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Two-Level Factorial How-To

This walkthrough builds a full 2^3 factorial for three factors, then builds the half fraction of a four factor design so you can see what a generator costs you and what it buys.

Nothing here needs data. A design is built before the experiment is run, so every number below is determined by the design itself and you can check each one against your own sheet.

Build the full factorial

  1. Start the builder

    QXL DOE New > Create Design. Choose 2 Level Factorial and press Next.

  2. Choose the design

    The grid shows factors along the top and the fraction down the side. Pick the cell for 3 factors and Full, which is 8 runs, and press Next.

  3. Name the factors

    On Enter Factor Names and Coding, type three factors and their settings:

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

    These are the settings every other DOE walkthrough uses, so the sheet this builds is the one Modify Design How-To replicates and the Analyze walkthroughs fit.

    Leave Categorical clear on all three, and press Next.

  4. Define the output

    On Define outputs (responses) set Select number of outputs: to 1, name it Strength, and press Next.

  5. Leave the last page alone

    On Enter number of replicates and number of blocks, leave Select number of replicates: at 1 and the blocking list at No blocking. Leave Number of center points per block: at 0 and Fold design: clear. Press Finish.

What you get

Eight runs, in standard order with the first factor changing fastest. In coded units:

Press Copy for Excel and paste the block into an empty part of the worksheet, beside your design sheet, to compare it row by row rather than by eye.

Run Temp Press Time
1 -1 -1 -1
2 +1 -1 -1
3 -1 +1 -1
4 +1 +1 -1
5 -1 -1 +1
6 +1 -1 +1
7 -1 +1 +1
8 +1 +1 +1

In the units you typed, run 1 is 180, 40, 10 and run 8 is 220, 60, 20.

Every column is balanced, four runs at each setting, which is what lets each effect be estimated independently of the others. Add the column up and it comes to zero.

All seven effects are estimable and none is aliased with another: the three main effects, the three two-factor interactions, and the three-factor interaction. That is what a full factorial buys, and it is why eight runs is the smallest design that separates everything for three factors.

Now the half fraction

Run the builder again, choose 4 factors and the 1/2 fraction, which is 8 runs. Name a fourth factor and finish as before.

You get eight runs for four factors instead of sixteen. In coded units:

Run A B C D
1 -1 -1 -1 -1
2 +1 -1 -1 +1
3 -1 +1 -1 +1
4 +1 +1 -1 -1
5 -1 -1 +1 +1
6 +1 -1 +1 -1
7 -1 +1 +1 -1
8 +1 +1 +1 +1

The fourth column is not free. It is the product of the first three: multiply A, B and C on any row and you get D. Check run 2: \((+1)(-1)(-1) = +1\), which is D. That relationship is the generator, and it is what the eight saved runs cost.

What the generator costs

The generator D = ABC makes the defining word ABCD, and the shortest defining word is the resolution. This one has four letters, so the design is resolution IV, which the sheet reports.

Resolution IV means:

Effect Aliased with
each main effect a three-factor interaction
each two-factor interaction another two-factor interaction

So A is clear of the other main effects and clear of every two-factor interaction, but AB and CD cannot be told apart. If the experiment finds a large AB effect, the design cannot say whether it is AB, CD, or some of both. The alias table on the design sheet lists every pair.

Things to try next

  • Build the same four factors at Full instead of 1/2. You get sixteen runs, the alias table disappears, and every two-factor interaction becomes estimable on its own. Comparing the two alias tables is the clearest way to see what a fraction gives up.
  • Set Number of center points per block: to 3 on the first design. The run count goes from 8 to 11, the three added runs sit at the middle of every factor, and the fit gains a check for curvature that a two-level design cannot otherwise make.
  • Set Select number of replicates: to 2. The run count doubles to 16 and the regression gains a pure error estimate, which is what makes a lack-of-fit test possible.

See Also