Solution 2.3  Chemometrics: Data Analysis for the Laboratory and Chemical Plant
Education Article
 Published: Jan 1, 2000
 Channels: Chemometrics & Informatics
1. The design matrix is as follows.

x_{1}
x_{2}
x_{3}
x_{1}x_{2}
x_{1}x_{3}
x_{2}x_{3}
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0.5
0.5
0
0.25
0
0
0.5
0
0.5
0
0.25
0
0
0.5
0.5
0
0
0.25
The six coefficients are as follows. This model is commonly called the Sheffé model.
b_{1}
41
b_{2}
12
b_{3}
18
b_{12}
10
b_{13}
22
b_{23}
8
2. The design matrix using the alternative (Cox) model is, together with the coefficients are as follows.

x_{0}
x_{1}
x_{2}
x_{1}x_{2}
1
1
0
1
0
0
1
0
1
0
1
0
1
0
0
0
0
0
1
0.5
0.5
0.25
0.25
0.25
1
0.5
0
0.25
0
0
1
0
0.5
0
0.25
0
Coefficients
a_{0}
18
a_{1}
1
a_{2}
2
a_{11}
22
a_{22}
8
a_{12}
24
3. The algebra is as follows.
The equivalence of the coefficients is, therefore,
a_{0}  b_{3}  
a_{1}  b_{1}  b_{3} + b_{13}  
a_{2}  b_{2}  b_{3} + b_{23}  
a_{11}  b_{13}  
a_{22}  b_{23}  
a_{12}  b_{12}  b_{13}  b_{23} 
This is easy to show numerically, using the models in questions 1 and 2. For example, a_{0} for the second model equals 18, which is the same as b_{3} for the first model, and so on.
The two models are equivalent. The model of question 1 is more common and more symmetrical when there are several factors. Note that it is impossible to include both intercept and all three single factor mixture terms in one model, because the proportion of the third component is dependent on the proportions of the first two.