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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

Choosing the base design¶

The first page is headed Select the base design and holds a grid of the factorial parts, with factors along the top and the fraction down the side. Each cell shows Full or a resolution as a Roman numeral, and hovering it reports the factors, the runs and the resolution.

The page divides the grid into Recommended Designs and Use with caution, and the rule behind the split is checkable: a cell counts as recommended when its factorial part is a full factorial or its resolution is V or above. Anything at resolution IV or below is in the caution group, because a CCD built on it has main effects or two-factor interactions aliased before the axial points are added.

Two lines under the grid describe the selection. This page words the fraction differently from the two-level factorial page, using a word rather than a number, so for a quarter-fraction base design with 8 factors they read:

Quarter factorial, 8 Factors in 64 Runs.

Resolution: V. Main effects are aliased with four-ways, and two-ways are aliased with three-ways.

The fraction word is one of Full, Half, Quarter, Eighth or Sixteenth. The resolution descriptions are the same ones used on the two-level factorial page, including the second clause about two-way interactions from resolution IV upward. See Two Level Factorial Designs.

Both lines are shown in red when the selected design is in the caution group.

The axial value¶

Axial and center points is the last page rather than the first. The alpha values it shows are recomputed as you type, from the factor count, the number of factorial runs and the number of center points. Blocking and replication do not enter that calculation, so changing either leaves the offered alpha unchanged. It asks for the number of center points and offers four ways to set alpha:

Choice	Alpha
Face CCD Alpha = 1	exactly 1, so the axial points sit on the faces of the cube rather than outside it
*Rotatable (value)*	computed, and the dialog shows the number in the label
*Optimal orthogonality (value)*	computed, and the dialog shows the number in the label
Manual	whatever you type. Selecting it reveals the alpha entry box, which is hidden for the other three choices, and the radio's own label gains a colon, becoming Manual:

The axial distance of a central composite design, in coded units, is written \(\alpha_{\text{axial}}\) here and grouped under Alpha value in the dialog; it is subscripted here because \(\alpha\) on its own is the type I error rate. It is set one of four ways. Writing \(F\) for the number of factorial runs, \(k\) for the number of factors and \(c\) for the number of centre points:

\[ \alpha_{\text{face}} = 1 \qquad \alpha_{\text{rotatable}} = F^{1/4} \]

\[ t = 2k + c \qquad q = \left(\sqrt{F + t} - \sqrt{F}\right)^{2} \qquad \alpha_{\text{orthogonal}} = \left(\tfrac{1}{4} q F\right)^{1/4} \]

The fourth way is to type the value in.

A face centred design puts the axial points level with the factorial faces, so no factor is ever run outside its stated low and high. The rotatable value makes the prediction variance depend only on distance from the centre. The orthogonal value makes the quadratic terms orthogonal to the rest of the model.

For the recognised combinations of factors and factorial runs, the dialog opens with a standard axial value and a standard centre-point count already filled in, and the alpha type set to Manual. That is a starting value rather than an alternative to the formulas: as soon as Rotatable, Optimal orthogonality or Face CCD Alpha = 1 is selected, the value used is what that formula computes, rounded to four decimal places, for a recognised design as much as for any other. The two computed captions are recalculated as the centre-point count changes.

The prefilled values assume an unblocked design and are not revised when you choose a block count. They are filled in on the base design step, before the replicates and blocks step is reached, and are looked up as though the design had one block. The stored table does hold different standard values for the same factors and factorial runs at different block counts, so on a blocked CCD the prefilled axial value and centre-point count are the unblocked entry rather than the one for the design you end up with. Type the values you want, or pick one of the computed alpha types, which are unaffected.

The standard split of centre points between the factorial and axial portions is used only when the total you ask for equals the standard total. Otherwise the centre points are distributed across the blocks instead.

Full details: Power and Sample Size.

The page also reports the totals as you type, and the design sheet records the value it used as Alpha = value. With a single replicate the two lines read Total number of center points: and Total number of runs:; with more than one they show the arithmetic instead, as Total center points: n replicates * m center points = t and Total runs: n replicates * m = t.

Selecting Manual is what reveals the alpha entry box, and the radio's own label gains a colon and reads Manual: at that point. For the other three choices there is no box to type in.

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.

The full page sequence¶

Select the base design > Enter Factor Names and Coding > Define outputs (responses) > Enter number of replicates and number of blocks > Axial and center points.