Full Factorial Design: Estimate Every Main Effect and Interaction
What is a Full Factorial Design?
A full factorial design runs every combination of every factor level: k two-level factors require 2^k runs (2 factors give 4, 3 give 8, 4 give 16), and three-level factors require 3^k runs. It is the most complete design, estimating all main effects and all orders of interaction without confounding, making it the first choice when the factor count is small, generally 4 or fewer. Run count grows exponentially with the number of factors, so cost must be weighed as factors increase.
When to Use It
Use a full factorial when the factor count is small enough that the run count is affordable and you need unconfounded estimates of main effects and interactions. It is the standard design for characterizing a process before optimization and for confirming which factors and interactions matter. For five or more factors the run count climbs quickly, so screening with a fractional design is usually preferred first.
How to Use It (Step by Step)
Choose two-level (-1/+1) or three-level (-1/0/+1) designs: two levels estimate linear effects and interactions and suit screening and most industrial experiments; three levels add the center level and can detect curvature, but 3^k grows fast, with 3 factors giving 27 runs. The tool generates the complete plan with coded levels and actual values, adds replicates for pure error and center points for curvature, and randomizes the run order. Split into blocks labeled with a block variable when runs cannot be completed in one session.
Key Formulas / Example
For k two-level factors the design matrix has 2^k rows of +/-1 coding; the effect of a factor is the difference in average response between its high and low levels. ANOVA decomposes the total sum of squares into main effects and interactions, and Pareto and normal probability plots highlight significant effects. If center-point responses deviate clearly from the corner-point plane, a quadratic term is needed and a response surface design should follow.