---
title: "Block design"
chunk: 5/6
source: "https://en.wikipedia.org/wiki/Block_design"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T09:49:03.148970+00:00"
instance: "kb-cron"
---

  
    
      
        
          R
          
            0
          
        
        =
        {
        (
        x
        ,
        x
        )
        :
        x
        ∈
        X
        }
      
    
    {\displaystyle R_{0}=\{(x,x):x\in X\}}
  
 and is called the Identity relation.
Defining 
  
    
      
        
          R
          
            ∗
          
        
        :=
        {
        (
        x
        ,
        y
        )
        ∣
        (
        y
        ,
        x
        )
        ∈
        R
        }
      
    
    {\displaystyle R^{*}:=\{(x,y)\mid (y,x)\in R\}}
  
, if R in S, then R* in S
If 
  
    
      
        (
        x
        ,
        y
        )
        ∈
        
          R
          
            k
          
        
      
    
    {\displaystyle (x,y)\in R_{k}}
  
, the number of 
  
    
      
        z
        ∈
        X
      
    
    {\displaystyle z\in X}
  
 such that 
  
    
      
        (
        x
        ,
        z
        )
        ∈
        
          R
          
            i
          
        
      
    
    {\displaystyle (x,z)\in R_{i}}
  
 and 
  
    
      
        (
        z
        ,
        y
        )
        ∈
        
          R
          
            j
          
        
      
    
    {\displaystyle (z,y)\in R_{j}}
  
 is a constant 
  
    
      
        
          p
          
            i
            j
          
          
            k
          
        
      
    
    {\displaystyle p_{ij}^{k}}
  
 depending on i, j, k but not on the particular choice of x and y.
An association scheme is commutative if 
  
    
      
        
          p
          
            i
            j
          
          
            k
          
        
        =
        
          p
          
            j
            i
          
          
            k
          
        
      
    
    {\displaystyle p_{ij}^{k}=p_{ji}^{k}}
  
 for all i, j and k. Most authors assume this property.
A partially balanced incomplete block design with n associate classes (PBIBD(n)) is a block design based on a v-set X with b blocks each of size k and with each element appearing in r blocks, such that there is an association scheme with n classes defined on X where, if elements x and y are ith associates, 1 ≤ i ≤ n, then they are together in precisely λi blocks.
A PBIBD(n) determines an association scheme but the converse is false.

=== Example ===
Let A(3) be the following association scheme with three associate classes on the set X = {1,2,3,4,5,6}. The (i,j) entry is s if elements i and j are in relation Rs.

The blocks of a PBIBD(3) based on A(3) are:

The parameters of this PBIBD(3) are: v  =  6, b  =  8, k  =  3, r  =  4 and λ1 = λ2 = 2 and λ3 = 1. Also, for the association scheme we have n0  =  n2  =  1 and n1  =  n3  =  2. The incidence matrix M is

and the concurrence matrix MMT is

from which we can recover the λ and r values.

=== Properties ===
The parameters of a PBIBD(m) satisfy:

  
    
      
        v
        r
        =
        b
        k
      
    
    {\displaystyle vr=bk}
  

  
    
      
        
          ∑
          
            i
            =
            1
          
          
            m
          
        
        
          n
          
            i
          
        
        =
        v
        −
        1
      
    
    {\displaystyle \sum _{i=1}^{m}n_{i}=v-1}
  

  
    
      
        
          ∑
          
            i
            =
            1
          
          
            m
          
        
        
          n
          
            i
          
        
        
          λ
          
            i
          
        
        =
        r
        (
        k
        −
        1
        )
      
    
    {\displaystyle \sum _{i=1}^{m}n_{i}\lambda _{i}=r(k-1)}
  

  
    
      
        
          ∑
          
            u
            =
            0
          
          
            m
          
        
        
          p
          
            j
            u
          
          
            h
          
        
        =
        
          n
          
            j
          
        
      
    
    {\displaystyle \sum _{u=0}^{m}p_{ju}^{h}=n_{j}}
  

  
    
      
        
          n
          
            i
          
        
        
          p
          
            j
            h
          
          
            i
          
        
        =
        
          n
          
            j
          
        
        
          p
          
            i
            h
          
          
            j
          
        
      
    
    {\displaystyle n_{i}p_{jh}^{i}=n_{j}p_{ih}^{j}}
  

A PBIBD(1) is a BIBD and a PBIBD(2) in which λ1 = λ2 is a BIBD.

=== Two associate class PBIBDs ===
PBIBD(2)s have been studied the most since they are the simplest and most useful of the PBIBDs. They fall into six types based on a classification of the then known PBIBD(2)s by Bose & Shimamoto (1952):

group divisible;
triangular;
Latin square type;
cyclic;
partial geometry type;
miscellaneous.

== Applications ==
The mathematical subject of block designs originated in the statistical framework of design of experiments. These designs were especially useful in applications of the technique of analysis of variance (ANOVA). This remains a significant area for the use of block designs.
While the origins of the subject are grounded in biological applications (as is some of the existing terminology), the designs are used in many applications where systematic comparisons are being made, such as in software testing.
The incidence matrix of block designs provide a natural source of interesting block codes that are used as error correcting codes. The rows of their incidence matrices are also used as the symbols in a form of pulse-position modulation.

=== Statistical application ===
Suppose that skin cancer researchers want to test three different sunscreens. They coat two different sunscreens on the upper sides of the hands of a test person. After a UV radiation they record the skin irritation in terms of sunburn. The number of treatments is 3 (sunscreens) and the block size is 2 (hands per person).
A corresponding BIBD can be generated by the R-function design.bib of the R-package agricolae and is specified in the following table:

The investigator chooses the parameters v = 3, k = 2 and λ = 1 for the block design which are then inserted into the R-function. Subsequently, the remaining parameters b and r are determined automatically.
Using the basic relations we calculate that we need b = 3 blocks, that is, 3 test people in order to obtain a balanced incomplete block design. Labeling the blocks A, B and C, to avoid confusion, we have the block design,

A = {2, 3},    B = {1, 3} and C = {1, 2}.
A corresponding incidence matrix is specified in the following table:

Each treatment occurs in 2 blocks, so r = 2.
Just one block (C) contains the treatments 1 and 2 simultaneously and the same applies to the pairs of treatments (1,3) and (2,3). Therefore, λ = 1.
It is impossible to use a complete design (all treatments in each block) in this example because there are 3 sunscreens to test, but only 2 hands on each person.

== See also ==
Incidence geometry
Steiner system
Fractional factorial design

== Notes ==