---
title: "Aliasing (factorial experiments)"
chunk: 3/7
source: "https://en.wikipedia.org/wiki/Aliasing_(factorial_experiments)"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T09:48:49.670316+00:00"
instance: "kb-cron"
---

=== The 2 × 2 × 2 (or 2³) experiment ===
This is a "two-level" experiment with factors 
  
    
      
        A
        ,
        B
      
    
    {\displaystyle A,B}
  
 and 
  
    
      
        C
      
    
    {\displaystyle C}
  
. In such experiments the factor levels are often denoted by 0 and 1, for reasons explained below. A treatment combination is then denoted by an ordered triple such as 101 (more formally, (1, 0, 1), denoting the cell in which 
  
    
      
        A
      
    
    {\displaystyle A}
  
 and 
  
    
      
        C
      
    
    {\displaystyle C}
  
 are at level "1" and 
  
    
      
        B
      
    
    {\displaystyle B}
  
 is at level "0"). The following table lists the eight cells of the full 2 × 2 × 2 factorial experiment, along with a contrast vector representing each effect, including a three-factor interaction:

Suppose that only the fraction consisting of the cells 000, 011, 101, and 110 is observed. The original contrast vectors, when restricted to these cells, are now 4 × 1, and can be seen by looking at just those four rows of the table. (Sorting the table on 
  
    
      
        A
        B
        C
      
    
    {\displaystyle ABC}
  
 will bring these rows together and make the restricted contrast vectors easier to see. Sorting twice puts them at the top.) The following can be observed concerning these restricted vectors:

The 
  
    
      
        A
        B
        C
      
    
    {\displaystyle ABC}
  
 column consists just of the constant 1 repeated four times.
The other columns are contrast vectors, having two 1's and two −1s.
The columns for 
  
    
      
        C
      
    
    {\displaystyle C}
  
 and 
  
    
      
        A
        B
      
    
    {\displaystyle AB}
  
 are equal. The same holds for 
  
    
      
        A
      
    
    {\displaystyle A}
  
 and 
  
    
      
        B
        C
      
    
    {\displaystyle BC}
  
, and for 
  
    
      
        B
      
    
    {\displaystyle B}
  
 and 
  
    
      
        A
        C
      
    
    {\displaystyle AC}
  
.
All other pairs of columns are orthogonal. For example, the column for 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is orthogonal to that for 
  
    
      
        B
      
    
    {\displaystyle B}
  
, for 
  
    
      
        C
      
    
    {\displaystyle C}
  
, for 
  
    
      
        A
        B
      
    
    {\displaystyle AB}
  
, and for 
  
    
      
        A
        C
      
    
    {\displaystyle AC}
  
, as one can see by computing dot products.
Thus

the 
  
    
      
        A
        B
        C
      
    
    {\displaystyle ABC}
  
 interaction is completely lost in the fraction;
the other effects are preserved in the fraction;
the effects 
  
    
      
        A
      
    
    {\displaystyle A}
  
 and 
  
    
      
        B
        C
      
    
    {\displaystyle BC}
  
 are completely aliased with each other, as are 
  
    
      
        B
      
    
    {\displaystyle B}
  
 and 
  
    
      
        A
        C
      
    
    {\displaystyle AC}
  
, and 
  
    
      
        C
      
    
    {\displaystyle C}
  
 and 
  
    
      
        A
        B
      
    
    {\displaystyle AB}
  
.
all other pairs of effects are unaliased. For example, 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is unaliased with both 
  
    
      
        B
      
    
    {\displaystyle B}
  
 and 
  
    
      
        C
      
    
    {\displaystyle C}
  
 and with the 
  
    
      
        A
        B
      
    
    {\displaystyle AB}
  
 and 
  
    
      
        A
        C
      
    
    {\displaystyle AC}
  
 interactions.
Now suppose instead that the complementary fraction {001,010,100,111} is observed. The same effects as before are lost or preserved, and the same pairs of effects as before are mutually unaliased. Moreover, 
  
    
      
        A
      
    
    {\displaystyle A}
  
 and 
  
    
      
        B
        C
      
    
    {\displaystyle BC}
  
 are still aliased in this fraction since the 
  
    
      
        A
      
    
    {\displaystyle A}
  
 and 
  
    
      
        B
        C
      
    
    {\displaystyle BC}
  
 vectors are negatives of each other, and similarly for 
  
    
      
        B
      
    
    {\displaystyle B}
  
 and 
  
    
      
        A
        C
      
    
    {\displaystyle AC}
  
 and for 
  
    
      
        C
      
    
    {\displaystyle C}
  
 and 
  
    
      
        A
        B
      
    
    {\displaystyle AB}
  
.  Both of these fractions thus have maximum resolution 3.

== Aliasing in regular fractions ==
The two half-fractions of a 
  
    
      
        
          2
          
            3
          
        
      
    
    {\displaystyle 2^{3}}
  
 factorial experiment described above are of a special kind: Each is the solution set of a linear equation using modular arithmetic. More exactly:

The fraction 
  
    
      
        {
        000
        ,
        011
        ,
        101
        ,
        110
        }
      
    
    {\displaystyle \{000,011,101,110\}}
  
 is the solution set of the equation 
  
    
      
        
          t
          
            1
          
        
        +
        
          t
          
            2
          
        
        +
        
          t
          
            3
          
        
        =
        0
        
          
          (
          mod
          
          2
          )
        
      
    
    {\displaystyle t_{1}+t_{2}+t_{3}=0{\pmod {2}}}
  
. For example, 
  
    
      
        011
      
    
    {\displaystyle 011}
  
 is a solution because 
  
    
      
        0
        +
        1
        +
        1
        =
        0
        
          
          (
          mod
          
          2
          )
        
      
    
    {\displaystyle 0+1+1=0{\pmod {2}}}
  
.
Similarly, the fraction 
  
    
      
        {
        001
        ,
        010
        ,
        100
        ,
        111
        }
      
    
    {\displaystyle \{001,010,100,111\}}
  
 is the solution set to 
  
    
      
        
          t
          
            1
          
        
        +
        
          t
          
            2
          
        
        +
        
          t
          
            3
          
        
        =
        1
        
          
          (
          mod
          
          2
          )
        
      
    
    {\displaystyle t_{1}+t_{2}+t_{3}=1{\pmod {2}}}
  

Such fractions are said to be regular. This idea applies to fractions of "classical" 
  
    
      
        
          s
          
            k
          
        
      
    
    {\displaystyle s^{k}}
  
 designs, that is, 
  
    
      
        
          s
          
            k
          
        
      
    
    {\displaystyle s^{k}}
  
 (or "symmetric") factorial designs in which the number of levels, 
  
    
      
        s
      
    
    {\displaystyle s}
  
, of each of the 
  
    
      
        k
      
    
    {\displaystyle k}
  
 factors is a prime or the power of a prime.

A fractional factorial design is regular if it is the solution set of a system of one or more equations of the form

  
    
      
        
          a
          
            1
          
        
        
          t
          
            1
          
        
        +
        ⋯
        +
        
          a
          
            k
          
        
        
          t
          
            k
          
        
        =
        b
        ,
      
    
    {\displaystyle a_{1}t_{1}+\cdots +a_{k}t_{k}=b,}