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Central Composite How-To

This walkthrough builds a three factor central composite design and shows what the alpha choice does to the runs, which is the one decision this design asks of you that no other design does.

Nothing here needs data. Every number below is determined by the design.

Steps

  1. Start the builder

    QXL DOE New > Create Design. Choose Central Composite and press Next.

  2. Choose the base design

    The grid of factorial parts appears, split into Recommended Designs and Use with caution. Choose the 3 factor Full cell, which is 8 factorial runs, and press Next.

  3. Name the factors

    Factor Low High
    Temp 180 220
    Press 40 60
    Time 10 20

    Press Next, name one output Yield, and press Next again.

  4. Leave replicates and blocking alone

    One replicate, No blocking. Press Next.

  5. Set the axial points

    On Axial and center points, type 6 centre points and leave Rotatable selected. The label shows the value it computed. Press Finish.

What you get

Twenty runs: eight factorial, six axial and six centre. The rotatable alpha at three factors is

\[ \alpha = 8^{1/4} = 1.6818 \]

In coded units:

Press Copy for Excel and paste the block into an empty part of the worksheet, beside your design sheet. This is the one design in the family worth pasting rather than reading: the axial rows carry four decimal places and a transcription slip in them is invisible.

Run Temp Press Time Part
1 -1 -1 -1 factorial
2 +1 -1 -1 factorial
3 -1 +1 -1 factorial
4 +1 +1 -1 factorial
5 -1 -1 +1 factorial
6 +1 -1 +1 factorial
7 -1 +1 +1 factorial
8 +1 +1 +1 factorial
9 -1.6818 0 0 axial
10 +1.6818 0 0 axial
11 0 -1.6818 0 axial
12 0 +1.6818 0 axial
13 0 0 -1.6818 axial
14 0 0 +1.6818 axial
15 0 0 0 centre
16 0 0 0 centre
17 0 0 0 centre
18 0 0 0 centre
19 0 0 0 centre
20 0 0 0 centre

Read the structure off the table

The design is three parts and each does a different job.

Part What it is What it gives the fit
factorial the eight corners of the cube the main effects and the two-factor interactions
axial two points per factor, out along that factor's own axis the squared terms, which is what makes a curved model estimable
centre six runs at the middle of every factor an estimate of pure error, and a check on curvature

The axial points go outside the cube. At 1.6818 they sit further out than the corners at 1, which is the point: the design reaches beyond the factorial box along each axis.

That has a practical consequence you must check before running it. Temp's axial runs are at coded -1.6818 and +1.6818, and Temp is centred at 200 with a half-range of 20, so those two runs are at 166.4 and 233.6 degrees: outside the 180 to 220 range you typed. Press's are at 33.2 and 66.8, and Time's at 6.6 and 23.4. If the process cannot reach those settings the design cannot be run as built, and you want Face CCD Alpha = 1 instead.

Now change the alpha

Build it again and choose Face CCD Alpha = 1 on the last page.

The six axial runs move onto the faces of the cube, at coded -1 and +1 rather than 1.6818, so every run stays inside the box the factor settings define. Nothing else about the design changes: the same eight factorial runs, the same six centre points, twenty runs in total.

What you give up is rotatability. The rotatable alpha makes the prediction variance depend only on the distance from the centre, so the model is equally precise in every direction; at alpha 1 it is not.

Things to try next

  • Choose Manual on the last page. The alpha entry box appears, which is hidden for the other three choices, and the radio's own label gains a colon and reads Manual:. Any alpha you type is used as it stands.
  • Change the centre point count and watch the offered alpha. The rotatable and optimal-orthogonality values are recomputed from the factor count, the factorial run count and the centre points as you type. Blocking and replication do not enter that calculation, so changing either leaves the offered alpha alone.
  • Build the same three factors as a Box-Behnken. Fifteen runs instead of twenty, and no run outside the cube at all.

See Also