $2^6 = 64$ runs, $2^7 = 128$ runs). According to the Sparsity of Effects Principle (or Pareto principle in DOE), a system is primarily driven by main effects and low-order (2-factor) interactions. High-order interactions (3-factor or higher) are rarely statistically significant.

A Fractional Factorial design, denoted as $2^{k-p}$, executes a fraction ($1/2^p$) of the full factorial runs by setting $p$ factors equal to high-order interaction columns (called generators).

Defining Relations and Aliasing Structure

When factor columns are set equal to interaction columns, information becomes confounded (or aliased). Two effects are aliased if their calculated contrast estimates measure the exact same numerical quantity.

Example: $2^{3-1}$ Half-Fraction Design

• Factors: $A, B, C$ ($k=3$), Fraction: $p=1$ ($1/2$ fraction = $4$ runs).
• Generator choice: $C = AB$.
• Multiplying both sides by $C$ yields the Defining Relation: $I = ABC$
• To determine the aliasing of any effect, multiply that effect by the defining relation (using modulo-2 arithmetic where $A^2 = I$): $A \cdot I = A \cdot ABC \implies A = BC$ $B \cdot I = B \cdot ABC \implies B = AC$ $C \cdot I = C \cdot ABC \implies C = AB$

Thus, in a $2^{3-1}$ design with $I=ABC$, main effect $A$ is aliased with interaction $BC$, $B$ with $AC$, and $C$ with $AB$.

Design Resolution Classification

Design Resolution (denoted by Roman numerals III, IV, V) characterizes the degree of aliasing in a fractional factorial design. The resolution equals the length of the shortest word in the defining relation $I$.

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2. Analysis of Variance (ANOVA) Models for DOE

ANOVA partitions the total variation observed in the experimental response ($SS_{\text{Total}}$) into components attributable to specific design factors, interactions, and random experimental error ($SS_{\text{Error}}$).

Fundamental ANOVA Equation

$SS_{\text{Total}} = SS_{\text{Model}} + SS_{\text{Error}} = \sum_{i=1}^k SS_{\text{Factor}_i} + \sum SS_{\text{Interactions}} + SS_{\text{Error}}$

Degrees of Freedom and Mean Squares

For an experiment with $N$ total observations and $p$ estimated parameters:

• Total Degrees of Freedom: $df_{\text{Total}} = N - 1$
• Factor Degrees of Freedom (for a 2-level factor): $df_{\text{Factor}} = 2 - 1 = 1$
• Error Degrees of Freedom: $df_{\text{Error}} = N - p$

The Mean Square ($MS$) for any term is calculated by dividing its Sum of Squares ($SS$) by its degrees of freedom ($df$):

$MS_{\text{Effect}} = \frac{SS_{\text{Effect}}}{df_{\text{Effect}}}, \quad MS_{\text{Error}} = \frac{SS_{\text{Error}}}{df_{\text{Error}}}$

Test Statistic ($F_0$) and Hypothesis Testing

To test the null hypothesis $H_0: \beta_{\text{Effect}} = 0$, the $F$-ratio is computed:

$F_0 = \frac{MS_{\text{Effect}}}{MS_{\text{Error}}}$

If $F_0 > F_{\alpha, df_{\text{Effect}}, df_{\text{Error}}}$ (or $p\text{-value} < \alpha$), the factor effect is statistically significant.

Source of Variation | Sum of Squares (SS) | Degrees of Freedom (df) | Mean Square (MS) | F-Ratio (F0)
---------------------------------------------------------------------------------------------------
Factor A            | SSA                 | 1                       | MSA = SSA / 1    | MSA / MSE
Factor B            | SSB                 | 1                       | MSB = SSB / 1    | MSB / MSE
Interaction AB      | SSAB                | 1                       | MSAB = SSAB / 1  | MSAB / MSE
Residual Error      | SSE                 | N - 4                   | MSE = SSE / dfE  | -
Total               | SST                 | N - 1                   | -                | -

Model Fit Metrics: $R^2$, $R^2_{\text{adj}}$, and $R^2_{\text{pred}}$

• Ordinary $R^2$: $R^2 = 1 - \frac{SS_{\text{Error}}}{SS_{\text{Total}}}$. Measures proportion of variation explained. (Always increases when factors are added).
• Adjusted $R^2$: $R^2_{\text{adj}} = 1 - \frac{MS_{\text{Error}}}{MS_{\text{Total}} / df_{\text{Total}}}$. Penalizes model complexity.
• Predicted $R^2$: $R^2_{\text{pred}} = 1 - \frac{PRESS}{SS_{\text{Total}}}$. Measures predictive capability on new data using Prediction Error Sum of Squares ($PRESS$).

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3. Response Surface Methodology (RSM)

When factor screening reveals non-linear behavior (curvature) in the response, Response Surface Methodology (RSM) is employed to model second-order quadratic surfaces and locate optimal process operating conditions.

Second-Order Response Surface Model

$y = \beta_0 + \sum_{i=1}^k \beta_i x_i + \sum_{i=1}^k \beta_{ii} x_i^2 + \sum_{i<j} \beta_{ij} x_i x_j + \epsilon$

Where \beta_{ii} represents pure quadratic curvature terms.

Sequential Optimization Strategy

1. Screening Phase: Fractional Factorial ($2^{k-p}$) to eliminate inactive factors.
2. Steepest Ascent Phase: Follow the gradient of maximum response change to move toward the optimal region.
3. Response Surface Phase: Fit a full quadratic model using a second-order design (Central Composite or Box-Behnken).

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4. Central Composite Design (CCD) & Box-Behnken Design (BBD)

Fitting a full second-order model requires factors to be evaluated at a minimum of $3$ levels. The two principal RSM designs are CCD and BBD.

Central Composite Design (CCD)

A CCD is constructed by augmenting a standard $2^k$ factorial (or $2^{k-p}$ Resolution V) design with axial (star) points and center points:

• $2^k$ Factorial Points: Code values $(\pm 1, \pm 1, \dots, \pm 1)$. Estimates linear and 2-factor interaction terms.
• $2k$ Axial (Star) Points: Code values $(\pm \alpha, 0, \dots, 0), (0, \pm \alpha, \dots, 0)$, etc. Estimates pure quadratic terms $\beta_{ii}$.
• $n_c$ Center Points: Code value $(0, 0, \dots, 0)$. Provides pure error estimation and tests for overall curvature.

Total CCD Runs $N = 2^k + 2k + n_c$.

       Central Composite Design (CCD) for k=2
       
            (-1, 1)          (1, 1)
               o---------------o
               |               |
   (-α, 0) o   |    (0,0) o    |   o (α, 0)  <-- Axial Points
               |               |
               o---------------o
            (-1,-1)          (1,-1)

Rotatability Condition in CCD

A design is rotatable if the variance of the predicted response $\hat{y}$ is identical at all points equidistant from the design center. For a CCD, rotatability is achieved by choosing the axial distance $\alpha$:

$\alpha = (N_F)^{1/4}$

Where $N_F = 2^k$ is the number of factorial points.

• For $k = 2$ factors: $N_F = 4 \implies \alpha = (4)^{1/4} = \sqrt{2} \approx 1.414$.
• For $k = 3$ factors: $N_F = 8 \implies \alpha = (8)^{1/4} \approx 1.682$.

Box-Behnken Design (BBD)

A Box-Behnken Design is an independent, spherical 3-level design formed by combining $2^2$ factorials with central block points. Unlike CCD, BBD does not contain corner points (where all factors are simultaneously at extreme levels $\pm 1$) or extreme axial points ($\pm \alpha$).