Solution 2.6 - Chemometrics: Data Analysis for the Laboratory and Chemical Plant
Education Article
- Published: Jan 1, 2000
- Channels: Chemometrics & Informatics
1. The design matrix is given by:
x_{0} |
x_{1} |
x_{2} |
x_{3} |
x_{1}x_{2} |
x_{1}x_{3} |
x_{2}x_{3} |
x_{1}x_{2}x_{3} |
1 |
-1 |
-1 |
-1 |
1 |
1 |
1 |
-1 |
1 |
1 |
-1 |
-1 |
-1 |
-1 |
1 |
1 |
1 |
-1 |
1 |
-1 |
-1 |
1 |
-1 |
1 |
1 |
1 |
1 |
-1 |
1 |
-1 |
-1 |
-1 |
1 |
-1 |
-1 |
1 |
1 |
-1 |
-1 |
1 |
1 |
1 |
-1 |
1 |
-1 |
1 |
-1 |
-1 |
1 |
-1 |
1 |
1 |
-1 |
-1 |
1 |
-1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
2. The coefficients are as follows:
b_{0} |
b_{1} |
b_{2} |
b_{3} |
b_{12} |
b_{13} |
b_{23} |
b_{123} |
84.44 |
-5.74 |
-1.86 |
-3.24 |
-1.14 |
-4.26 |
0.31 |
-0.61 |
3. The most significant coefficients appear to be b_{1}, b_{3} and b_{13}. Certainly b_{123} and b_{23} can be ruled out. The interaction term between moisture and clorazepate is quite significant.
4. The fifth and seventh experiments result in slightly higher response to the first experiment. In these experiments, the factor 1 is high and factor 3 is low. Because the term for the interaction between factors 1 and 3 is larger in magnitude than the term for factor 3, a slightly more favourable response is obtained by setting these two factors at opposite values.
5. A fractional factorial design to study the three main factors would not obtain any information on interaction terms. Such a design would have missed the significant interaction between factors 1 and 3.
6. The inverse in the design matrix can be used because the number of experiments and number of terms in the model are equal. If the number of terms is less than the number of experiments, the pseudo-inverse must be employed instead.
7. This is because D^{-1} = (1/N) D'
which can be shown as follows.
The inverse of the design matrix is given by
0.125 |
0.125 |
0.125 |
0.125 |
0.125 |
0.125 |
0.125 |
0.125 |
-0.125 |
0.125 |
-0.125 |
0.125 |
-0.125 |
0.125 |
-0.125 |
0.125 |
-0.125 |
-0.125 |
0.125 |
0.125 |
-0.125 |
-0.125 |
0.125 |
0.125 |
-0.125 |
-0.125 |
-0.125 |
-0.125 |
0.125 |
0.125 |
0.125 |
0.125 |
0.125 |
-0.125 |
-0.125 |
0.125 |
0.125 |
-0.125 |
-0.125 |
0.125 |
0.125 |
-0.125 |
0.125 |
-0.125 |
-0.125 |
0.125 |
-0.125 |
0.125 |
0.125 |
0.125 |
-0.125 |
-0.125 |
-0.125 |
-0.125 |
0.125 |
0.125 |
-0.125 |
0.125 |
0.125 |
-0.125 |
0.125 |
-0.125 |
-0.125 |
0.125 |
which when multiplied by 8 gives
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
-1 |
1 |
-1 |
1 |
-1 |
1 |
-1 |
1 |
-1 |
-1 |
1 |
1 |
-1 |
-1 |
1 |
1 |
-1 |
-1 |
-1 |
-1 |
1 |
1 |
1 |
1 |
1 |
-1 |
-1 |
1 |
1 |
-1 |
-1 |
1 |
1 |
-1 |
1 |
-1 |
-1 |
1 |
-1 |
1 |
1 |
1 |
-1 |
-1 |
-1 |
-1 |
1 |
1 |
-1 |
1 |
1 |
-1 |
1 |
-1 |
-1 |
1 |
being the transpose of the design matrix. Hence, each value of the resultant 8´ 1 column vector b is the result of multiplying a row of D^{-1} by y. However, each row of D^{-1} corresponds to a row of D' divided by the total number of experiments (8), which in turn equals of a column of D.
For the parameter b_{1} an alternative method of calculation becomes
b_{1} = (-90.8+88.9-87.5+83.5-91.0+74.5-91.4+67.9)/8 = -5.74.
8. A rotatable central composite design is given below.
1 |
1 |
1 |
-1 |
-1 |
1 |
-1 |
-1 |
-1.41 |
0 |
1.41 |
0 |
0 |
-1.41 |
0 |
1.41 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
Note that the number of replicates in the centre could be varied. It is important to decide on the coding, clearly it is not possible to have less than 0% clorazepate, so the correspondence between coded an true levels of the two factors will have to differ from the factorial design, it is probably most sensible to keep to the true range, setting the extremes up as ± 1.41.
It is slightly preferable that factor 2 is at the low level. Since there are only two possible states for this factor, in the remaining experiments the drug is always taken as a powder.