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Fundamentals of Experimental Design 439

In Table 12.12 we can find that since all the contrast coefficients for
ABC are 1s, we will not be able to estimate the effect of ABC at all.
However, for other main effects and interactions, the first four runs
have equal numbers of 1s and 1s, so we can calculate their effects.
However, we can find that the contrast coefficients of A are identical
as those of BC, and the contrast coefficients of B are exactly the same
as those of AC, as well as C and AB. Since the effects are computed
using the contrast coefficient, there is no way to distinguish the
effects of A and BC, B and AC, and C and AB. For example, when we
estimate the effect of A, we are really estimating the combined effect
of A and BC. This mixup of main effects and interactions is called
aliases or confounding.
All alias relationships can be found from the defining relation: I
ABC. If we simply multiply A on both sides of the equation, we get AI
AABC. Since multiplying identical columns will give an I column, this
equation becomes A BC. Similarly, we can get B AC and C AB.
This half-fraction based on I ABC is called the principal fraction.
If we use the second half of Table 12.12, the defining relationship
will be I ABC. Because all ABC coefficients are equal to 1s, we
can easily determine that A BC, B AC, and C AB. Therefore
A is aliased with BC, B is aliased with AC, and C is aliased
with AB.
In summary, in the case of half-fractional two-level factorial experi-
ments, we will completely lose the information about the highest order
interaction effect and partially lose some information about lower-
order interactions.

12.4.2 How to lay out a general half
k
fractional 2 design
The half-fractional 2 design is also called 2 k 1 design, because it has
k
N 2 k 1 runs.
Using the definition relationship to lay out the experiment, we
describe the procedure to lay out 2 k 1 design, and illustrate it with an
example.

Step 1: Compute N 2 k 1 and determine the number of runs. For
Example 12.6, for k 4, N 2 k 1 2 8.
3
Step 2: Create a table with N runs and lay out the first k 1 factors
in standard order. For example, for k 4, the factors are A, B, C, and
D, and the first k 1 3 factors are A, B, and C, as shown in Table
12.13.
We will lay out the first three columns with A, B, and C in stan-
dard order.