Lesson 9: 3-Level and Mixed-Level Factorials and Fractional Factorials

This set covers 3-level factorial design structures, degrees of freedom calculation, confounding via pseudo-components, and the transition into response surface designs.

3-level factorial design

A design where every factor is set to three levels (low, middle, and high) to study curvature and the shape of the response function.

$3^k$ design

A full factorial involving $k$ factors at three levels each, consisting of $3^k$ treatment combinations.

Main-effect degrees of freedom ($df$) (3-level)

$$3 - 1 = 2$$

Two-way interaction degrees of freedom ($df$) (3-level)

$$(3 - 1) \times (3 - 1) = 4$$

Three-way interaction degrees of freedom ($df$) (3-level)

$$(3 - 1)^3 = 8$$

Pseudo-interaction components

Orthogonal $2-df$ partitions of an interaction used for efficient confounding in 3-level designs.

Modular Arithmetic (Mod 3)

A system where values are reduced to remainders of $0, 1, \text{ or } 2$ after division by $3$, used to create balanced pseudo-factor columns.

Blocking a $3 \times 3$ design

The process of partitioning nine runs into three blocks of three runs by confounding one $2-df$ pseudo-interaction component.

Partial Confounding

A strategy in replicated blocked designs where different replicates confound different interaction components to preserve information for all components.

Split-plot connection

A design type that arises when a main factor is deliberately confounded with blocks or whole plots.

$3^{k-p}$ fractional factorial

A subset of a full $3^k$ design created using $p$ defining pseudo-factor relationships.

Resolution III (3-level fraction)

A design resolution where main effects are clear of one another but aliased with components of two-way and higher-order interactions.

Latin Square (3-level context)

A 9-run design that is equivalent to a $\frac{1}{3}$ fraction of a $3^3$ factorial, where main effects are estimable but interactions are aliased with them.

Graeco-Latin Square (3-level context)

A 9-run design for four factors equivalent to a $\frac{1}{9}$ fraction of a $3^4$ factorial.

Mixed-level factorial design

An experimental layout where different factors have different numbers of levels, such as combining 2-level and 3-level factors.

Four-level factor representation

A single factor with four levels treated as the combination of two 2-level pseudo-factors.

Center point

A design run set at the middle level of all quantitative factors to check for curvature.

Central composite design (CCD)

A response-surface design consisting of factorial points, center points, and axial (star) points to estimate quadratic response functions.

Star point (Axial point)

A design point located at a distance of $\text{alpha}$ from the center along a single factor axis, outside the factorial range.

Response Surface Methodology (RSM)

An efficient approach using designs like CCD to fit curved quantitative response functions with fewer runs than full factorials.