Methods

What the AI does and does not do

A language model (Claude) reads your description and proposes factor names, units, low and high levels, a response name and a goal. That proposal is validated and sanitized by code, shown to you, and editable before anything is built. If the AI service is unavailable, a built-in keyword parser is used instead. The model never generates the design matrix and never computes a statistic.

Designs

Factor levels are coded so the low setting is −1, the high setting is +1 and the midpoint is 0. All fitting is done in coded units.

• 2-level full factorial: every combination of low and high, 2^k runs.
• Fractional factorial (4 to 6 factors): 2⁴⁻¹ with D = ABC (resolution IV), 2⁵⁻¹ with E = ABCD (resolution V), and 2⁶⁻² with E = ABC and F = BCD (resolution IV).
• Box-Behnken (3 or more factors): runs at the midpoints of the edges of the factor cube, 2k(k−1) runs plus centre points.
• Central composite: the full factorial corners, plus axial points on the faces of the cube at ±1 (a face-centred design), 2^k + 2k runs plus centre points.
• Centre points: 1 to 10 replicates at the midpoint, 3 by default. They provide the pure-error estimate used by the lack-of-fit test.
• Run order: randomized with a seeded pseudo-random shuffle, so a given seed reproduces the same order. The standard order is kept alongside.

Up to six factors are supported. The design is chosen by deterministic rules from the number of factors and the kind of study, and you can change it.

The model

The fitted terms depend on the design:

• Full factorial: main effects and all two-factor interactions.
• Box-Behnken and central composite: main effects, two-factor interactions and pure quadratic terms (a full second-order model).
• Fractional factorial: main effects only, because its two-factor interactions are aliased with each other and cannot be separated.

Coefficients are ordinary least squares estimates, solved directly from the normal equations. The model is fitted to the runs that have a response entered, and updates as you add more.

ANOVA and lack of fit

The ANOVA table splits variation into model and residual, with an F-test for the model as a whole. Where there are replicated runs (typically the centre points), the residual is split further into pure error, from the spread among replicates, and lack of fit, the part the model's shape cannot explain. A significant lack-of-fit F-test says the chosen model form is too simple for the data. Without replicates, lack of fit cannot be tested and is not reported.

p-values come from the F and t distributions, computed via the regularized incomplete beta function.

R², adjusted R² and predicted R²

R² is the share of variation the model explains. Adjusted R² penalizes terms that add little. Predicted R² is based on PRESS, the sum of squared leave-one-out prediction errors, and is the one to trust for judging how the model will do on new runs: it can be low or negative even when R² looks excellent, which is a sign of overfitting.

Residual diagnostics

Residuals are scaled to studentized residuals using each run's leverage, then plotted against fitted values (a pattern suggests a missing term or unequal variance), in run order (a trend suggests drift), and on a normal probability plot. Leverage is computed for every run.

The optimum

The optimum finder evaluates the fitted model on a grid across the coded factor space (41 points per factor for two factors, 21 for three, 11 for four, 7 for five or six) and reports the best predicted response for your goal: maximize, minimize, or get closest to a target value. It is a grid search of the fitted model, so it is only as good as the model, and it is restricted to the region you tested. Confirm any predicted optimum with a real run.

How it is tested

• The probability functions (t, F and normal distributions) are compared against standard published critical values.
• The regression code is checked against Anscombe's quartet, a widely published OLS benchmark.
• Designs are tested against their defining properties: run counts, balance and orthogonality.
• Models fitted to noise-free data generated from known effects must recover those effects.

We have not yet validated against other statistical packages on a broad set of published DoE datasets. If you find a result that disagrees with another tool, please tell us.

Limits

• Continuous numeric factors only: no categorical factors, mixture designs or split-plot structure yet.
• No blocking, and no D-optimal or custom designs.
• The analysis assumes independent errors with roughly constant variance. Check the diagnostics before acting on a result.