# 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

 b0 b1 b2 b11 b22 b12 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.

## Microsites

Suppliers Selection
Societies Selection