Factorial Design

A factorial design is an experimental design in which two or more factors are investigated together through combinations of their levels. A factor is an input or treatment variable, and a level is a particular setting or category. A full factorial design includes every possible combination; a fractional factorial design includes a systematically selected subset. In statistics, these designs distinguish individual factor effects from interactions, in which the effect of one factor depends on another. (itl.nist.gov)

Structure and notation

The treatment combinations in a full factorial design form the Cartesian product of the factors’ level sets. If there are $k$ factors with $l_1,\ldots,l_k$ levels, the number of distinct combinations is

Thus, a $2\times3$ design has six combinations, while a $2^k$ design has $k$ two-level factors and $2^k$ combinations. With $r$ observations per combination, the total is $N=rM$. Equal replication produces a balanced design. (itl.nist.gov)

Factors may be quantitative, such as temperature, or categorical, such as material type. In two-level designs, settings are commonly coded as $-1$ and $+1$. These codes identify the chosen low and high settings rather than physical units. A three-factor design therefore has eight combinations, interpretable as the corners of a cube. (itl.nist.gov)

Main effects and interactions

A main effect describes the change associated with one factor, averaged over the levels of the others. An interaction describes how that change varies with another factor. Unlike an experiment that changes one factor at a time while holding others fixed, a factorial experiment observes combinations needed to identify such dependencies. (itl.nist.gov)

For an illustrative $2\times2$ experiment, let the cell means be:

The effect of increasing $A$ is $2$ when $B$ is low but $8$ when $B$ is high. Their difference, $8-2=6$, is a difference-of-differences measure of interaction. The averaged main effect of $A$ is $5$. Under the conventional two-level factorial-effect definition, the interaction effect is half the difference of differences, or $3$. These values are calculations from the hypothetical table, not empirical findings. (online.stat.psu.edu)

An interaction plot displays response means against one factor, with separate lines for another. Nonparallel lines indicate interaction in the plotted means, although graphical appearance alone does not establish statistical significance. A substantial interaction makes the averaged main effect an incomplete description of the response. (online.stat.psu.edu)

Statistical models and analysis

For two fixed factors with $a$ and $b$ levels and $r$ observations per combination, a standard model is

Here $\mu$ is the overall mean, $\alpha_i$ and $\beta_j$ are main-effect terms, and $(\alpha\beta)_{ij}$ represents interaction. Conventional constraints, such as zero sums of effects, make the parameterization identifiable. Classical inference assumes independent errors with zero mean, common variance, and a normal distribution. (online.stat.psu.edu)

Analysis of variance partitions variation into factor, interaction, and error components. In this balanced model, the respective degrees of freedom are $a-1$, $b-1$, $(a-1)(b-1)$, and $ab(r-1)$. Hypothesis tests assess whether specified effects are zero. (online.stat.psu.edu)

The same analysis can be expressed as linear regression and fitted using ordinary least squares. For two factors coded $\pm1$,

In a balanced full design, these model columns are mutually orthogonal: their cross-products sum to zero. Each non-intercept coefficient equals half its corresponding factorial effect. Residual analysis checks whether the fitted model adequately describes the observations. (itl.nist.gov)

Randomization, replication, and blocking

Factorial structure specifies treatment combinations, not their execution order. Randomization governs allocation or run order, reducing systematic associations between treatments and uncontrolled conditions. Blocking groups observations under relatively homogeneous conditions, such as a common production batch, so nuisance variation can be accounted for separately. (itl.nist.gov)

Replication repeats treatment combinations and supplies an estimate of experimental error independent of model lack of fit. Merely repeating a measurement on the same treated unit does not necessarily provide an independent experimental replicate. With one observation per combination, a model containing every interaction is saturated: it fits the observed responses exactly and leaves no residual degrees of freedom. Error estimation then requires additional information or assumptions about negligible effects. (itl.nist.gov)

Reduced designs and curvature

Full factorial run counts grow rapidly: ten two-level factors require $1{,}024$ combinations before replication. Fractional designs reduce this burden by deliberately introducing aliasing, whereby distinct effects cannot be estimated separately. This form of confounding is determined by the selected combinations rather than arising accidentally during execution. (itl.nist.gov)

Design resolution describes the separation of effects in regular two-level fractions. Resolution III designs can alias main effects with two-factor interactions; resolution IV separates those classes but can alias two-factor interactions with one another. Interpretation therefore depends on assumptions about which effects are negligible. Additional foldover runs can break selected alias relationships. (itl.nist.gov)

Two-level factorial designs estimate changes across chosen settings but cannot separately identify pure quadratic effects. For quantitative factors, center-point runs can test for curvature by comparing responses at the center with those at factorial settings. Identifying individual quadratic terms requires additional settings, as used in response surface methodology. (itl.nist.gov)