Solution 2.9 - Chemometrics: Data Analysis for the Laboratory and Chemical Plant

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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.

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