# 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.

1.  x1 x2 x3 x1x2 x1x3 x2x3 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.

 b1 41 b2 12 b3 18 b12 10 b13 -22 b23 8

2. The design matrix using the alternative (Cox) model is, together with the coefficients are as follows.

1.  x0 x1 x2 x1x2 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

 a0 18 a1 1 a2 2 a11 22 a22 -8 a12 24

3. The algebra is as follows.

The equivalence of the coefficients is, therefore,

 a0 b3 a1 b1 - b3 + b13 a2 b2 - b3 + b23 a11 -b13 a22 -b23 a12 b12 - b13 - b23

This is easy to show numerically, using the models in questions 1 and 2. For example, a0 for the second model equals 18, which is the same as b3 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.

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