---
title: "Aliasing (factorial experiments)"
chunk: 6/7
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"
---

In the 
  
    
      
        
          2
          
            3
            −
            1
          
        
      
    
    {\displaystyle 2^{3-1}}
  
 fraction with defining word 
  
    
      
        A
        B
        C
      
    
    {\displaystyle ABC}
  
, the maximum resolution is 3 (the length of that word), while the fraction with defining word 
  
    
      
        A
        B
      
    
    {\displaystyle AB}
  
 has maximum resolution 2.
The defining words of the 
  
    
      
        
          3
          
            4
            −
            1
          
        
      
    
    {\displaystyle 3^{4-1}}
  
 fraction were 
  
    
      
        A
        B
        C
        
          D
          
            2
          
        
      
    
    {\displaystyle ABCD^{2}}
  
 and 
  
    
      
        
          A
          
            2
          
        
        
          B
          
            2
          
        
        
          C
          
            2
          
        
        D
      
    
    {\displaystyle A^{2}B^{2}C^{2}D}
  
, both of length 4, so that the fraction has maximum resolution 4, as indicated.
In the 
  
    
      
        
          2
          
            5
            −
            2
          
        
      
    
    {\displaystyle 2^{5-2}}
  
 fraction with defining words 
  
    
      
        A
        B
        D
        ,
        A
        C
        E
      
    
    {\displaystyle ABD,ACE}
  
 and 
  
    
      
        B
        C
        D
        E
      
    
    {\displaystyle BCDE}
  
, the maximum resolution is 3, which is the shortest "wordlength".
One could also construct a 
  
    
      
        
          2
          
            5
            −
            2
          
        
      
    
    {\displaystyle 2^{5-2}}
  
 fraction from the defining words 
  
    
      
        A
        B
        C
        D
      
    
    {\displaystyle ABCD}
  
 and 
  
    
      
        B
        C
        D
        E
      
    
    {\displaystyle BCDE}
  
, but the defining subgroup 
  
    
      
        
          H
        
      
    
    {\displaystyle \mathbb {H} }
  
 will also include 
  
    
      
        A
        E
      
    
    {\displaystyle AE}
  
, their product, and so the fraction will only have resolution 2 (the length of 
  
    
      
        A
        E
      
    
    {\displaystyle AE}
  
). This is true starting with any two words of length 4. Thus resolution 3 is the best one can hope for in a fraction of type 
  
    
      
        
          2
          
            5
            −
            2
          
        
      
    
    {\displaystyle 2^{5-2}}
  
.
As these examples indicate, one must consider all the elements of the defining subgroup in applying the theorem above.  This theorem is often taken to be a definition of resolution, but the Box-Hunter definition given earlier applies to arbitrary fractional designs and so is more general.

== Aliasing in general fractions ==
Nonregular fractions are common, and have certain advantages.  For example, they are not restricted to having size a power of 
  
    
      
        s
      
    
    {\displaystyle s}
  
, where 
  
    
      
        s
      
    
    {\displaystyle s}
  
 is a prime or prime power. While some methods have been developed to deal with aliasing in particular nonregular designs, no overall algebraic scheme has emerged.
There is a universal combinatorial approach, however, going back to Rao. If the treatment combinations of the fraction are written as rows of a table, that table is an orthogonal array. These rows are often referred to as "runs".  The columns will correspond to the 
  
    
      
        k
      
    
    {\displaystyle k}
  
 factors, and the entries of the table will simply be the symbols used for factor levels, and need not be numbers. The number of levels need not be prime or prime-powered, and they may vary from factor to factor, so that the table may be a mixed-level array. In this section fractional designs are allowed to be mixed-level unless explicitly restricted.
A key parameter of an orthogonal array is its strength, the definition of which is given in the article on orthogonal arrays. One may thus refer to the strength of a fractional design. Two important facts flow immediately from its definition:

If an array (or fraction) has strength 
  
    
      
        t
      
    
    {\displaystyle t}
  
 then it also has strength 
  
    
      
        
          t
          ′
        
      
    
    {\displaystyle t'}
  
 for every 
  
    
      
        
          t
          ′
        
        <
        t
      
    
    {\displaystyle t'<t}
  
.  The array's maximum strength is of particular importance.
In a fixed-level array, all factors having 
  
    
      
        s
      
    
    {\displaystyle s}
  
 levels, the number of runs is a multiple of 
  
    
      
        
          s
          
            t
          
        
      
    
    {\displaystyle s^{t}}
  
, where 
  
    
      
        t
      
    
    {\displaystyle t}
  
 is the strength.  Here 
  
    
      
        s
      
    
    {\displaystyle s}
  
 need not be a prime or prime power.
To state the next result, it is convenient to enumerate the factors of the experiment by 1 through 
  
    
      
        k
      
    
    {\displaystyle k}
  
, and to let each nonempty subset 
  
    
      
        I
      
    
    {\displaystyle I}
  
 of 
  
    
      
        {
        1
        ,
        …
        ,
        k
        }
      
    
    {\displaystyle \{1,\ldots ,k\}}
  
 correspond to a main effect or interaction in the following way:  
  
    
      
        I
        =
        {
        i
        }
      
    
    {\displaystyle I=\{i\}}
  
 corresponds to the main effect of factor

  
    
      
        i
      
    
    {\displaystyle i}
  
, 
  
    
      
        I
        =
        {
        i
        ,
        j
        }
      
    
    {\displaystyle I=\{i,j\}}
  
 corresponds to the interaction of factors 
  
    
      
        i
      
    
    {\displaystyle i}
  
 and 
  
    
      
        j
      
    
    {\displaystyle j}
  
, and so on.

Example: Consider a fractional factorial design with factors 
  
    
      
        A
        ,
        B
        ,
        C
        ,
        D
        ,
        E
      
    
    {\displaystyle A,B,C,D,E}
  
 and maximum strength 
  
    
      
        t
        =
        3
      
    
    {\displaystyle t=3}
  
. Then:

All effects up to three-factor interactions are preserved in the fraction.
Main effects are unaliased with each other and with two-factor interactions.
Two-factor interactions are unaliased with each other if they share a factor. For example, the 
  
    
      
        A
        ×
        B
      
    
    {\displaystyle A\times B}
  
 and 
  
    
      
        A
        ×
        C
      
    
    {\displaystyle A\times C}
  
 interactions are unaliased, but the 
  
    
      
        A
        ×
        B
      
    
    {\displaystyle A\times B}
  
 and 
  
    
      
        C
        ×
        D
      
    
    {\displaystyle C\times D}
  
 interactions may be at least partly aliased as the set 
  
    
      
        {
        A
        ,
        B
        ,
        C
        ,
        D
        }
      
    
    {\displaystyle \{A,B,C,D\}}
  
 contains 4 elements but the strength of the fraction is only 3.
The Fundamental Theorem has a number of important consequences. In particular, it follows almost immediately that if a fraction has strength 
  
    
      
        t
      
    
    {\displaystyle t}
  
 then it has resolution 
  
    
      
        t
        +
        1
      
    
    {\displaystyle t+1}
  
.  With additional assumptions, a stronger conclusion is possible:

This result replaces the group-theoretic condition (minimum wordlength) in regular fractions with a combinatorial condition (maximum strength) in arbitrary ones.
Example.  An important class of nonregular two-level designs are Plackett-Burman designs.  As with all fractions constructed from Hadamard matrices, they have strength 2, and therefore resolution 3.  The smallest such design has 11 factors and 12 runs (treatment combinations), and is displayed in the article on such designs.  Since 2 is its maximum strength, 3 is its maximum resolution. Some detail about its aliasing pattern is given in the next section.