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N-Level Factorial Design

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A full factorial in which each factor can have its own number of levels. Two factors with 3 and 4 levels give 12 runs, not 9 or 16.

What this design gives you

Every combination of every level is run, so the design is fully crossed and nothing is aliased. It is the design to build when the factors do not all have the same number of settings, which is common when some factors are categorical with an awkward number of levels.

The run count is the product of the level counts, so it grows quickly: three factors with 4, 3 and 3 levels is 36 runs before any replication.

How the model treats the levels

A factor is either Quantitative or Categorical, and the choice changes what the software can do with it.

A Quantitative factor takes numeric values on a scale, so the model can interpolate between the settings you ran and predict at a value you did not run. It is coded to a common scale before the regression, and it can carry an axis on a surface or contour chart.

A Categorical factor takes named levels with no scale between them. It is never coded to a numeric range; instead each non-reference level gets its own coefficient. There is nothing between two levels to predict at, so a chart that needs to evaluate the model between settings either refuses a categorical factor on that axis or draws it as separate levels rather than as a line. Which of the two depends on the chart, and each chart page says.

For a categorical factor one level is the reference. The other levels each get a coefficient measured against it, and the reference itself has none, which is why the regression table shows one fewer coefficient than the factor has levels. The reference level is not missing from the model: it is the baseline the others are expressed relative to.

A quantitative factor with four or more levels can still carry a quadratic term, and its coded values come from its own minimum and maximum. See Design Coding.

Every control

Every control on every page this builder shows for this design is on Options.

See Also