Solution 2.9 - Chemometrics: Data Analysis for the Laboratory and Chemical Plant
Education Article
- Published: Jan 1, 2000
- Channels: Chemometrics & Informatics
1. The coefficients are calculated in the usual way using the pseudo-inverse, namely
b = (D'.D)^{-1} .D'.y and are as follows
b_{0} |
b_{1} |
b_{2} |
b_{11} |
b_{22} |
b_{12} |
5.11 |
3.10 |
-1.94 |
4.08 |
-1.19 |
2.05 |
2. The predicted values and the error calculation are given below.
Observed response |
Estimated response |
Residual |
5.4384 |
5.1123 |
-0.3262 |
4.9845 |
5.1123 |
0.1278 |
4.3228 |
5.1123 |
0.7894 |
5.2538 |
5.1123 |
-0.1415 |
8.7288 |
8.8903 |
0.1615 |
0.7971 |
0.9160 |
0.1189 |
10.8833 |
10.9892 |
0.1059 |
11.1540 |
11.2173 |
0.0633 |
12.4607 |
12.2915 |
-0.1692 |
6.3716 |
6.0913 |
-0.2803 |
6.1280 |
5.8606 |
-0.2673 |
2.1698 |
1.9876 |
-0.1822 |
Sum of squares | ||
658.5233 |
657.4900 |
1.0333 |
Hence the residual error sum of squares is 1.0333. Note that this number could be obtained in one of two ways, either by take the sum of squares of the residuals, or subtracting the sum of squares for the estimated from the observed response.
3. The mean of the four replicates is 4.9999, and the sum of square replicate error is 0.7154. An example of this calculation is presented in Table 2.3(a) in the printed text.
4. The lack-of-fit error is 0.3178.
5. The number of degrees of freedom can be calculated as follows:
- N=12 (total number of experiments)
- P=6 (parameters in the model)
- R=3 (replicates)
- D=N-P-R=3 (lack-of-fit)
Hence, the mean replicate error is 0.2385 and the mean lack-of-fit error is 0.1060. Therefore, the lack-of-fit is substantially less than the replication error, and so it is very likely that there is a good fit to the model.