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Three-Level Factorial Designs

QXL DOE New > Create Design > Modeling > Create 3-Level Factorial Design

A full factorial with three levels on every factor, so the design has 3^k runs for k factors.

What this design gives you

Three settings on a factor is enough to estimate a quadratic term for it, so this design can hold curvature for every factor at once without axial points. It is a full factorial, so nothing is aliased. The model it builds stops at 7-way interactions, though: higher-order interactions are never created, so with eight or more factors the design has degrees of freedom the model does not use.

The cost is the run count, which is 3 to the power of the number of factors: 9 runs for two factors, 27 for three, 81 for four, 243 for five.

The pages

This design type has the shortest build sequence of any, three pages, because the first page does both the counting and the naming.

Page heading What it asks
Enter factor names and factor levels Select number of factors:, then a grid with one row per factor, headed Factor name, Categorical, Level 1, Level 2 and Level 3
Define outputs (responses) the responses and the design format
Enter number of replicates and number of blocks replicates and blocking

There is no center points page and no folding. Every quantitative factor already has a middle level, which is what a center point is for, and folding applies to fractional designs only. Add/Remove Center Points on the Modify Design menu refuses on this design type with the message center points are not available for the 3 Levels Factorial design.

Levels

Every factor has exactly three levels here, and the count cannot be changed. The N-level factorial page carries a # Levels column for that; this page does not show one.

The layout does not change with the factor type either. A quantitative factor takes numbers in the Level 1, Level 2 and Level 3 cells and a categorical factor takes names in the same three cells, with Categorical ticked.

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.

A categorical factor with three levels contributes two coefficients to the model rather than one, because one level is the reference.

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.

See Also