Home / DOE / Additional Information / Categorical vs. Quantitative
Categorical vs. Quantitative¶
Every factor in a Quantum XL design is one of two kinds, and the Categorical checkbox on the factor page is where the choice is made. This page collects what changes between them.
Not every design offers the checkbox. Categorical factors are available on the 2-level factorial, Plackett-Burman, 3-level and N-level factorial, Taguchi and custom designs. CCD and Box-Behnken designs have no Categorical control at all: their factor page carries plain Low and High columns, no Categorical column and no categorical hint line, so every factor in them is quantitative. The Design Wizard applies the same rule.
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.
The dialog states the input rule on the factor page itself, in two hint lines: For Quantitative Inputs enter the low and high values (e.g. low=10 high=20) and For Categorical Inputs enter the name of each level (e.g. Red,Blue).
What changes, in one table¶
| Quantitative | Categorical | |
|---|---|---|
| What you enter | a low and a high value, or a list of numeric levels | a list of level names |
| Coding | scaled to a common range before the regression | never coded; values carried through as entered |
| Coefficients in the model | one per term | one per non-reference level |
| Reference level | not applicable | one level is the reference and has no coefficient. Which level that is by default depends on the design type, see below, and you can change it |
| Can carry a surface axis | yes, either axis | yes, but only the series axis, and at least one axis must be quantitative |
| Can carry a contour axis | yes, both axes | no, both contour axes must be quantitative |
| Can carry a cube axis | yes, with exactly two distinct values in the design | yes, with exactly two levels |
| Can carry a quadratic term | yes, with three or more distinct settings | no |
| Center points | available, at the middle setting | not available; an all-categorical design is refused |
| Prediction | the coded value enters the equation | the level selects which coefficient is read |
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.
The reference level is yours to set. For each categorical factor the design sheet carries a reference-level row below the coded-name row, marked <-- reference level. Type a level name there and that level becomes the reference. A blocking factor is the exception: its reference is always its highest block number.
Leave that row empty and the default depends on the design type, which is worth knowing because it is not always the level you entered first:
| Design type | The default reference level |
|---|---|
| 2-level factorial, Plackett-Burman | the level typed into the factor's Low cell |
| 3-level factorial, N-level factorial, Taguchi, custom | whichever of the entered levels sorts first: alphabetically for names, ascending for numbers, earliest for dates |
So on a 3-level factorial a factor entered as Red then Blue takes Blue as its reference, because Blue sorts first. Type the level you want into the reference-level row if it matters which one it is.
Consequences you will meet in the reports¶
The Pareto chart draws one bar per non-reference level of a categorical term, named term - level, rather than one bar for the term. See Pareto of Regression Coefficients.
A categorical term occupies several columns of the design matrix and is treated as one unit, which is why the ANOVA table has one row per term rather than one row per column. Its sequential sum of squares is the total of its columns'. Its adjusted sum of squares is the drop in the regression sum of squares when all of its columns are removed together, which is not in general that total. See Sums of Squares.
An interaction involving a categorical factor is indexed by level, so it has a coefficient per combination of levels rather than a single slope. See Prediction Equation.
Center points are refused on an all-categorical two-level factorial. The label above the centre points box is replaced by Center points are not available when all inputs are qualitative., and the box shows 0 greyed out.
A note on the word¶
The dialogs use both Categorical and Qualitative for the same thing. The factor page's checkbox says Categorical; the wizard's question says Do you have any Qualitative or Categorical Factors (Inputs)? and its explanation covers both words; some messages say qualitative. They mean the same kind of factor.