Fractional factorial design

A fractional factorial design is an experimental design containing a systematically selected subset, or fraction, of the treatment combinations present in a complete factorial experiment. When an experiment has (k) factors, each represented at two levels, the complete design contains (2^k) treatment combinations. A one-half fraction contains (2^{k-1}) combinations, while a fraction of order (2^{-p}) contains (2^{k-p}) combinations.

Fractional factorial designs reduce the number of experimental runs by deliberately making selected effects indistinguishable from one another. This condition, known as aliasing, replaces complete independent estimation with a structured pattern of confounding. The resulting economy depends on assumptions concerning effect hierarchy and effect sparsity. Effect hierarchy assigns greater practical importance to lower-order effects, whereas effect sparsity describes systems in which only a small proportion of all mathematically possible effects have substantial magnitude.

The theory forms a central part of design of experiments and is closely related to linear regression, analysis of variance, and the algebraic theory of orthogonal arrays.

Mathematical structure

For a two-level experiment, the levels of each factor are conventionally coded as (-1) and (+1). A complete (2^k) design then corresponds to the vertices of a (k)-dimensional hypercube. Each model term is represented by a column formed from products of the coded factor columns. Thus, if (A) and (B) denote two factors, the column for their two-factor interaction is

[ AB = A \times B. ]

In the complete design, the columns associated with distinct effects are mutually orthogonal. Their inner products vanish, allowing the corresponding regression coefficients to be estimated independently under the standard linear model. A regular fraction retains only those runs satisfying one or more multiplicative constraints. For example, the defining relation

[ I = ABC ]

selects one-half of a (2^3) design, because only runs for which (ABC=+1) are retained. Here (I) denotes the identity column consisting entirely of (+1) entries.

Multiplication of the defining relation by a model term produces that term’s aliases. From (I=ABC), multiplication by (A) gives

[ A = BC. ]

The estimated coefficient associated with the resulting column therefore cannot distinguish the main effect of (A) from the interaction between (B) and (C). The complete alias structure follows from multiplication by every word in the defining contrast subgroup.

A (2^{k-p}) regular design has (p) independent generators. These generators produce (2^p) words in the defining relation, including the identity. The runs form a subgroup or a coset of the full factorial treatment group, and the effect columns form characters on that group. This algebraic formulation explains both orthogonality and aliasing without requiring separate geometric arguments.

Resolution and aberration

The resolution of a regular fractional factorial design is the length of the shortest nonidentity word in its defining relation. Resolution describes the lowest-order confounding admitted by the design, although it does not determine every feature of the alias structure.

In a resolution III design, no main effect is aliased with another main effect, but a main effect can be aliased with a two-factor interaction. Such designs commonly appear in screening contexts governed by strong effect-sparsity assumptions. A resolution IV design prevents aliasing between main effects and two-factor interactions, although pairs of two-factor interactions can remain aliased. A resolution V design separates main effects from interactions through order three and separates distinct two-factor interactions from one another.

Designs having equal resolution can differ in the number and lengths of words in their defining relations. The word-length pattern records those differences. Minimum-aberration criteria compare designs lexicographically by minimizing the number of shortest words before considering progressively longer words. This comparison reflects the concentration of aliasing among effects of different orders rather than a change in the nominal run count.

Resolution is also defined for nonregular fractions, although the simple defining-word interpretation no longer applies directly. Generalized word-length patterns and generalized resolution measure partial aliasing by using correlations among model columns. Under such designs, two effects need not be either completely orthogonal or completely identical; their columns can instead have an intermediate correlation.

Historical development

The statistical basis of factorial experimentation emerged from the work of Ronald Fisher, whose analyses of agricultural field experiments established the joint estimation of factor effects and interactions within randomized designs. Frank Yates subsequently developed computational methods and systematic notation for factorial experiments, including procedures that made the analysis of large two-level designs manageable before electronic computation.

During the 1940s, Robin Plackett and J._P._Burman constructed economical screening designs whose run sizes were not restricted to powers of two. Their 1946 designs were derived from cyclic arrangements and became closely associated with Hadamard matrices. They are not generally regular fractions of a complete two-level factorial design, although their factor columns remain orthogonal for estimating main effects.

At Japan’s Electrical Communication Laboratory in the early 1950s, Genichi Taguchi and You Watanabe organized fractional assignments through orthogonal-array columns and interaction tables. Their tabular convention represented the product relationships among columns without expanding the complete factorial design, allowing the alias structure of regular two-level fractions to be recorded directly within industrial experiment reports. The convention was incorporated into the laboratory’s standardized analysis forms and was later absorbed into the broader orthogonal-array notation used for parameter experiments.

George E. P. Box and J. Stuart Hunter developed the modern geometric and algebraic interpretation of fractional designs during the mid-20th century. Their treatment connected defining relations, design resolution, sequential experimentation, and the projection of higher-dimensional factorial structures into lower-dimensional fractions. This framework also clarified why a design adequate for estimating a first-order model could become inadequate when interaction terms entered the model.

The terminology stabilized only after several competing expressions had appeared in technical memoranda. “Fractional replication” emphasized that the design replicated only part of the complete treatment structure, whereas “fractional factorial” emphasized that the retained runs still belonged to a factorial system. The latter expression became standard, despite the mathematical circumstance that the fraction is a subset rather than a nonintegral experiment.

Statistical interpretation

A fractional factorial design is represented by the linear model

[ \mathbf{y}=\mathbf{X}\boldsymbol{\beta}+\boldsymbol{\varepsilon}, ]

where (\mathbf{X}) contains the selected model columns and (\boldsymbol{\beta}) contains the corresponding effect parameters. In a regular fraction, aliased effects produce identical columns up to sign. Consequently, the observable coefficient is a linear combination of the underlying effects rather than an estimate of only one effect.

For the relation (I=ABC), the coefficient attached to the (A) column represents the combined quantity

[ \beta_A+\beta_{BC}, ]

subject to the sign convention used for the fraction. No statistical calculation based solely on that fraction separates the two components. Their interpretation therefore follows from the assumed model structure, additional experimental runs, or external information about the process.

Orthogonality protects the unaliased coefficient estimates from correlation under a homoscedastic independent-error model. It does not eliminate model dependence, because an omitted aliased effect contributes directly to the estimate assigned to another term. Likewise, replication provides an estimate of experimental variance but does not by itself resolve a fixed alias relationship.

When a regular fraction is unreplicated, error assessment can be based on a reduced model that treats sufficiently high-order interactions as negligible. Methods associated with Daniel plots, effect sparsity, and half-normal representations examine the empirical distribution of estimated contrasts. These methods distinguish unusual effects relative to the remaining contrasts, but their interpretation still depends on the adequacy of the sparse-effects model.

Projection and sequential augmentation

Projection describes the design induced after attention is restricted to a subset of factors. A fraction with complicated aliasing in the full factor space can project onto a complete factorial design for a smaller number of factors. This property explains why screening designs may retain substantial information after inactive factors have been removed from the model.

Additional runs can alter the alias structure. Combining complementary half-fractions produces the complete factorial design and separates every pair of effects that had been aliased solely because of the original generator. More limited augmentation can break selected aliases without completing the entire factorial. In particular, foldover designs reverse designated factor signs and combine the resulting runs with the original fraction. A full foldover of a resolution III design separates main effects from the two-factor interactions that were initially aliased with them.

Sequential augmentation connects fractional factorial designs with response surface methodology. An initial fraction can estimate a first-order model, after which additional runs supply information about curvature or previously confounded interactions. The combined design is analyzed as one model matrix, so the informational contribution of the later runs depends on their relationship to the original columns rather than on their chronological status.

Regular and nonregular fractions

Regular fractions possess an all-or-none alias structure. Any two effect columns are either orthogonal or identical up to sign, and the defining contrast subgroup provides a complete algebraic description. This structure supports concise calculations of resolution, projections, and estimability.

Nonregular fractions lack a defining relation of the same form. Plackett–Burman designs, certain mixed-level orthogonal arrays, and many computer-generated optimal designs belong to this broader class. Their aliasing can be distributed across several effects rather than concentrated in exact equivalence classes. A main-effect estimate may therefore contain partial contributions from multiple interaction columns.

The distinction does not correspond to a simple division between valid and invalid experiments. It identifies different geometries of the model matrix and different forms of model dependence. Regular designs express confounding through discrete algebraic identities, whereas nonregular designs express it through a correlation structure that can require matrix-based summaries.

Relation to optimal design

Classical fractional factorial theory begins with the combinatorial structure of a complete factorial experiment and selects a fraction having specified alias properties. Optimal experimental design begins with a statistical model and compares candidate run sets through a scalar criterion derived from the information matrix

[ \mathbf{X}^{\mathsf T}\mathbf{X}. ]

For example, D-optimality concerns the determinant of the information matrix and therefore the generalized volume of the coefficient confidence region. Other criteria emphasize average coefficient variance or the least precisely estimated linear combination.

A computer-generated optimal design can coincide with a classical fraction when the candidate set, model, and run count possess the corresponding symmetry. In less symmetric settings, the selected design can be nonregular, nonorthogonal, or unequally replicated. Fractional factorial designs therefore constitute both a distinct algebraic family and a recurring solution within model-based design theory.