where 
is the portion of the design matrix,
that contains the effects for which
the aliases need to be calculated, and 
contains the remaining columns of
the design matrix, other than those included in 
.In Chapter 7, Calculation
of Aliases, the alias structure for fractional factorial designs was
obtained using the defining relation. However, this method of obtaining
the alias structure is not very efficient when the alias structure is
very complex or when partial aliasing is involved. [Note
]
One of the ways to obtain the alias structure for any design, regardless
of its complexity, is to use the alias matrix. The alias matrix for a
design is calculated using where is the portion of the design matrix,
that contains the effects for which
the aliases need to be calculated, and contains the remaining columns of
the design matrix, other than those included in .
To illustrate the use of the alias matrix, consider the design matrix
for the 2
design (using the defining relation
) shown next:
The alias structure for this design can be obtained by defining 
using eight columns since the 2
design estimates eight effects. If
the first eight columns of 
are used then 
is:
is obtained using the remaining columns
as:
Then the alias matrix 
is:
The alias relations can be easily obtained by observing the alias matrix
as:
