Central Composite Design

Five-level design combining factorial points, axial points, and center points to enable second-order response surface modeling and optimization.

Design Overview & Structure

What it accomplishes:

Central Composite Designs (CCDs) extend factorial designs by adding axial (star) points and center points, creating a five-level design that enables fitting full second-order polynomial models for response surface optimization.

Advantages:

• • Fits complete quadratic models
• • Rotatable or orthogonal properties
• • Efficient for 2-5 factors
• • Sequential from factorial designs
• • Well-established theory and analysis

Design Components:

• • Factorial points (2^k or 2^(k-p))
• • Axial points (2k star points)
• • Center points (n₀ replicates)
• • Alpha distance determines properties

CCD Types & Alpha (α) Formulas

Rotatable CCD (CCC):

Rotatable axial distance

= number of factors

Example: k = 2 factors gives α = 1.414.

• Property: Prediction variance depends only on distance from center

• Use: When all directions from center are equally important

Orthogonal CCD (CCO):

• Alpha value: Chosen to make design orthogonal

• Property: Independent parameter estimates

• Use: When factor effects need to be estimated independently

• Trade-off: Loses rotatability but gains orthogonality

Face-Centered CCD (CCF):

• Alpha value: α = 1 (axial points on cube faces)

• Property: Three-level design, no extreme points

• Use: When factor ranges are constrained

• Advantage: Safer when extreme conditions risky

Key Assumptions

• • The underlying response surface is adequately approximated by a second-order polynomial
• • Errors are independent, normally distributed with constant variance
• • Factor levels are set accurately at the coded design points (±1, ±α, 0)

Reading the Central Composite Design Layout

Example Model & Optimization Interpretation

Model Fitting:

Second-order response surface model

= intercept

= linear coefficient for factor i

βii

= quadratic coefficient for factor i

βij

= two-factor interaction coefficient

= random error

• Coefficient estimation: Least squares regression

• Model testing: ANOVA and lack-of-fit tests

• Diagnostics: Residual analysis and validation

Worked Example:

A rotatable CCD for temperature (X1) and pressure (X2) fits Yield = 84.7 + 4.8·X1 + 2.1·X2 − 1.6·X1² − 0.9·X2² + 0.5·X1·X2. The ANOVA shows a significant quadratic term (p < 0.01) and no significant lack-of-fit, so a stationary point search identifies the maximum yield near X1 = 1.5, X2 = 1.2.

Response Surface Analysis:

• • Contour plots and 3D surfaces
• • Canonical analysis (stationary point)
• • Ridge analysis for constrained optimization
• • Confidence regions for optimum

Optimization Methods:

• • Steepest ascent/descent paths
• • Constrained optimization
• • Desirability function approach
• • Multi-response optimization

Design Selection Guidelines:

• 2-3 factors: CCC or CCO (13-20 runs)

• 4-5 factors: CCC with fractional factorial (25-32 runs)

• > 5 factors: Consider Box-Behnken or optimal designs

• Constrained regions: Use CCF or custom designs

sixsigmacenter.com Implementation Guide

Design Construction Steps:

1. Factorial Block:

• 2^k factorial (or 2^(k-p) fractional)

• All factors at ±1 levels

• Provides interaction estimates

2. Axial Block:

• 2k axial points at (±α, 0, 0, ..., 0)

• One factor at ±α, others at 0

• Enables quadratic estimation

3. Center Points:

• Multiple replicates at (0, 0, 0, ..., 0)

• Provides pure error estimate

• Tests for curvature

Platform Implementation:

The sixsigmacenter.com platform automatically constructs optimal CCDs based on your requirements, handles rotatable/orthogonal/face-centered variants, performs complete RSM analysis, and provides optimization with prediction intervals.

Example Data Layout for Central Composite Design (CCC)

Rotatable CCD with α=1.414 for 2 factors. Block 1: factorial + center, Block 2: axial + centers. Sample Size Recommendation: 2^k + 2k + n₀ runs, typically n₀ = 3-6 center points.

This is how your spreadsheet or CSV file should be structured before you upload it. Match the column names, keep one row per observation, and leave the analysis to the platform.