Modern Analytical Cheymistry - Chapter 14 potx

666 CChhaapptteerr 1 4 Developing a Standard Method In Chapter 1 we made a distinction between analytical chemistry and chemical analysis. The goals of analytical chemistry are to improve established methods of analysis, to extend existing methods of analysis to new types of samples, and to develop new analytical methods. Once a method has been developed and tested, its application is best described as chemical analysis. We recognize the status of such methods by calling them standard methods. A standard method may be unique to a particular laboratory, which developed the method for their specific purpose, or it may be a widely accepted method used by many laboratories. Numerous examples of standard methods have been presented and discussed in the preceding six chapters. What we have yet to consider, however, is what constitutes a standard method. In this chapter we consider how a standard method is developed, including optimizing the experimental procedure, verifying that the method produces acceptable precision and accuracy in the hands of a single analyst, and validating the method for general use. 1400-CH14 9/8/99 4:35 PM Page 666 Figure 14.1 Example of a one-factor response surface. Chapter 14 Developing a Standard Method 667 Absorbance 01234 0.000 0.500 0.450 0.400 0.350 0.300 0.250 0.200 0.150 0.100 0.050 Analyte (ppm) 5 response The property of a system that is measured (R). factor A property of a system that is experimentally varied and that may affect the response. factor level A factor’s value. response surface A graph showing how a system’s response changes as a function of its factors. 1 4 A Optimizing the Experimental Procedure In the presence of H 2 O 2 and H 2 SO 4 , solutions of vanadium form a reddish brown color that is believed to be a compound with the general formula (VO) 2 (SO 4 ) 3 . Since the intensity of the color depends on the concentration of vanadium, the ab- sorbance of the solution at a wavelength of 450 nm can be used for the quantitative analysis of vanadium. The intensity of the color, however, also depends on the amount of H 2 O 2 and H 2 SO 4 present. In particular, a large excess of H 2 O 2 is known to decrease the solution’s absorbance as it changes from a reddish brown to a yel- lowish color. 1 Developing a standard method for vanadium based on its reaction with H 2 O 2 and H 2 SO 4 requires that their respective concentrations be optimized to give a max- imum absorbance. Using terminology adapted by statisticians, the absorbance of the solution is called the response. Hydrogen peroxide and sulfuric acid are factors whose concentrations, or factor levels, determine the system’s response. Optimiza- tion involves finding the best combination of factor levels. Usually we desire a max- imum response, such as maximum absorbance in the quantitative analysis for vana- dium as (VO) 2 (SO 4 ) 3 . In other situations, such as minimizing percent error, we seek a minimum response. 1 4 A.1 Response Surfaces One of the most effective ways to think about optimization is to visualize how a sys- tem’s response changes when we increase or decrease the levels of one or more of its factors. A plot of the system’s response as a function of the factor levels is called a response surface. The simplest response surface is for a system with only one fac- tor. In this case the response surface is a straight or curved line in two dimensions. A calibration curve, such as that shown in Figure 14.1, is an example of a one-factor response surface in which the response (absorbance) is plotted on the y-axis versus the factor level (concentration of analyte) on the x-axis. Response surfaces can also be expressed mathematically. The response surface in Figure 14.1, for example, is A = 0.008 + 0.0896C A where A is the absorbance, and C A is the analyte’s concentration in parts per million. For a two-factor system, such as the quantitative analysis for vanadium de- scribed earlier, the response surface is a flat or curved plane plotted in three dimen- sions. For example, Figure 14.2a shows the response surface for a system obeying the equation R = 3.0 – 0.30A + 0.020AB where R is the response, and A and B are the factor levels. Alternatively, we may represent a two-factor response surface as a contour plot, in which contour lines in- dicate the magnitude of the response (Figure 14.2b). The response surfaces in Figure 14.2 are plotted for a limited range of factor levels (0 ≤ A ≤ 10, 0 ≤ B ≤ 10), but can be extended toward more positive or more negative values. This is an example of an unconstrained response surface. Most re- sponse surfaces of interest to analytical chemists, however, are naturally constrained by the nature of the factors or the response or are constrained by practical limits set by the analyst. The response surface in Figure 14.1, for example, has a natural con- straint on its factor since the smallest possible concentration for the analyte is zero. Furthermore, an upper limit exists because it is usually undesirable to extrapolate a calibration curve beyond the highest concentration standard. standard method A method that has been identified as providing acceptable results. 1400-CH14 9/8/99 4:35 PM Page 667 Figure 14.3 Mountain-climbing analogy to using a searching algorithm to find the optimum response for a response surface. The path on the left leads to the global optimum, and the path on the right leads to a local optimum. If the equation for the response surface is known, then the optimum response is easy to locate. Unfortunately, the response surface is usually unknown; instead, its shape, and the location of the optimum response must be determined experimen- tally. The focus of this section is on useful experimental designs for optimizing ana- lytical methods. These experimental designs are divided into two broad categories: searching methods, in which an algorithm guides a systematic search for the opti- mum response; and modeling methods, in which we use a theoretical or empirical model of the response surface to predict the optimum response. 1 4 A.2 Searching Algorithms for Response Surfaces Imagine that you wish to climb to the top of a mountain. Because the mountain is covered with trees that obscure its shape, the shortest path to the summit is un- known. Nevertheless, you can reach the summit by always walking in a direction that moves you to a higher elevation. The route followed (Figure 14.3) is the result of a systematic search for the summit. Of course, many routes are possible leading from the initial starting point to the summit. The route taken, therefore, is deter- mined by the set of rules (the algorithm) used to determine the direction of each step. For example, one algorithm for climbing a mountain is to always move in the direction that has the steepest slope. A systematic searching algorithm can also be used to locate the optimum re- sponse for an analytical method. To find the optimum response, we select an initial set of factor levels and measure the response. We then apply the rules of the search- ing algorithm to determine the next set of factor levels. This process is repeated until the algorithm indicates that we have reached the optimum response. Two common searching algorithms are described in this section. First, however, we must consider how to evaluate a searching algorithm. Effectiveness and Efficiency A searching algorithm is characterized by its effec- tiveness and its efficiency. To be effective, the experimentally determined optimum should closely coincide with the system’s true optimum. A searching algorithm may fail to find the true optimum for several reasons, including a poorly designed algo- rithm, uncertainty in measuring the response, and the presence of local optima. A poorly designed algorithm may prematurely end the search. For example, an algo- rithm for climbing a mountain that allows movement to the north, south, east, or west will fail to find a summit that can only be reached by moving to the northwest. When measuring the response is subject to relatively large random errors, or noise, a spuriously high response may produce a false optimum from which the 668 Modern Analytical Chemistry Start 0 10 8 6 4 2 3 2.5 2 1.5 1 0.5 0 0 1.5 3 4.5 6 7.5 9 Factor A Factor B Response 0 0 1 10 9 8 7 6 5 4 3 2 Factor A Factor B 10 9 8 7 6 5 4 3 2 1 Figure 14.2 Example of a two-factor response surface displayed as (a) a pseudo-three-dimensional graph and (b) a contour plot. Contour lines are shown for intervals of 0.5 response units. (a) (b) 1400-CH14 9/8/99 4:35 PM Page 668 Figure 14.4 Factor effect plot for two independent factors. searching algorithm cannot move. When climbing a mountain, boulders encoun- tered along the way are examples of “noise” that must be avoided if the true opti- mum is to be found. The effect of noise can be minimized by increasing the size of the individual steps such that the change in response is larger than the noise. Finally, a response surface may contain several local optima, only one of which is the system’s true, or global, optimum. This is a problem because a set of initial conditions near a local optimum may be unable to move toward the global opti- mum. The mountain shown in Figure 14.3, for example, contains two peaks, with the peak on the left being the true summit. A search for the summit beginning at the position identified by the dot may find the local peak instead of the true sum- mit. Ideally, a searching algorithm should reach the global optimum regardless of the initial set of factor levels. One way to evaluate a searching algorithm’s effective- ness, therefore, is to use several sets of initial factor levels, finding the optimum re- sponse for each, and comparing the results. A second desirable characteristic for a searching algorithm is efficiency. An effi- cient algorithm moves from the initial set of factor levels to the optimum response in as few steps as possible. The rate at which the optimum is approached can be in- creased by taking larger steps. If the step size is too large, however, the difference between the experimental optimum and the true optimum may be unacceptably large. One solution is to adjust the step size during the search, using larger steps at the beginning, and smaller steps as the optimum response is approached. One-Factor-at-a-Time Optimization One approach to optimizing the quantitative method for vanadium described earlier is to select initial concentrations for H 2 O 2 and H 2 SO 4 and measure the absorbance. We then increase or decrease the concentration of one reagent in steps, while the second reagent’s concentration remains constant, until the absorbance decreases in value. The concentration of the second reagent is then adjusted until a decrease in ab- sorbance is again observed. This process can be stopped after one cycle or repeated until the absorbance reaches a maximum value or exceeds an acceptable threshold value. A one-factor-at-a-time optimization is consistent with a commonly held belief that to determine the influ- ence of one factor it is necessary to hold constant all other factors. This is an effective, although not necessarily an ef- ficient, experimental design when the factors are indepen- dent. 2 Two factors are considered independent when changing the level of one factor does not influence the ef- fect of changing the other factor’s level. Table 14.1 provides an example of two in- dependent factors. When factor B is held at level B 1 , changing factor A from level A 1 to level A 2 increases the response from 40 to 80; thus, the change in response, ∆R, is ∆R =80–40=40 In the same manner, when factor B is at level B 2 , we find that ∆R =100–60=40 when the level of factor A changes from A 1 to A 2 . We can see this independence graphically by plotting the response versus the factor levels for factor A (Figure 14.4). The parallel lines show that the level of factor B does not influence the effect on the response of changing factor A. In the same manner, the effect of changing factor B’s level is independent of the level of factor A. Chapter 14 Developing a Standard Method 669 Response Level for factor A Factor B 2 constant Factor B 1 constant Table 1 4 .1 Example of Two Independent Factors Factor A Factor B Response A 1 B 1 40 A 2 B 1 80 A 1 B 2 60 A 2 B 2 100 1400-CH14 9/8/99 4:35 PM Page 669 Figure 14.5 Two views of a two-factor response surface for which the factors are independent. The optimum response in (b) is indicated by the • at (2, 8). Contour lines are shown for intervals of 0.5 response units. Mathematically, two factors are independent if they do not appear in the same term in the algebraic equation describing the response surface. For example, factors A and B are independent when the response, R, is given as R = 2.0 + 0.12A + 0.48B – 0.03A 2 – 0.03B 2 14.1 The resulting response surface for equation 14.1 is shown in Figure 14.5. The progress of a searching algorithm is best followed by mapping its path on a contour plot of the response surface. Positions on the response surface are identified as (a, b) where a and b are the levels for factors A and B. Four examples of a one- factor-at-a-time optimization of the response surface for equation 14.1 are shown in Figure 14.5b. For those paths indicated by a solid line, factor A is optimized first, followed by factor B. The order of optimization is reversed for paths marked by a dashed line. The effectiveness of this algorithm for optimizing independent factors is shown by the fact that the optimum response at (2, 8) is reached in a single cycle from any set of initial factor levels. Further- more, it does not matter which factor is optimized first. Although this algorithm is effective at locating the opti- mum response, its efficiency is limited by requiring that only a single factor can be changed at a time. Unfortunately, it is more common to find that two factors are not independent. In Table 14.2, for instance, changing the level of factor B from level B 1 to level B 2 has a significant effect on the response when factor A is at level A 1 ∆R =60–20=40 but has no effect when factor A is at level A 2 . ∆R =80–80=0 This effect is seen graphically in Figure 14.6. Factors that are not independent are said to interact. In this case the equation for the response includes an interaction term in which both factors A and B are present. Equation 14.2, for example, con- tains a final term accounting for the interaction between the factors A and B. R = 5.5 + 1.5A + 0.6B – 0.15A 2 – 0.0245B 2 – 0.0857AB 14.2 The resulting response surface for equation 14.2 is shown in Figure 14.7a. 670 Modern Analytical Chemistry 0 10 8 6 4 2 4.5 4 3.5 3 2 2.5 1.5 0.5 1 0 0 1.5 3 4.5 6 7.5 9 Factor A Factor B Response 0 0 1 10 9 8 7 6 5 4 3 2 Factor A Factor B 10 9 8 7 6 5 4 3 2 1 (a) (b) Table 1 4 .2 Example of Two Dependent Factors Factor A Factor B Response A 1 B 1 20 A 2 B 1 80 A 1 B 2 60 A 2 B 2 80 Response Level for factor B Factor A 2 constant Factor A 1 constant Figure 14.6 Factor effect plot for two interacting factors. 1400-CH14 9/8/99 4:35 PM Page 670 Figure 14.7 Two views of a two-factor response surface for which the factors interact. The optimum response in (b) is indicated by the • at (3, 7). The response at the end of the first cycle is shown in (b) by the ♦. Contour lines are shown for intervals of 1.0 response units. The progress of a one-factor-at-a-time optimization for the response surface given by equation 14.2 is shown in Figure 14.7b. In this case the optimum response of (3, 7) is not reached in a single cycle. If we start at (0, 0), for example, optimizing factor A follows the solid line to the point (5, 0). Optimizing factor B completes the first cycle at the point (5, 3.5). If our algorithm allows for only a single cycle, then the optimum response is not found. The optimum response usually can be reached by continuing the search through additional cycles, as shown in Figure 14.7b. The efficiency of a one-factor-at-a-time searching algorithm is significantly less when the factors interact. An additional complication with interacting factors is the possi- bility that the search will end prematurely on a ridge of the response surface, where a change in level for any single factor results in a smaller response (Figure 14.8). Simplex Optimization The efficiency of a searching algorithm is improved by al- lowing more than one factor to be changed at a time. A convenient way to accom- plish this with two factors is to select three sets of initial factor levels, positioned as the vertices of a triangle (Figure 14.9), and to measure the response for each. The set of factor levels giving the smallest response is rejected and replaced with a new set of factor levels using a set of rules. This process is continued until no further optimiza- tion is possible. The set of factor levels is called a simplex. In general, for k factors a simplex is a (k + 1)-dimensional geometric figure. 3,4 The initial simplex is determined by choosing a starting point on the response surface and selecting step sizes for each factor. Ideally the step sizes for each factor should produce an approximately equal change in the response. For two factors a convenient set of factor levels is (a, b), (a + s A , b), and (a + 0.5s A , b + 0.87s B ), where s A and s B are the step sizes for factors A and B. 5 Optimization is achieved using the following set of rules: Rule 1. Rank the response for each vertex of the simplex from best to worst. Rule 2. Reject the worst vertex, and replace it with a new vertex generated by reflecting the worst vertex through the midpoint of the remaining vertices. The factor levels for the new vertex are twice the average factor levels for the retained vertices minus the factor levels for worst vertex. Rule 3. If the new vertex has the worst response, then reject the vertex with the second-worst response, and calculate the new vertex using rule 2. This rule ensures that the simplex does not return to the previous simplex. Rule 4. Boundary conditions are a useful way to limit the range of possible factor levels. For example, it may be necessary to limit the concentration of a factor Chapter 14 Developing a Standard Method 671 (a) (b) 0 10 8 6 4 2 10 9 8 7 5 6 4 1 2 3 0 0 1.5 3 4.5 6 7.5 9 Factor A Factor B Response 0 0 1 10 9 8 7 6 5 4 3 2 Factor A Factor B 10 9 8 7 6 5 4 3 2 1 True optimum False optimum X Figure 14.8 Example of a false optimum for a one- factor-at-a-time searching algorithm. Factor B Factor A Figure 14.9 Simplex for two factors. 1400-CH14 9/8/99 4:35 PM Page 671 672 Modern Analytical Chemistry for solubility reasons or to limit temperature due to a reagent’s thermal stability. If the new vertex exceeds a boundary condition, then assign it a response lower than all other responses, and follow rule 3. Because the size of the simplex remains constant during the search, this algorithm is called a fixed-sized simplex optimization. Example 14.1 illustrates the application of these rules. EXAMPLE 1 4 .1 Find the optimum response for the response surface in Figure 14.7 using the fixed-sized simplex searching algorithm. Use (0, 0) for the initial factor levels, and set the step size for each factor to 1.0. SOLUTION Letting a = 0, b = 0, s a = 1 and s b = 1 gives the vertices for the initial simplex as Vertex 1: (a, b) = (0, 0) Vertex 2: (a + s A , b) = (1, 0) Vertex 3: (a + 0.5s A , b + 0.87s B ) = (0.5, 0.87) The responses (calculated using equation 14.2) for the three vertices are shown in the following table Vertex Factor A Factor B Response V 1 0 0 5.50 V 2 1.00 0 6.85 V 3 0.50 0.87 6.68 with V 1 giving the worst response and V 3 the best response (rule 1). We reject V 1 and replace it with a new vertex whose factor levels are calculated using rule 2; thus The new simplex, therefore, is Vertex Factor A Factor B Response V 2 1.00 0 6.85 V 3 0.50 0.87 6.68 V 4 1.50 0.87 7.80 The worst response is for vertex 3, which we replace with the following new vertex ab=× + −= =× + −=2 100 150 2 05 200 2 0087 2 087 0 . . New for V + for V 2 for V ) = 2 0 + 0.87 2 1 b bb b=×       −×−=20087 23 (. New for V + for V 2 for V ) = 2 1.00 + 0.50 2 3 1 a aa a=×       −×−=20150 2 (. simplex optimization An efficient optimization method that allows several factors to be optimized at the same time. 1400-CH14 9/8/99 4:35 PM Page 672 The resulting simplex now consists of the following vertices Vertex Factor A Factor B Response V 2 1.00 0 6.85 V 4 1.50 0.87 7.80 V 5 2.00 0 7.90 The calculation of the remaining vertices is left as an exercise. The progress of the completed optimization is shown in Table 14.3 and in Figure 14.10. The optimum response of (3, 7) first appears in the twenty-fourth simplex, but a total of 29 steps is needed to verify that the optimum has been found. Chapter 14 Developing a Standard Method 673 Table 1 4 . 3 Progress of Fixed-Sized Simplex Optimization for Response Surface in Figure 14.10 Simplex Vertices Notes 1 1, 2, 3 2 2, 3, 4 3 2, 4, 5 4 4, 5, 6 5 5, 6, 7 6 6, 7, 8 7 7, 8, 9 8 8, 9, 10 9 8, 10, 11 10 10, 11, 12 11 11, 12, 13 12 12, 13, 14 follow rule 3 13 13, 14, 15 14 13, 15, 16 15 13, 16, 17 follow rule 3 16 16, 17, 18 17 16, 18, 19 18 16, 19, 20 follow rule 3 19 19, 20, 21 20 19, 21, 22 follow rule 3 21 21, 22, 23 22 21, 23, 24 follow rule 3 23 23, 24, 25 24 23, 25, 26 25 23, 26, 27 follow rule 3 26 26, 27, 28 follow rule 3 27 26, 28, 29 28 26, 29, 30 follow rule 3 29 26, 30, 31 vertex 31 same as vertex 25 1400-CH14 9/8/99 4:35 PM Page 673 The fixed-size simplex searching algorithm is effective at locating the optimum response for both independent and interacting factors. Its efficiency, however, is limited by the simplex’s size. We can increase its efficiency by allowing the size of the simplex to expand or contract in response to the rate at which the optimum is being approached. 3,6 Although the algorithm for a variable-sized simplex is not pre- sented here, an example of its increased efficiency is shown Figure 14.11. The refer- ences and suggested readings may be consulted for further details. 1 4 A. 3 Mathematical Models of Response Surfaces Earlier we noted that a response surface can be described mathematically by an equation relating the response to its factors. If a series of experiments is carried out in which we measure the response for several combinations of factor levels, then lin- ear regression can be used to fit an equation describing the response surface to the data. The calculations for a linear regression when the system is first-order in one factor (a straight line) were described in Chapter 5. A complete mathematical treat- ment of linear regression for systems that are second-order or that contain more than one factor is beyond the scope of this text. Nevertheless, the computations for 674 Modern Analytical Chemistry 6 0 8 0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3 3.3 3.6 3.9 4.2 4.5 4.8 5.1 5.4 5.7 7.6 7.2 6.8 6.4 6 5.6 5.2 4.8 4.4 4 3.6 3.2 2.8 2.4 2 1.6 1.2 0.8 0.4 1 34 6 8 25 79 10 11 12 13 16 14 17 1518 19 20 21 22 27 28 25, 31 29 30 26 23 24 Figure 14.10 Progress of a fixed-sized simplex optimization for the response surface of Example 14.1. The optimum response at (3, 7) corresponds to vertex 26. 1400-CH14 9/8/99 4:35 PM Page 674 Figure 14.11 Progress of a variable-sized simplex optimization for the response surface of Example 14.1. The optimum response is at (3, 7). a few special cases are straightforward and are considered in this section. A more comprehensive treatment of linear regression can be found in several of the sug- gested readings listed at the end of this chapter. Theoretical Models of the Response Surface Mathematical models for response surfaces are divided into two categories: those based on theory and those that are empirical. Theoretical models are derived from known chemical and physical rela- tionships between the response and the factors. In spectrophotometry, for example, Beer’s law is a theoretical model relating a substance’s absorbance, A, to its concen- tration, C A A = εbC A where ε is the molar absorptivity, and b is the pathlength of the electromagnetic ra- diation through the sample. A Beer’s law calibration curve, therefore, is a theoretical model of a response surface. Empirical Models of the Response Surface In many cases the underlying theoreti- cal relationship between the response and its factors is unknown, making impossi- ble a theoretical model of the response surface. A model can still be developed if we make some reasonable assumptions about the equation describing the response sur- face. For example, a response surface for two factors, A and B, might be represented by an equation that is first-order in both factors Chapter 14 Developing a Standard Method 675 6 0 8 0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3 3.3 3.6 3.9 4.2 4.5 4.8 5.1 5.4 5.7 7.6 7.2 6.8 6.4 6 5.6 5.2 4.8 4.4 4 3.6 3.2 2.8 2.4 2 1.6 1.2 0.8 0.4 1 3 4 6 8 2 5 7 9 10 11 13 16 14 17 15 18 19 20 21 12 theoretical model A model describing a system’s response that has a theoretical basis and can be derived from theoretical principles. 1400-CH14 9/8/99 4:35 PM Page 675 [...]... equation 14. 10 SOLUTION We begin by calculating the estimated parameters using equations 14. 6 14. 9 and 14. 11 14. 14 b0 = 1 (137.25 + 54.75 + 73.75 + 30.25 + 61.75 + 30.25 + 41.25 + 18.75) = 56.0 8 ba = 1 (137.25 + 54.75 + 73.75 + 30.25 − 61.75 − 30.25 − 41.25 − 18.75) = 18.0 8 bb = 1 (137.25 + 54.75 − 73.75 − 30.25 + 61.75 + 30.25 − 41.25 − 18.75) = 15.0 8 679 140 0-CH14 9/8/99 4:35 PM Page 680 680 Modern Analytical. .. βabAB + βacAC + βbcBC + βabcABC 14. 10 where A, B, and C are the factors The terms β0, βa, βb, and βab are estimated using equations 14. 6 14. 9 The remaining parameters are estimated using the following equations 1 ∑ Ci*R i n 14. 11 β ac ≈ bac = 1 ∑ Ai*Ci*R i n 14. 12 βbc ≈ bbc = 1 ∑ Bi*Ci*R i n 14. 13 1 ∑ Ai*Bi*Ci*R i n 14. 14 β c ≈ bc = β abc ≈ babc = EXAMPLE 14. 4 Table 14. 5 lists the uncoded factor levels,... following equations 1 ∑ Ri n 14. 5 β a ≈ ba = 1 ∑ Ai*R i n 14. 6 βb ≈ bb = 1 ∑ Bi*R i n 14. 7 β0 ≈ b0 = Table 14. 4 Example of Uncoded and Coded Factor Levels and Responses for a 22 Factorial Design Run A B A* B* Response 1 2 3 4 15 15 5 5 30 10 30 10 +1 +1 –1 –1 +1 –1 +1 –1 22.5 11.5 17.5 8.5 677 140 0-CH14 9/8/99 4:35 PM Page 678 678 Modern Analytical Chemistry β ab ≈ bab = 1 ∑ Ai*Bi*R i n 14. 8 where n is the... respectively, for systems in- 140 0-CH14 9/8/99 4:35 PM Page 683 683 Factor A Factor A Chapter 14 Developing a Standard Method Factor B (a) Factor B (b) Figure 14. 15 Central composite designs for (a) k = 2 and (b) k = 3 volving three or four factors A discussion of central composite designs, including computational considerations, can be found in the suggested readings at the end of the chapter 14B Verifying the... example, a two-way ANOVA can be used in a collaborative study to determine the importance to an analytical method of both the analyst and the instrumentation used The treatment of multivariable ANOVA is beyond the scope of this text, but is covered in several of the texts listed as suggested readings at the end of the chapter 697 140 0-CH14 9/8/99 4:36 PM Page 698 698 Modern Analytical Chemistry 14C.3 What... first-order empirical model are inappropriate for this system A complete empirical model for this system is presented in problem 10 in the end-of -chapter problem set Many systems that cannot be represented by a first-order empirical model can be described by a full second-order polynomial equation, such as that for two factors Factor A R = β0 + βaA + βbB + βaaA2 + βbbB2 + βabAB Factor B Figure 14. 14... 0.01AB + 0.02AC – 0.01BC + 0.005ABC 140 0-CH14 9/8/99 4:35 PM Page 681 681 Response Response Chapter 14 Developing a Standard Method Figure 14. 13 Level for factor A (a) Level for factor A (b) A 2k factorial design is limited to models that include only a factor’s first-order effects on the response Thus, for a 22 factorial design, it is possible to determine the first-order effect for each factor (βa... equation 14. 18 σ sys = σ2 − σ2 tot rand ≈ 2 2 s 2 − sD T = 2 (13.3)2 − (5.95)2 = 8.41 2 *Here is a short-cut that simplifies the calculation of sD and sT Enter the values for Di into your calculator, and use its built-in functions to find the standard deviation Divide this result by 2 to obtain sD You can use the same approach to calculate sT 691 140 0-CH14 9/8/99 4:36 PM Page 692 692 Modern Analytical. .. data 140 0-CH14 9/8/99 4:36 PM Page 694 694 Modern Analytical Chemistry Table 14. 7 Results of Four Analysts for the %Purity of a Preparation of Sulfanilamide Replicate Analyst A Analyst B Analyst C Analyst D 1 2 3 4 5 6 94.09 94.64 95.08 94.54 95.38 93.62 99.55 98.24 101.1 100.4 100.1 95 .14 94.62 95.28 94.59 94.24 93.88 94.23 96.05 93.89 94.95 95.49 h s2 = ni ∑ ∑  X ij  − X  i =1 j =1 2 14. 21 N... errors and, therefore, provides an estimate for both σ2 and σ2 rand sys – s2 = σ2 + nσ2 14. 24 b rand sys – where n is approximated as the average number of replicates per analyst h n = ∑ ni i =1 h 140 0-CH14 9/8/99 4:36 PM Page 695 695 Chapter 14 Developing a Standard Method Table 14. 8 Summary of Calculations for a One-Way Analysis of Variance Source Between sample Within sample Total Sum of Squares Degrees . optimum X Figure 14. 8 Example of a false optimum for a one- factor-at-a-time searching algorithm. Factor B Factor A Figure 14. 9 Simplex for two factors. 140 0-CH14 9/8/99 4:35 PM Page 671 672 Modern Analytical. β ab are estimated using equations 14. 6 14. 9. The remaining parameters are estimated using the following equations. 14. 11 14. 12 14. 13 14. 14 EXAMPLE 1 4 . 4 Table 14. 5 lists the uncoded factor levels,. analyst, and validating the method for general use. 140 0-CH14 9/8/99 4:35 PM Page 666 Figure 14. 1 Example of a one-factor response surface. Chapter 14 Developing a Standard Method 667 Absorbance 01234 0.000 0.500 0.450 0.400 0.350 0.300 0.250 0.200 0.150 0.100 0.050 Analyte