14.E: Developing a Standard Method (Exercises)
14.1
For each of the following equations determine the optimum response using the onefactoratatime searching algorithm. Begin the search at (0,0) by first changing factor A, using a stepsize of 1 for both factors. The boundary conditions for each response surface are 0 ≤ A ≤ 10 and 0 ≤ B ≤ 10. Continue the search through as many cycles as necessary until you find the optimum response. Compare your optimum response for each equation to the true optimum.
 \(R = 1.68 + 0.24A + 0.56B  0.04A^2  0.04B^2 \hspace{30px} \mu_\ce{opt} = (3, 7)\)
 \(R = 4.0  0.4A + 0.08B \hspace{30px} \mu_\ce{opt} = (10, 10)\)
 \(R = 3.264 + 1.537A + 0.5664B  0.1505A^2  0.02734B^2  0.05785AB \hspace{30px} \mu_\ce{opt} = (391, 6.22)\)
Note: These equations are from Deming, S. N.; Morgan, S. L. Experimental Design: A Chemometric Approach, Elsevier: Amsterdam, 1987, and pseudothree dimensional plots of the response surfaces can be found in their Figures 11.4, 11.5 and 11.14.
14.2.
Determine the optimum response for the equation in Problem 14.1c, using the fixedsized simplex searching algorithm. Compare your optimum response to the true optimum.
14.3
Show that equation 14.3 and equation 14.4 are correct.
14.4
A 2^{k} factorial design was used to determine the equation for the response surface in Problem 1b. The uncoded levels, coded levels, and the responses are shown in the following table.
A  B  A*  B*  response 

8  8  +1  +1  5.92 
8  2  +1  –1  2.08 
2  8  –1  +1  4.48 
2  2  –1  –1  3.52 
Determine the uncoded equation for the response surface.
14.5
Koscielniak and Parczewski investigated the influence of Al on the determination of Ca by atomic absorption spectrophotometry using the 2^{k} factorial design shown in the following table.^{11}
Ca^{2+} (ppm)  Al^{3+} (ppm)  Ca*  Al*  response 

10  160  +1  +1  54.92 
10  0  +1  –1  98.44 
4  160  –1  +1  19.18 
4  0  –1  –1  38.52 
 Determine the uncoded equation for the response surface.
 If you wish to analyze a sample that is 6.0 ppm Ca^{2}^{+}, what is the maximum concentration of Al^{3}^{+} that can be present if the error in the response must be less than 5.0%?
14.6
Strange reports the following information for a 2^{3} factorial design used to investigate the yield of a chemical process.^{12}
factor  high (+1) level  low (–1) level 

X: temperature  140^{o}C  120^{o}C 
Y: catalyst  type B  type A 
Z: [reactant]  0.50 M  0.25 M 
run  X*  Y*  Z*  % yield 

1  –1  –1  –1  28 
2  +1  –1  –1  17 
3  –1  +1  –1  41 
4  +1  +1  –1  34 
5  –1  –1  +1  56 
6  +1  –1  +1  51 
7  –1  +1  +1  42 
8  +1  +1  +1  36 
 Determine the coded equation for this data.
 If β terms of less than ±1 are insignificant, what main effects and interaction terms in the coded equation are important? Write down this simpler form for the coded equation.
 Explain why the coded equation for this data can not be transformed into an uncoded form.
 Which is the better catalyst, A or B?
 What is the yield using this catalyst if the temperature is set to 125^{o}C and the concentration of the reactant is 0.45 M?
14.7
Pharmaceutical tablets coated with lactose often develop a brown discoloration. The primary factors affecting the discoloration are temperature, relative humidity, and the presence of a base acting as a catalyst. The following data have been reported for a 2^{3} factorial design.^{13}
factor  high (+1) level  low (–1) level 

X: benzocaine 
present  absent 
Y: temperature 
40^{o}C  25^{o}C 
Z: relative humidity 
75%  50% 
run 
X* 
Y* 
Z* 
color (arb. units) 

1  –1  –1  –1  1.55 
2  +1  –1  –1  5.40 
3  –1  +1  –1  3.50 
4  +1  +1  –1  6.75 
5  –1  –1  +1  2.45 
6  +1  –1  +1  3.60 
7  –1  +1  +1  3.05 
8  +1  +1  +1  7.10 
 Determine the coded equation for this data.
 If β terms of less than 0.5 are insignificant, what main effects and interaction terms in the coded equation are important? Write down this simpler form for the coded equation.
14.8
The following data for a 2^{3} factorial design were collected during a study of the effect of temperature, pressure, and residence time on the % yield of a reaction.^{14}
factor 
high (+1) level 
low (–1) level 

X: temperature 
200^{o}C 
100^{o}C 
Y: pressure 
0.6 MPa 
0.2 MPa 
Z: residence time 
20 min 
10 min 




percent 

1 
–1 
–1 
–1 
2 
2 
+1 
–1 
–1 
6 
3 
–1 
+1 
–1 
4 
4 
+1 
+1 
–1 
8 
5 
–1 
–1 
+1 
10 
6 
+1 
–1 
+1 
18 
7 
–1 
+1 
+1 
8 
8 
+1 
+1 
+1 
12 
 Determine the coded equation for this data.
 If β terms of less than 0.5 are insignificant, what main effects and interaction terms in the coded equation are important? Write down this simpler form for the coded equation.
 Three runs at the center of the factorial design—a temperature of 150^{o}C, a pressure of 0.4 MPa, and a residence time of 15 min—give percent yields of 12%, 8%, 9%, and 8.8%. Determine if a firstorder empirical model is appropriate for this system at α = 0.05.
14.9
Duarte and colleagues used a factorial design to optimize a flowinjection analysis method for determining penicillin.^{15} Three factors were studied: reactor length, carrier flow rate, and sample volume, with the high and low values summarized in the following table.
factor  high (+1) level  low (–1) level 

X: reactor length 
1.5 cm  2.0 cm 
Y: carrier flow rate 
1.6 mL/min  2.2 mL/min 
Z: sample volume 
100 μL  150 μL 
The authors determined the optimum response using two criteria: the greatest sensitivity, as determined by the change in potential for the potentiometric detector, and the largest sampling rate. The following table summarizes their optimization results.
run  X*  Y*  Z*  ∆E(mV)  samples/h 

1  –1  –1  –1  37.45  21.5 
2  +1  –1  –1  31.70  26.0 
3  –1  +1  –1  32.10  30.0 
4  +1  +1  –1  27.20  33.0 
5  –1  –1  +1  39.85  21.0 
6  +1  –1  +1  32.85  19.5 
7  –1  +1  +1  35.00  30.0 
8  +1  +1  +1  32.15  34.0 
 Determine the coded equation for the response surface where ∆E is the response.
 Determine the coded equation for the response surface where sample/h is the response.
 Based on the coded equations, do conditions favoring sensitivity also improve the sampling rate?
 What conditions would you choose if your goal is to optimize both sensitivity and sampling rate?
14.10
Here is a challenge! McMinn, Eatherton, and Hill investigated the effect of five factors for optimizing an H_{2}atmosphere flame ionization detector using a 2^{5} factorial design.^{16} The factors and their levels were
factor  high (+1) level  low (–1) level 

A: H_{2} flow rate  1460 mL/min  1382 mL/min 
B: SiH_{4}  20.0 ppm  12.2 ppm 
C: O_{2} + N_{2} flow rate  255 mL/min  210 mL/min 
D: O_{2}/N_{2}  1.36  1.19 
E: electrode height  75 (arb. unit)  55 (arb. unit) 
The coded (“+” = +1, “–” = –1) factor levels and responses, R, for the 32 experiments are shown in the following table
run  A*  B*  C*  D*  E*  run  A*  B*  C*  D*  E* 

1  –  –  –  –  –  17  –  –  –  –  + 
2  +  –  –  –  –  18  +  –  –  –  + 
3  –  +  –  –  –  19  –  +  –  –  + 
4  +  +  –  –  –  20  +  +  –  –  + 
5  –  –  +  –  –  21  –  –  +  –  + 
6  +  –  +  –  –  22  +  –  +  –  + 
7  –  +  +  –  –  23  –  +  +  –  + 
8  +  +  +  –  –  24  +  +  +  –  + 
9  –  –  –  +  –  25  –  –  –  +  + 
10  +  –  –  +  –  26  +  –  –  +  + 
11  –  +  –  +  –  27  –  +  –  +  + 
12  +  +  –  +  –  28  +  +  –  +  + 
13  –  +  +  +  –  29  –  –  +  +  + 
14  +  –  +  +  –  30  +  –  +  +  + 
15  –  +  +  +  –  31  –  +  +  +  + 
16  +  +  +  +  –  32  +  +  +  +  + 
 Determine the coded equation for this response surface, ignoring β terms less than ±0.03.
 A simplex optimization of this system finds optimal values for the factors of A = 2278 mL/min, B = 9.90 ppm, C = 260.6 mL/min, and D = 1.71. The value of E was maintained at its high level. Are these values consistent with your analysis of the factorial design?
14.11
A good empirical model provides an accurate picture of the response surface over the range of factor levels within the experimental design. The same model, however, may yield an inaccurate prediction for the response at other factor levels. For this reason, an empirical model, is tested before extrapolating to conditions other than those used in determining the model. For example, Palasota and Deming studied the effect of the relative amounts of H_{2}SO_{4} and H_{2}O_{2} on the absorbance of solutions of vanadium using the following central composite design.^{17}
run  drops 1% H_{2}SO_{4}  drops 20% H_{2}O_{2} 

1  15  22 
2  10  20 
3  20  20 
4  8  15 
5  15  15 
6  15  15 
7  15  15 
8  15  15 
9  22  15 
10  10  10 
11  20  10 
12  15  8 
The reaction of H_{2}SO_{4} and H_{2}O_{2} generates a redbrown solution whose absorbance is measured at a wavelength of 450 nm. A regression analysis on their data yielded the following uncoded equation for the response (absorbance × 1000).
\[R = 835.90  36.82X_1  21.34X_2 + 0.52(X_1)^2 + 0.15(X_2)^2 + 0.98X_1X_2\]
where X_{1} is the drops of H_{2}O_{2}, and X_{2} is the drops of H_{2}SO_{4}. Calculate the predicted absorbances for 10 drops of H_{2}O_{2} and 0 drops of H_{2}SO_{4}, 0 drops of H_{2}O_{2} and 10 drops of H_{2}SO_{4}, and for 0 drops of each reagent. Are these results reasonable? Explain. What does your answer tell you about this empirical model?
14.12
A newly proposed method is to be tested for its singleoperator characteristics. To be competitive with the standard method, the new method must have a relative standard deviation of less than 10%, with a bias of less than 10%. To test the method, an analyst performs 10 replicate analyses on a standard sample known to contain 1.30 ppm of analyte. The results for the 10 trials are
1.25 1.26 1.29 1.56 1.46 1.23 1.49 1.27 1.31 1.43
Are the singleoperator characteristics for this method acceptable?
14.13
A proposed gravimetric method was evaluated for its ruggedness by varying the following factors.
Factor A: sample size 
A = 1 g 
a = 1.1 g 

Factr B: pH 
B = 6.5 
b = 6.0 
Factor C: digestion time 
C = 3 h 
c = 1 h 
Factor D: number rinses 
D = 3 
d = 5 
Factor E: precipitant 
E = reagent 1 
e = reagent 2 
Factor F: digestion temperature 
F = 50^{o}C 
f = 60^{o}C 
Factor G: drying temperature 
G = 110^{o}C 
g = 140^{o}C 
A standard sample containing a known amount of analyte was carried through the procedure using the experimental design in Table 14.5. The percentage of analyte actually found in the eight trials were found to be
R_{1} = 98.9 R_{5} = 98.8 
R_{2} = 98.5 R_{6} = 98.5 
R_{3} = 97.7 R_{7} = 97.7 
R_{4} = 97.0 R_{8} = 97.3 
Determine which factors, if any, appear to have a significant affect on the response, and estimate the expected standard deviation for the method.
14.14
The twosample plot for the data in Example 14.6 is shown in Figure 14.21. Identify the analyst whose work is (a) the most accurate, (b) the most precise, (c) the least accurate, and (d) the least precise.
14.15
Chichilo reports the following data for the determination of the %w/w Al in two samples of limestone.^{18}
analyst  sample 1  sample 2 

1  1.35  1.57 
2  1.35  1.33 
3  1.34  1.47 
4  1.50  1.60 
5  1.52  1.62 
6  1.39  1.52 
7  1.30  1.36 
8  1.32  1.53 
Construct a twosample plot for this data and estimate values for σ_{rand} and σ_{syst}.
14.16
The importance of betweenlaboratory variability on the results of an analytical method can be determined by having several laboratories analyze the same sample. In one such study, seven laboratories analyzed a sample of homogenized milk for a selected alfatoxin.^{19} The results, in ppb, are summarized below.
lab A  lab B  lab C  lab D  lab E  lab F  lab G 

1.6  4.6  1.2  1/5  6.0  6.2  3.3 
2.9  2.8  1.9  2.17  3.9  3.8  3.8 
3.5  3.0  2.9  3.4  4.3  5.5  5.5 
1.8  4.5  1.1  2.0  5.8  4.2  4.9 
2.2  3.1  2.9  3.4  4.0  5.3  4.5 
 Determine if the betweenlaboratory variability is significantly greater than the withinlaboratory variability at α = 0.05. If the betweenlaboratory variability is significant, then determine the source(s) of that variability.
 Estimate values for σ_{rand}^{2} and σ_{syst}^{2}.
14.17
Show that the total sumofsquares (SS_{t}) is the sum of the withinsample sumofsquares (SS_{w}) and the betweensample sumofsquares (SS_{b}). See Table 14.7 for the relevant equations.
14.18
Eighteen analytical students are asked to determine the %w/w Mn in a sample of steel, with the results shown here.
0.26% 
0.28% 
0.27% 
0.24% 
0.26% 
0.25% 

0.26% 
0.28% 
0.25% 
0.24% 
0.26% 
0.25% 
0.29% 
0.24% 
0.27% 
0.23% 
0.26% 
0.24% 
(a) Given that the steel sample is 0.26% w/w Mn, estimate the expected relative standard deviation for the class’ results.
(b) The actual results obtained by the students are shown here. Are these results consistent with the estimated relative standard deviation?
14.5.3 Solutions to Practice Exercises
Practice Exercise 14.1
If we hold factor A at level A_{1}, changing factor B from level B_{1} to level B_{2} increases the response from 40 to 60, or a change ∆R, of
\[R = 60  40 = 20\]
If we hold factor A at level A_{2}, we find that we have the same change in response when the level of factor B changes from B_{1} to B_{2}.
\[R = 100  80 = 20\]
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Practice Exercise 14.2
If we hold factor B at level B_{1}, changing factor A from level A_{1} to level A_{2} increases the response from 20 to 80, or a change ∆R, of
\[R = 80  20 = 60\]
If we hold factor B at level B_{2}, we find that the change in response when the level of factor A changes from A_{1} to A_{2} is now 20.
\[R = 80  20 = 60\]
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Practice Exercise 14.3
Answers will vary here depending on the options you decided to explore. The last response surface is an interesting one to explore. Figure 14.26 shows the response surface as a level plot and a contour plot. The interesting feature of this surface is the saddle point on a ridge connecting a local optimum (maximum response of 4.45) and the global optimum (maximum response of 10.0). All three optimization strategies are very sensitive to the initial position and the stepsize.
Figure 14.26 Level plot and contour plot for the fifth response surface in Practice Exercise 14.3.
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