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Create Central Composite (CCD) Design

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A central composite design is a two-level factorial with center points and two extra runs per factor placed beyond the corners, at a distance called the axial value or alpha.

What this design gives you

The factorial corners estimate main effects and interactions. The axial points give each quantitative factor a third and fourth distinct setting, which is what makes its quadratic term estimable. So a CCD can fit a curved surface at far fewer runs than a three-level full factorial: five factors is 32 factorial runs plus 10 axial runs plus center points, against 243 for the three-level design.

Every factor in a central composite design is quantitative. Its factor page offers no Categorical column and no categorical hint line, so there is no way to enter a qualitative factor for this design type.

The design is built in three parts and each row is tagged with which part it belongs to, because coding, blocking and the lack-of-fit calculation all treat them differently:

Part Rows
Factorial 2 raised to the power of the number of non-aliased factors
Axial 2 per factor, one at -alpha and one at +alpha
Center as many as you ask for

Axial points and coding

Only rows whose point type is factorial or edge centroid set the scale. Centre points, both kinds of axial point, and any extra runs added with Add/Remove Runs are all excluded. That applies to the minimum and maximum behind Autocode and equally to the mean and sample standard deviation behind Standardize.

Two consequences are worth knowing. Centre points and replicated centre runs do not move the centre or the scale. And an extra run entered outside the factorial range does not widen the minimum or the maximum, so it codes outside the usual range with nothing on the sheet to explain why.

A factor needs at least two distinct values among those rows. With only one the scale would be zero, and the coding stops instead, for Autocode as well as for Standardize.

For a central composite design under Autocode this has a visible consequence: the corners code to \(\pm 1\) as usual, and axial points beyond the corners code to values outside \(\pm 1\), at \(\pm \alpha_{\text{axial}}\). Under Standardize the same rows set a mean and a standard deviation instead, so neither the corners nor the axial points land on those values exactly.

This is worth checking on the sheet the first time: an autocoded CCD has corners at -1 and +1 and, when alpha is greater than 1, axial points beyond them. That is correct and not a coding error. With Face CCD Alpha = 1 the axial points code to exactly -1 and +1 and sit on the faces of the cube, and a manual alpha below 1 puts them inside it.

Blocking

In-design blocking splits a CCD so that the factorial part and the axial part are separate blocks. With two blocks, one block is the axial portion and the other the factorial portion; with more blocks, one takes the axial portion and the rest divide the factorial portion.

That split is why the standard designs carry their own center point counts for the two portions separately.

In-design blocking is not available for every base design. The block counts on offer come from the standard design table, matched on the exact combination of factor count and factorial run count. Where there is an entry they are 2, 3 or 5; where there is none, and several cells of the chooser grid have none, the list offers no blocking at all.

Choosing a block-with-replicate entry does something different. Those entries appear for any divisor of the replicate count once the design has more than one replicate, and they make groups of whole replicates into blocks. The factorial and axial split does not happen at all in that case.

Every control

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

See Also