Solution 2.17 - Chemometrics: Data Analysis for the Laboratory and Chemical Plant
Education Article
- Published: Jan 1, 2000
- Channels: Chemometrics & Informatics
1. The equation is given by
y = b_{1}x_{1} + b_{2}x_{2} + b_{3}x_{3} + b_{12}x_{1}x_{2} + b_{23}x_{2}x_{3} + b_{123}x_{1}x_{2}x_{3}
The design matrix and coefficients are as follows
2. The coefficients are as follows.
b_{1} |
b_{2} |
b_{3} |
8.133 |
8.933 |
18.933 |
Note interestingly, that these are not equal, or even approximately equal to the first three coefficients of the model in question 1, because the factors are dependent on each other, so the model is somewhat unstable.
The sum of square residual error is 65.467, and the root mean square residual error is 4.045 (if dividing by 4 which equals the number of degrees of freedom rather than 7). This is quite large in percentage terms (33.7%), so the other terms are likely to be quite significant.
3. This can be obtained by substituting x_{3} = (1-x_{1}-x_{2}) into the equation of question 1, to give
The two sets of coefficients, from questions 1 and 3 can then be compared. For example the coefficient of x_{1} in question 3 equals (b_{1} -b_{3} + b_{13}) in question 1.
4. The design matrix and coefficients are as follows.
The relationships in question 4 can easily be verified. For example, the coefficient of x_{1} equals (b_{1} -b_{3} + b_{13}) = 9 – 17 + 20 = 12.
5. The reason for this is that knowing any two x's and the sum (=1) provides full information. The information can be represented either by the three proportions or by two proportions and the sum. Having a matrix of four columns that consists of one column that is simply the sum of the other three columns, results in a matrix that has no inverse, and it is impossible to obtain a unique answer.