Full factorial
Every factor combination, full information.
A full factorial runs every combination of factor levels, so every main effect and every interaction is estimable with no confounding. It is the most complete design, and the most expensive: the run count doubles with each factor you add.
What it measures
A factorial design varies several factors together, deliberately, instead of one at a time, which is what lets it see how factors interact:
- The geometry: for k factors at two levels each, the design is the 2^k corners of a cube. Three factors is 8 runs, four is 16, five is 32: the run count doubles per added factor.
- What it estimates: every main effect (the average effect of each factor) and every interaction (where the effect of one factor depends on another), all of them, unconfounded, because every combination is actually run.
- Why interactions are the point: one-factor-at-a-time testing holds everything else fixed and so is blind to interactions, the most common way real processes behave. A factorial finds them because it changes factors together.
Effects are read from the difference in average response between a factor's high and low settings, and interactions from how that difference itself changes with another factor. The completeness is the value; the exponential run count is the cost.
How to read the output
Read the effects by size and the interactions first. A large interaction means you cannot talk about either factor's effect in isolation: the right setting for one depends on the other, and a main-effects-only summary would mislead. The trade-off to manage is breadth versus depth. With a few factors, a full factorial is the gold standard: complete information, no confounding. As factors grow, the doubling run count forces a choice, a fractional factorial that gives up high-order interactions for far fewer runs to screen many factors, or a response-surface design once the important few are known and curvature matters. Use a full factorial when the factor count is small enough that completeness is affordable.
A real use case
An electrode slurry team wants to understand three factors on coating quality: solids content, binder ratio, and mixing time. A full 2^3 factorial is just 8 runs, cheap enough to do completely. The result is more than three main effects: it reveals that binder ratio and mixing time interact, a high binder ratio only helps if mixing time is long enough to disperse it, and at short mixing time more binder actually hurts. A one-factor-at-a-time study, holding mixing time at its default while varying binder, would have found the wrong binder optimum and never seen the interaction. Eight deliberate runs gave the team the interaction that decided the formulation.
Common mistakes
- Reaching for a full factorial with many factors. The run count doubles per factor; at six or more, screen with a fractional design first.
- Running one factor at a time to save runs, which is blind to interactions, the very thing a factorial exists to find.
- Reading main effects while ignoring a large interaction, which makes the main-effect numbers misleading on their own.
- Skipping center points, so curvature goes undetected and a two-level design is trusted in a region where the response is actually nonlinear.
- Not replicating or blocking, leaving no estimate of noise and no protection against a lurking time or batch effect.
The complete design, run-budget aware
Niobia builds the full factorial for the factors and levels you specify, runs the analysis to estimate every main effect and interaction, and reports them with their significance so the interactions that matter are not buried under the main effects. When the factor count makes a full factorial too expensive, it steers toward the right alternative, a fractional design to screen, or a central composite or Box-Behnken design once curvature is the question, and it can carry the results into a prediction profiler to explore the fitted model. The design choice is matched to the run budget rather than defaulting to the most expensive option.
Frequently asked
When should I use a full factorial instead of a fractional one?
When the factor count is small (roughly two to five) and you can afford the 2^k runs, because a full factorial estimates every interaction with no confounding. With many factors, a fractional factorial trades high-order interactions you rarely need for a large reduction in runs, which is the right screening move.
Why not just change one factor at a time?
Because one-factor-at-a-time testing holds everything else fixed and is structurally blind to interactions, where the effect of one factor depends on another. Real processes interact constantly, and a factorial finds those interactions precisely because it varies factors together.
Do I need center points in a two-level factorial?
They are strongly recommended: a few center-point runs let you detect curvature, telling you whether the two-level linear model is trustworthy or whether you need to augment to a response-surface design to capture a nonlinear optimum.