kb/data/en.wikipedia.org/wiki/Design_matrix-1.md

---
title: "Design matrix"
chunk: 2/2
source: "https://en.wikipedia.org/wiki/Design_matrix"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T09:49:58.034133+00:00"
instance: "kb-cron"
---

  
    
      
        
          
            [
            
              
                
                  
                    y
                    
                      1
                    
                  
                
              
              
                
                  
                    y
                    
                      2
                    
                  
                
              
              
                
                  
                    y
                    
                      3
                    
                  
                
              
              
                
                  
                    y
                    
                      4
                    
                  
                
              
              
                
                  
                    y
                    
                      5
                    
                  
                
              
              
                
                  
                    y
                    
                      6
                    
                  
                
              
              
                
                  
                    y
                    
                      7
                    
                  
                
              
            
            ]
          
        
        =
        
          
            [
            
              
                
                  1
                
                
                  0
                
                
                  0
                
              
              
                
                  1
                
                
                  0
                
                
                  0
                
              
              
                
                  1
                
                
                  0
                
                
                  0
                
              
              
                
                  0
                
                
                  1
                
                
                  0
                
              
              
                
                  0
                
                
                  1
                
                
                  0
                
              
              
                
                  0
                
                
                  0
                
                
                  1
                
              
              
                
                  0
                
                
                  0
                
                
                  1
                
              
            
            ]
          
        
        
          
            [
            
              
                
                  
                    μ
                    
                      1
                    
                  
                
              
              
                
                  
                    μ
                    
                      2
                    
                  
                
              
              
                
                  
                    μ
                    
                      3
                    
                  
                
              
            
            ]
          
        
        +
        
          
            [
            
              
                
                  
                    ε
                    
                      1
                    
                  
                
              
              
                
                  
                    ε
                    
                      2
                    
                  
                
              
              
                
                  
                    ε
                    
                      3
                    
                  
                
              
              
                
                  
                    ε
                    
                      4
                    
                  
                
              
              
                
                  
                    ε
                    
                      5
                    
                  
                
              
              
                
                  
                    ε
                    
                      6
                    
                  
                
              
              
                
                  
                    ε
                    
                      7
                    
                  
                
              
            
            ]
          
        
      
    
    {\displaystyle {\begin{bmatrix}y_{1}\\y_{2}\\y_{3}\\y_{4}\\y_{5}\\y_{6}\\y_{7}\end{bmatrix}}={\begin{bmatrix}1&0&0\\1&0&0\\1&0&0\\0&1&0\\0&1&0\\0&0&1\\0&0&1\end{bmatrix}}{\begin{bmatrix}\mu _{1}\\\mu _{2}\\\mu _{3}\end{bmatrix}}+{\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{3}\\\varepsilon _{4}\\\varepsilon _{5}\\\varepsilon _{6}\\\varepsilon _{7}\end{bmatrix}}}
  

In this model 
  
    
      
        
          μ
          
            i
          
        
      
    
    {\displaystyle \mu _{i}}
  
 represents the mean of the 
  
    
      
        i
      
    
    {\displaystyle i}
  
th group.

=== One-way ANOVA (offset from reference group) ===
The ANOVA model could be equivalently written as each group parameter 
  
    
      
        
          τ
          
            i
          
        
      
    
    {\displaystyle \tau _{i}}
  
 being an offset from some overall reference.  Typically this reference point is taken to be one of the groups under consideration. This makes sense in the context of comparing multiple treatment groups to a control group and the control group is considered the "reference". In this example, group 1 was chosen to be the reference group. As such the model to be fit is

  
    
      
        
          y
          
            i
            j
          
        
        =
        μ
        +
        
          τ
          
            i
          
        
        +
        
          ε
          
            i
            j
          
        
      
    
    {\displaystyle y_{ij}=\mu +\tau _{i}+\varepsilon _{ij}}
  

with the constraint that 
  
    
      
        
          τ
          
            1
          
        
      
    
    {\displaystyle \tau _{1}}
  
 is zero.

  
    
      
        
          
            [
            
              
                
                  
                    y
                    
                      1
                    
                  
                
              
              
                
                  
                    y
                    
                      2
                    
                  
                
              
              
                
                  
                    y
                    
                      3
                    
                  
                
              
              
                
                  
                    y
                    
                      4
                    
                  
                
              
              
                
                  
                    y
                    
                      5
                    
                  
                
              
              
                
                  
                    y
                    
                      6
                    
                  
                
              
              
                
                  
                    y
                    
                      7
                    
                  
                
              
            
            ]
          
        
        =
        
          
            [
            
              
                
                  1
                
                
                  0
                
                
                  0
                
              
              
                
                  1
                
                
                  0
                
                
                  0
                
              
              
                
                  1
                
                
                  0
                
                
                  0
                
              
              
                
                  1
                
                
                  1
                
                
                  0
                
              
              
                
                  1
                
                
                  1
                
                
                  0
                
              
              
                
                  1
                
                
                  0
                
                
                  1
                
              
              
                
                  1
                
                
                  0
                
                
                  1
                
              
            
            ]
          
        
        
          
            [
            
              
                
                  μ
                
              
              
                
                  
                    τ
                    
                      2
                    
                  
                
              
              
                
                  
                    τ
                    
                      3
                    
                  
                
              
            
            ]
          
        
        +
        
          
            [
            
              
                
                  
                    ε
                    
                      1
                    
                  
                
              
              
                
                  
                    ε
                    
                      2
                    
                  
                
              
              
                
                  
                    ε
                    
                      3
                    
                  
                
              
              
                
                  
                    ε
                    
                      4
                    
                  
                
              
              
                
                  
                    ε
                    
                      5
                    
                  
                
              
              
                
                  
                    ε
                    
                      6
                    
                  
                
              
              
                
                  
                    ε
                    
                      7
                    
                  
                
              
            
            ]
          
        
      
    
    {\displaystyle {\begin{bmatrix}y_{1}\\y_{2}\\y_{3}\\y_{4}\\y_{5}\\y_{6}\\y_{7}\end{bmatrix}}={\begin{bmatrix}1&0&0\\1&0&0\\1&0&0\\1&1&0\\1&1&0\\1&0&1\\1&0&1\end{bmatrix}}{\begin{bmatrix}\mu \\\tau _{2}\\\tau _{3}\end{bmatrix}}+{\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{3}\\\varepsilon _{4}\\\varepsilon _{5}\\\varepsilon _{6}\\\varepsilon _{7}\end{bmatrix}}}
  

In this model 
  
    
      
        μ
      
    
    {\displaystyle \mu }
  
 is the mean of the reference group and 
  
    
      
        
          τ
          
            i
          
        
      
    
    {\displaystyle \tau _{i}}
  
 is the difference from group 
  
    
      
        i
      
    
    {\displaystyle i}
  
 to the reference group. 
  
    
      
        
          τ
          
            1
          
        
      
    
    {\displaystyle \tau _{1}}
  
 is not included in the matrix because its difference from the reference group (itself) is necessarily zero.

== See also ==
Moment matrix
Projection matrix
Jacobian matrix and determinant
Scatter matrix
Gram matrix
Vandermonde matrix

== References ==

== Further reading ==
Verbeek, Albert (1984). "The Geometry of Model Selection in Regression". In Dijkstra, Theo K. (ed.). Misspecification Analysis. New York: Springer. pp. 20–36. ISBN 0-387-13893-5.