Modern Analytical Cheymistry - Chapter 14 potx

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 c

C

Developing a Standard Method

I n 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.

14A Optimizing the Experimental Procedure

In the presence of H2O2and H2SO4, solutions of vanadium form a reddish brown

color that is believed to be a compound with the general formula (VO)2(SO4)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 H2O2and H2SO4present In particular, a large excess of H2O2is 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 H2O2

and H2SO4requires 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 combinaOptimiza-tion of factor levels Usually we desire a

max-imum response, such as maxmax-imum absorbance in the quantitative analysis for

vana-dium as (VO)2(SO4)3 In other situations, such as minimizing percent error, we

seek a minimum response

14A.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 + 0.0896CA

where A is the absorbance, and CAis 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.

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, itsshape, 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 empiricalmodel of the response surface to predict the optimum response

14A.2 Searching Algorithms for Response Surfaces

Imagine that you wish to climb to the top of a mountain Because the mountain iscovered 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 directionthat 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 leadingfrom 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 eachstep For example, one algorithm for climbing a mountain is to always move in thedirection that has the steepest slope

A systematic searching algorithm can also be used to locate the optimum sponse for an analytical method To find the optimum response, we select an initialset 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 repeateduntil the algorithm indicates that we have reached the optimum response Twocommon searching algorithms are described in this section First, however, we mustconsider how to evaluate a searching algorithm

re-Effectiveness and Efficiency A searching algorithm is characterized by its tiveness and its efficiency To be effective, the experimentally determined optimumshould closely coincide with the system’s true optimum A searching algorithm mayfail 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 Apoorly designed algorithm may prematurely end the search For example, an algo-rithm for climbing a mountain that allows movement to the north, south, east, orwest 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, ornoise, a spuriously high response may produce a false optimum from which the

effec-Start

0

10 8 6 4 2

3 2.5 2 1.5 1 0.5 0

10 9 8 7 6 5 4 3 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

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 H2O2and

H2SO4and 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 B1, changing factor A from level A1

to level A2increases 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 B2, we find that

∆R = 100 – 60 = 40

when the level of factor A changes from A1to A2 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

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 sameterm 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 – 0.03A2– 0.03B2 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 acontour 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 inFigure 14.5b For those paths indicated by a solid line, factor A is optimized first,

followed by factor B The order of optimization is reversedfor paths marked by a dashed line The effectiveness of thisalgorithm for optimizing independent factors is shown bythe fact that the optimum response at (2, 8) is reached in asingle 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 thatonly a single factor can be changed at a time

Unfortunately, it is more common to find that twofactors are not independent In Table 14.2, for instance,

changing the level of factor B from level B1to level B2has a significant effect on the

response when factor A is at level A1

R = 5.5 + 1.5A + 0.6B – 0.15A2– 0.0245B2– 0.0857AB 14.2

The resulting response surface for equation 14.2 is shown in Figure 14.7a

0

10 8 6 4 2

4.5 4 3.5 3 2 2.5 1.5 0.5 1 0

10 9 8 7 6 5 4 3 2

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 + sA, b), and (a + 0.5sA, b + 0.87sB), where

sAand sBare the step sizes for factors A and B.5Optimization 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

0

10 8 6 4 2

10 9 8 7 5 6 4 1 2 3 0

10 9 8 7 6 5 4 3 2

Simplex for two factors.

for solubility reasons or to limit temperature due to a reagent’s thermalstability If the new vertex exceeds a boundary condition, then assign it aresponse 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 14.1

Find the optimum response for the response surface in Figure 14.7 using thefixed-sized simplex searching algorithm Use (0, 0) for the initial factor levels,and set the step size for each factor to 1.0

in the following table

with V1giving the worst response and V3the best response (rule 1) We reject

V1and replace it with a new vertex whose factor levels are calculated using rule2; thus

The new simplex, therefore, is

An efficient optimization method that

allows several factors to be optimized at

the same time.

The resulting simplex now consists of the following vertices

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

Table 14.3 Progress of Fixed-Sized Simplex

Optimization for Response Surface in Figure 14.10

The fixed-size simplex searching algorithm is effective at locating the optimumresponse for both independent and interacting factors Its efficiency, however, islimited by the simplex’s size We can increase its efficiency by allowing the size ofthe simplex to expand or contract in response to the rate at which the optimum isbeing approached.3,6Although 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.

14A.3 Mathematical Models of Response Surfaces

Earlier we noted that a response surface can be described mathematically by anequation 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 ear regression can be used to fit an equation describing the response surface to thedata The calculations for a linear regression when the system is first-order in onefactor (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 morethan one factor is beyond the scope of this text Nevertheless, the computations for

lin-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

16

14 17

15

18

21 22

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, CA

A =εbCA

where εis 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

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

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.

de-are empirical models of the response surface because they have no basis in a

theo-retical understanding of the relationship between the response and its factors Anempirical model may provide an excellent description of the response surface over awide range of factor levels It is more common, however, to find that an empiricalmodel only applies to the range of factor levels for which data have been collected

To develop an empirical model for a response surface, it is necessary to collectthe right data using an appropriate experimental design Two popular experimentaldesigns are considered in the following sections

Factorial Designs To determine a factor’s effect on the response, it is necessary tomeasure the response for at least two factor levels For convenience these levels are

labeled high, Hf, and low, Lf, where f is the factor; thus HAis the high level for factor

A, and LBis the low level for factor B When more than one factor is included in theempirical model, then each factor’s high level should be paired with both the highand low levels for all other factors In the same way, the low level for each factorshould be paired with the high and low levels for all other factors (Figure 14.12) Alltogether, a minimum of 2k experiments is necessary, where k is the number of fac-

tors This experimental design is known as a 2kfactorial design

5

6

7 8

1

1 2 3 4 5 6 7 8

(b)

empirical model

A model describing a system’s response

that is not derived from theoretical

principles.

The linear regression calculations for a 2kfactorial design are straightforward

and can be done without the aid of a sophisticated statistical software package To

simplify the computations, factor levels are coded as +1 for the high level, and –1

for the low level The relationship between a factor’s coded level, xf*, and its actual

value, xf, is given as

where cf is the factor’s average level, and dfis the absolute difference between the

factor’s average level and its high and low values Equation 14.3 is used to transform

coded results back to their actual values

EXAMPLE 14.2

The factor A has coded levels of +1 and –1 with an average factor level of 100,

and dAequal to 5 What are the actual factor levels?

SOLUTION

The actual factor levels are

HA= 100 + (1)(5) = 105 LA= 100 + (–1)(5) = 95

Let’s start by considering a simple example involving two factors, A and B, to

which we wish to fit the following empirical model

A 2kfactorial design with two factors requires four runs, or sets of experimental

conditions, for which the uncoded levels, coded levels, and responses are shown in

Table 14.4 The terms β0, βa,βb, and βabin equation 14.4 account for, respectively,

the mean effect (which is the average response), first-order effects due to factors A

and B, and the interaction between the two factors Estimates for these parameters

are given by the following equations

Table 14.4 Example of Uncoded and Coded

Factor Levels and Responses for

where n is the number of runs, and A i*and B i*are the coded factor levels for the ith

run Solving for the estimated parameters using the data in Table 14.4

leaves us with the following empirical model for the response surface

The suitability of this model can be evaluated by substituting values for A* and B*

from Table 14.4 and comparing the calculated response to the known response.Using the values for the first run as an example gives

SOLUTION

To convert the equation to its uncoded form, it is necessary to solve equation

14.3 for each factor Values for cfand dfare determined from the high and lowlevels for each factor; thus

Substituting known values into equation 14.3,

R = 15.0 + 2(0.2A – 2) + 5(0.1B – 2) + 0.5(0.2A – 2)(0.1B – 2)

= 15.0 + 0.4A – 4 + 0.5B – 10 + 0.01AB – 0.2A – 0.1B + 2

= 3.0 + 0.2A + 0.4B + 0.01AB

We can verify this equation by substituting values for A and B from Table 14.4

and solving for the response Using values for the first run, for example, gives

R = 3.0 + (0.2)(15) + (0.4)(30) + (0.01)(15)(30) = 22.5

which agrees with the expected value

The computation just outlined is easily extended to any number of factors For

a system with three factors, for example, a 23factorial design can be used to

deter-mine the parameters for the empirical model described by the following equation

R =β0+βaA +βbB +βcC +βabAB +βacAC +βbcBC +βabcABC 14.10

where A, B, and C are the factors The terms β0, βa,βb, and βabare estimated using

equations 14.6–14.9 The remaining parameters are estimated using the following

Table 14.5 lists the uncoded factor levels, coded factor levels, and responses for

a 23 factorial design Determine the coded and uncoded empirical model for

the response surface based on equation 14.10

The coded empirical model, therefore, is

R = 56 + 18A* + 15B* + 22.5C * + 7A*B* + 9A*C * + 6B*C * + 3.75A*B*C *

To check the result we substitute the coded factor levels for the first run intothe coded empirical model, giving

R = 56 + (18)(+1) + (15)(+1) + (22.5)(+1) + (7)(+1)(+1) + (9)(+1)(+1)

+ (6)(+1)(+1) + (3.75)(+1)(+1)(+1) = 137.25

which agrees with the measured response

To transform the coded empirical model into its uncoded form, it is

necessary to replace A*, B*, and C * with the following relationships

the derivations of which are left as an exercise Substituting these relationshipsinto the coded empirical model and simplifying (which also is left as anexercise) gives the following result for the uncoded empirical model

R = 3 + 0.2A + 0.4B + 0.5C – 0.01AB + 0.02AC – 0.01BC + 0.005ABC

Table 14.5 Uncoded and Coded Factor Levels and Responses

for the 2 3 Factorial Design of Example 14.4

Figure 14.13

Curved one-factor response surface showing (a) the limitation of a 2kfactorial design for modeling second-order effects; and (b) the application of a 3kfactorial design for modeling second-order effects.

A 2kfactorial design is limited to models that include only a factor’s first-order

effects on the response Thus, for a 22factorial design, it is possible to determine the

first-order effect for each factor (βaand βb), as well as the interaction between the

factors (βab) There is insufficient information in the factorial design, however, to

determine any higher order effects or interactions This limitation is a consequence

of having only two levels for each factor Consider, for example, a system in which

the response is a function of a single factor Figure 14.13a shows the experimentally

measured response for a 21factorial design in which only two levels of the factor are

used The only empirical model that can be fit to the data is that for a straight line

R =β0+βaA

If the actual response is that represented by the dashed curve, then the empirical

model is in error To fit an empirical model that includes curvature, a minimum of

three levels must be included for each factor The 31factorial design shown in

Fig-ure 14.13b, for example, can be fit to an empirical model that includes second-order

effects for the factor

R =β0+βaA +βaaA2

In general, an n-level factorial design can include single-factor and interaction

terms up to the (n – 1)th order.

The effectiveness of a first-order empirical model can be judged by measuring

the response at the center of the factorial design If there are no higher order effects,

the average response of the runs in a 2kfactorial design should be equal to the

mea-sured response at the center of the factorial design The influence of random error

can be accounted for by making several determinations of the response at the center

of the factorial design and establishing a suitable confidence interval If the

differ-ence between the two responses is significant, then a first-order empirical model is

probably not appropriate

EXAMPLE 14.5

At the beginning of this section we noted that the concentration of vanadium

can be determined spectrophotometrically by making the solution acidic with

H2SO4 and reacting with H2O2to form a reddish brown compound with the

general formula (VO)2(SO4)3 Palasota and Deming7studied the effect on the

absorbance of the relative amounts of H2SO4and H2O2, reporting the following

results for a 22factorial design

We begin by determining the confidence interval for the response at the center

of the factorial design The mean response is 0.335, with a standard deviation of0.0094 The 90% confidence interval, therefore, is

The average response, R, from the factorial design is–

Because R exceeds the confidence interval’s upper limit of 0.346, there is reason–

to believe that a 22factorial design and a first-order empirical model areinappropriate for this system A complete empirical model for this system ispresented 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

R =β0+βaA +βbB +βaaA2+βbbB2+βabAB

Because each factor must be measured for at least three levels, a convenient mental design is a 3kfactorial design A 32factorial design for two factors, for exam-ple, is shown in Figure 14.14 The computations for 3kfactorial designs are not aseasily generalized as those for a 2kfactorial design and are not considered in thistext Further details about the calculations are found in the suggested readings atthe end of the chapter

experi-Central Composite Designs One limitation to a 3kfactorial design is the ber of trials that must be run For two factors, as shown in Figure 14.14, a total

num-of nine trials is needed This number increases to 27 for three factors and 81 forfour factors A more efficient experimental design for systems containing morethan two factors is the central composite design, two examples of which areshown in Figure 14.15 The central composite design consists of a 2kfactorial de-sign, which provides data for estimating the first-order effects for each factor and

interactions between the factors, and a “star” design consisting of 2k + 1 points,

which provides data for estimating second-order effect Although a central posite design for two factors requires the same number of trials, 9, as a 32facto-rial design, it requires only 15 trials and 25 trials, respectively, for systems in-

volving three or four factors A discussion of central composite designs,

includ-ing computational considerations, can be found in the suggested readinclud-ings at the

end of the chapter

14B Verifying the Method

After developing and optimizing a method, it is necessary to determine the quality

of results that can reasonably be expected when the method is used by a single

ana-lyst Generally, three steps are included in the process of verifying a method:

mining single-operator characteristics, the blind analysis of standards, and

deter-mining the method’s ruggedness In addition, if an alternative standard method

exists, both the standard method and the new method can be used to analyze the

same sample, and the results compared If the quality of the results is unacceptable,

the method is not suitable for consideration as a standard method

14B.1 Single-Operator Characteristics

The first step in verifying a method is to determine the precision, accuracy, and

de-tection limit when a single analyst uses the method to analyze a standard sample of

known composition The detection limit, which was discussed in Chapter 4, is

de-termined by analyzing a reagent blank for each type of sample matrix for which the

method will be used Precision is determined by analyzing replicate portions,

preferably more than ten, of a standard sample Finding the method’s accuracy is

evaluated by a t-test, as described in Chapter 4 Precision and accuracy should be

evaluated for several different concentration levels of analyte, including at least one

concentration near the detection limit, and for each type of sample matrix that will

be encountered The analysis of several concentrations allows for the detection of

constant sources of determinate error and establishes the range of concentrations

for which the method is applicable

14B.2 Blind Analysis of Standard Samples

Single-operator characteristics are determined by analyzing a sample whose

concen-tration of analyte is known to the analyst The second step in verifying a method is

the blind analysis of standard samples where the analyte’s concentration remains

unknown to the analyst The standard sample is analyzed several times, and the

av-erage concentration of the analyte is determined This value should be within three,

and preferably two standard deviations (as determined from the single-operator

characteristics) of the analyte’s known concentration

blind analysis

The analysis of a standard sample whose composition is unknown to the analyst.

14B.3 Ruggedness Testing

In many cases an optimized method may produce excellent results in the laboratorydeveloping the method, but poor results in other laboratories This is not surprisingsince a method is often optimized by a single analyst under an ideal set of condi-tions, in which the sources of reagents, equipment, and instrumentation remain thesame for each trial The procedure might also be influenced by environmental fac-tors, such as the temperature or relative humidity in the laboratory, whose levels arenot specified in the procedure and which may differ between laboratories Finally,when optimizing a method the analyst usually takes particular care to perform theanalysis in exactly the same way during every trial

An important step in developing a standard method is to determine which tors have a pronounced effect on the quality of the analytical method’s result Theprocedure can then be written to specify the degree to which these factors must becontrolled A procedure that, when carefully followed, produces high-quality results

in different laboratories is considered rugged The method by which the critical

fac-tors are discovered is called ruggedness testing.8

Ruggedness testing is often performed by the laboratory developing the dard method Potential factors are identified and their effects evaluated by perform-ing the analysis while the factors are held at two levels Normally one level for eachfactor is that given in the procedure, and the other is a level likely to be encounteredwhen the procedure is used by other laboratories

stan-This approach to ruggedness testing can be time-consuming If seven tial factors are identified, for example, ruggedness testing can be accomplishedwith a 27factorial design This requires a total of 128 trials, which is a prohibitivelylarge amount of work A simpler experimental design is shown in Table 14.6,

poten-in which the two factor levels are identified by upper case and lower case letters.This design, which is similar to that for the 23 factorial design discussed in theprevious section, is called a fractional factorial design and provides informationabout the first-order effect of each factor It does not, however, provide suffi-cient information to evaluate higher order effects or potential interactions be-tween factors, both of which are assumed to be of less importance than first-order effects

The experimental design for ruggedness testing is balanced in that each factorlevel is paired an equal number of times with the upper case and lower case levels

ruggedness testing

The process of evaluating a method to

determine those factors for which a small

change in value has a significant effect on

the method’s results.

Table 14.6 Experimental Design for a Ruggedness Test

Involving Seven Factors