Solution 2.5 - Chemometrics: Data Analysis for the Laboratory and Chemical Plant
Education Article
- Published: Jan 1, 2000
- Channels: Chemometrics & Informatics
1. Squared coefficients cannot be used in the model because all experiments are at only two levels. Central composite designs are usually employed when squared terms are to be included.
2. The design matrix and coefficients are as follows.
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 |
coefficients | |||||||
10.3625 |
-0.1875 |
0.9625 |
0.0625 |
0.0625 |
0.0125 |
0.4125 |
0.0125 |
The coefficients are calculated by b = D^{-1}.y as the design matrix is a square matrix.
Rank (p) |
Coefficient |
Size |
(p-0.5)/7 |
Normal probabilities | |
1 |
b_{1} |
-0.1875 |
0.071429 |
-1.46523 | |
2 |
b_{13} |
0.0125 |
0.214286 |
-0.79164 | |
3 |
b_{123} |
0.0125 |
0.357143 |
-0.36611 | |
4 |
b_{3} |
0.0625 |
0.5 |
0 | |
5 |
b_{12} |
0.0625 |
0.642857 |
0.366106 | |
6 |
b_{23} |
0.4125 |
0.785714 |
0.791638 | |
7 |
b_{2} |
0.9625 |
0.928571 |
1.465232 |
3. The various steps are indicated below.
- The first column indicates the rank, from 1 (lowest) to 7 (highest)
- The second column is the actual coefficient.
- The third column is the size of the coefficient.
- The fourth column represents the expected area in standard deviations from the mean for a random factor of a given rank.
- The final column is the normal probability.
A graph is plotted of the fifth against third column. Note that in the past, normal probability paper could be purchased (often a great expense), but it is now easy to perform such calculations and graphs computationally.
4. The two most positive coefficients (b_{23} and b_{2}) and the most negative coefficient (b_{1}) appear significant. The other four coefficient fall roughly on a straight-line in the centre of the graph.
5. An explanation is given on page 43 of the printed text. Note such plots are only really useful if there a several factors, and it is assumed that a good proportion of the factors are not significant.