Brereton Chemometrics

EXPERIMENTAL DESIGN	69

Provided that coded values are used throughout, since there are no interaction or squared terms, the size of the coefficients is directly related to importance. An alternative method of calculation is to multiply the response by each column, dividing by the number of experiments:

N

bj = xij yi /N

i=1

as in normal full factorial designs where xij is a number equal to +1 or −1 according to the value in the experimental matrix. If one or more dummy factor is included, it is easy to compare the size of the real factors with that of the dummy factor, and factors that are demonstrably larger in magnitude have significance.

An alternative approach comes from the work of Glenichi Taguchi. His method of quality control was much used by Japanese industry, and only fairly recently was it recognised that certain aspects of the theory are very similar to Western practices. His philosophy was that consumers desire products that have constant properties within narrow limits. For example, a consumer panel may taste the sweetness of a product, rating it from 1 to 10. A good marketable product may result in a taste panel score of 8: above this value the product is too sickly, and below it the consumer expects the product to be sweeter. There will be a huge number of factors in the manufacturing process that might cause deviations from the norm, including suppliers of raw materials, storage and preservation of the food and so on. Which factors are significant? Taguchi developed designs for screening large numbers of potential factors.

His designs are presented in the form of a table similar to that of Plackett and Burman, but with a 1 for a low and 2 for a high level. Superficially, Taguchi designs might appear different, but by changing the notation, and swapping rows and columns around, it is possible to show that both types of design are identical and, indeed, the simpler designs are the same as the well known partial factorial designs. There is a great deal of controversy surrounding Taguchi’s work; while many statisticians feel that he has ‘reinvented the wheel’, he was an engineer, but his way of thinking had a major and positive effect on Japanese industrial productivity. Before globalisation and the Internet, there was less exchange of ideas between different cultures. His designs are part of a more comprehensive approach to quality control in industry.

Taguchi designs can be extended to three or more levels, but construction becomes fairly complicated. Some texts do provide tables of multilevel screening designs, and it is also possible to mix the number of levels, for example having one factor at two levels and another at three levels. This could be useful, for example, if there are three alternative sources of one raw material and two of another. Remember that the factors can fall into discrete categories and do not have to be numerical values such as temperature or concentrations. A large number of designs have been developed from Taguchi’s work, but most are quite specialist, and it is not easy to generalise. The interested reader is advised to consult the source literature.

2.3.4 Partial Factorials at Several Levels: Calibration Designs

Two level designs are useful for exploratory purposes and can sometimes result in useful models, but in many areas of chemistry, such as calibration (see Chapter 5 for more details), it is desirable to have several levels, especially in the case of spectra of

70	CHEMOMETRICS

mixtures. Much of chemometrics is concerned primarily with linearly additive models of the form

X = C .S

where X is an observed matrix, such as a set of spectra, each row consisting of a spectrum, and each column of a wavelength, C is a matrix of, e.g., concentrations, each row consisting of the concentration of a number of compounds in a spectrum, and S could consist of the corresponding spectra of each compound. There are innumerable variations on this theme, in some cases where all the concentrations of all the components in a mixture are known, the aim being to develop a calibration model that predicts the concentrations from an unknown spectrum, to cases where the concentrations of only a few components in a mixture are known. In many situations, it is possible to control the experiments by mixing up components in the laboratory but in other cases this is not practicable, samples being taken from the field. A typical laboratory based experiment might involve recording a series of four component mixtures at five concentration levels.

A recommended strategy is as follows:

1.perform a calibration experiment, by producing a set of mixtures on a series of compounds of known concentrations to give a ‘training set’;

2.then test this model on an independent set of mixtures called a ‘test set’;

3.finally, use the model on real data to produce predictions.

More detail is described in Chapter 5, Section 5.6. Many brush aside the design of training sets, often employing empirical or random approaches. Some chemometricians recommend huge training sets of several hundred samples so as to get a representative distribution, especially if there are known to be half a dozen or more significant components in a mixture. In large industrial calibration models, such a procedure is often considered important for robust predictions. However, this approach is expensive in time and resources, and rarely possible in routine laboratory studies. More seriously, many instrumental calibration models are unstable, so calibration on Monday might vary significantly from calibration on Tuesday; hence if calibrations are to be repeated at regular intervals, the number of spectra in the training set must be limited. Finally, very ambitious calibrations can take months or even years, by which time the instruments and often the detection methods may have been replaced.

For the most effective calibration models, the nature of the training set must be carefully thought out using rational experimental design. Provided that the spectra are linearly additive, and there are no serious baseline problems or interactions there are standard designs that can be employed to obtain training sets. It is important to recognise that the majority of chemometric techniques for regression and calibration assume linear additivity. In the case where this may not be so, either the experimental conditions can be modified (for example, if the concentration of a compound is too high so that the absorbance does not obey the Beer–Lambert law, the solution is simply diluted) or various approaches for multilinear modelling are required. It is important to recognise that there is a big difference between the application of chemometrics to primarily analytical or physical chemistry, where it is usual to be able to attain conditions of linearity, and organic or biological chemistry (e.g. QSAR), where often this is not possible. The designs in this section are most applicable in the former case.

EXPERIMENTAL DESIGN	71

In calibration it is normal to use several concentration levels to form a model. Indeed, for information on lack-of-fit and so predictive ability, this is essential. Hence two level factorial designs are inadequate and typically four or five concentration levels are required for each compound. However, chemometric techniques are most useful for multicomponent mixtures. Consider an experiment carried out in a mixture of methanol and acetone. What happens if the concentrations of acetone and methanol in a training set are completely correlated? If the concentration of acetone increases, so does that of methanol, and similarly with a decrease. Such an experimental arrangement is shown in Figure 2.25. A more satisfactory design is given in Figure 2.26, in which the two

	0.4
(mM)methanol		0.35
	0.3

0.5

Figure 2.25

Poorly designed calibration experiment

0.4

	(mM)	0.35
	methanol
		0.3

0.5

Figure 2.26

Well designed calibration experiment

0

0.7 0.9

acetone (mM)

0

0.7 0.9

acetone (mM)

72	CHEMOMETRICS

concentrations are completely uncorrelated or orthogonal. In the former design there is no way of knowing whether a change in spectral characteristic results from change in concentration of acetone or of methanol. If this feature is consciously built into the training set and expected in all future samples, there is no problem, but if a future sample arises with a high acetone and low methanol concentration, calibration software will give a wrong answer for the concentration of each component. This is potentially very serious, especially when the result of chemometric analysis of spectral data is used to make decisions, such as the quality of a batch of pharmaceuticals, based on the concentration of each constituent as predicted by computational analysis of spectra. Some packages include elaborate diagnostics for so-called outliers, which may in many cases be perfectly good samples but ones whose correlation structure differs from that of the training set. In this chapter we will emphasize the importance of good design. In the absence of any certain knowledge (for example, that in all conceivable future samples the concentrations of acetone and methanol will be correlated), it is safest to design the calibration set so that the concentrations of as many compounds as possible in a calibration set are orthogonal.

A guideline to designing a series of multicomponent mixtures for calibration is described below.

1.Determine how many components in the mixture (=k) and the maximum and minimum concentration of each component. Remember that, if studied by spectroscopy or chromatography, the overall absorbance when each component is at a maximum should be within the Beer–Lambert limit (about 1.2 AU for safety).

2.Decide how many concentration levels are required each compound (=l), typically four or five. Mutually orthogonal designs are only possible if the number of concentration levels is a prime number or a power of a prime number, meaning that they are possible for 3, 4, 5, 7, 8 and 9 levels but not 6 or 10 levels.

3.Decide on how many mixtures to produce. Designs exist involving N = mlp mixtures, where l equals the number of concentration levels, p is an integer at least equal to 2, and m an integer at least equal to 1. Setting both m and p at their

minimum values, at least 25 experiments are required to study a mixture (of more than one component) at five concentration levels, or l2 at l levels.

4.The maximum number of mutually orthogonal compound concentrations in a mixture design where m = 1 is four for a three level design, five for a four level design and 12 for a five level design, so using five levels can dramatically increase the number of compounds that we can study using calibration designs. We will discuss how to extend the number of mutually orthogonal concentrations below. Hence choose the design and number of levels with the number of compounds of interest in mind.

The method for setting up a calibration design will be illustrated by a five level, eight compound, 25 experiment, mixture. The theory is rather complicated so the design will be presented as a series of steps.

1.The first step is to number the levels, typically from −2 (lowest) to +2 (highest), corresponding to coded concentrations, e.g. the level −2 = 0.7 mM and level +2 = 1.1 mM; note that the concentration levels can be coded differently for each component in a mixture.

EXPERIMENTAL DESIGN	73

−2

−1

1

2

Figure 2.27

Cyclic permuter

2.Next, choose a repeater level, recommended to be the middle level, 0. For a 5 level design, and 7 to 12 factors (=components in a mixture), it is essential that this is 0. The first experiment is at this level for all factors.

3.Third, select a cyclical permuter for the remaining (l − 1) levels. This relates each

of these four levels as will be illustrated below; only certain cyclic generators can be used, namely −2 −−→ −1 −−→ 2 −−→ 1 −−→ −2 and −2 −−→ 1 −−→ 2 −−→ −1 −−→ −2 which have the property that factors j and j + l + 1 are orthogonal (these are listed in Table 2.30, as discussed below). For less than l + 2 (=7) factors, any permuter can be used so long as it includes all four levels. One such permuter is illustrated in Figure 2.27, and is used in the example below.

4.Finally, select a difference vector; this consists of l − 1 numbers from 0 to l − 2,

arranged in a particular sequence (or four numbers from 0 to 3 in this example). Only a very restricted set of such vectors are acceptable of which {0 2 3 1} is one. The use of the difference vector will be described below.

5.Then generate the first column of the design consisting of l2 (=25) levels in this case, each level corresponding to the concentration of the first compound in the mixture in each of 25 experiments.

(a)	The first experiment is at the repeater level for each factor.
(b)	The l − 1 (=4) experiments 2, 8, 14 and 20 are at the repeater level (=0 in
	this case). In general, the experiments 2, 2 + l + 1, 2 + 2(l + 1) up to 2 + (l −
	1) × (l + 1) are at this level. These divide the columns into “blocks” of 5 (=l)
	experiments.

(c)Now determine the levels for the first block, from experiments 3 to 7 (or in general, experiments 3 to 2 + l). Experiment 3 can be at any level apart from the repeater. In the example below, we use level −2. The key to determining the levels for the next four experiments is the difference vector. The conditions

for the fourth experiment are obtained from the difference vector and cyclic generator. The difference vector {0 2 3 1} implies that the second experiment of the block is zero cyclical differences away from the third experiment or

74	CHEMOMETRICS

−2 using the cyclic permuter of Figure 2.27. The next number in the difference vector is 2, making the fifth experiment at level 2 which is two cyclic differences from −2. Continuing, the sixth experiment is three cyclic differences from the fifth experiment or at level −1, and the final experiment of the block is at level 2.

(d)For the second block (experiments 9 to 13), simply shift the first block by one cyclic difference using the permuter of Figure 2.27 and continue until the last (or fourth) block is generated.

6.Then generate the next column of the design as follows:

(a)the concentration level for the first experiment is always at the repeater level;

(b)the concentration for the second experiment is at the same level as the third experiment of the previous column, up to the 24th [or (l2 − 1)th etc.] experiment;

(c)the final experiment is at the same level as the second experiment for the previous column.

7.Finally, generate successive columns using the principle in step 6 above.

The development of the design is illustrated in Table 2.29. Note that a full five level factorial design for eight compounds would require 58 or 390 625 experiments, so there has been a dramatic reduction in the number of experiments required.

There are a number of important features to note about the design in Table 2.29.

• In each column there are an equal number of −2, −1, 0, +1 and +2 levels.

Table 2.29 Development of a multilevel partial factorial design.

9 10 11 12

13

14 15 16

17

18

19

20 21 22 23 24

25

EXPERIMENTAL DESIGN	75

(a) Factors 1 vs 2

2

1

		0
–2	–1	0	1	2

–1

–2

(b) Factors 1 vs 7

Figure 2.28

Graph of factor levels for the design in Table 2.29

•Each column is orthogonal to every other column, that is the correlation coefficient is 0.

•A graph of the levels of any two factors against each other is given in Figure 2.28(a) for each combination of factors except factors 1 and 7, and 2 and 8, for which a graph is given in Figure 2.28(b). It can be seen that in most cases the levels of any two factors are distributed exactly as they would be for a full factorial design, which would require almost half a million experiments. The nature of the difference vector

is crucial to this important property. Some compromise is required between factors differing by l + 1 (or 6) columns, such as factors 1 and 7. This is unavoidable unless more experiments are performed.

Table 2.30 summarises information to generate some common designs, including the difference vectors and cyclic permuters, following the general rules above for different designs. Look for the five factor design and it can be seen that {0 2 3 1} is one possible difference vector, and also the permuter used above is one of two possibilities.

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Table 2.30 Parameters for construction of a multilevel calibration design.

Levels	Experiments	Max. no.	Repeater	Difference				Cyclic
		of orthogonal		vectors				permuters
		factors
3	9	4	Any	{01}, {10}
4	16	5	Any	{021}, {120}	−2 → −1 → 2			→ 1	→ −2,
5	25	12	0	{0231}, {1320},
				{2013}, {3102}	−2 → 1		→ 2 → −1		→ −2	→ −3,
7	49	16	0	{241 035}, {514 302},	−3	→ 2	→ 3 → −1		→ 1 → −2
				{451 023}, {124 350}, −3		→ 1	→ −1	→ 2	→ 3 → −2	→ −3,
				{530 142}, {203 415}, −3		→ −2 → 3		→ 2	→ −1 → 1	→ −3,
				{320 154}, {053 421}	−3	→ −2 → 1		→ −1 → 3 → 2		→ −3

It is possible to expand the number of factors using a simple trick of matrix algebra. If a matrix A is orthogonal, then the matrix

A A

A −A

is also orthogonal. Therefore, new matrices can be generated from the original orthogonal designs, to expand the number of compounds in the mixture.

2.4 Central Composite or Response Surface Designs

Two level factorial designs are primarily useful for exploratory purposes and calibration designs have special uses in areas such as multivariate calibration where we often expect an independent linear response from each component in a mixture. It is often important, though, to provide a more detailed model of a system. There are two prime reasons. The first is for optimisation – to find the conditions that result in a maximum or minimum as appropriate. An example is when improving the yield of synthetic reaction, or a chromatographic resolution. The second is to produce a detailed quantitative model: to predict mathematically how a response relates to the values of various factors. An example may be how the near-infrared spectrum of a manufactured product relates to the nature of the material and processing employed in manufacturing.

Most exploratory designs do not involve recording replicates, nor do they provide information on squared terms; some, such as Plackett–Burman and highly fractional factorials, do not even provide details of interactions. In the case of detailed modelling it is often desirable at a first stage to reduce the number of factors via exploratory designs as described in Section 2.3, to a small number of main factors (perhaps three or four) that are to be studied in detail, for which both squared and interaction terms in the model are of interest.

2.4.1 Setting Up the Design

Many designs for use in chemistry for modelling are based on the central composite design (sometimes called a response surface design), the main principles of which will be illustrated via a three factor example, in Figure 2.29 and Table 2.31. The first step,

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77

Fractional	Full Factorial	Star	Replication	Central
Factorial				Composite

Figure 2.29

Elements of a central composite design, each axis representing one factor

of course, is to code the factors, and it is always important to choose sensible physical values for each of the factors first. It is assumed that the central point for each factor is 0, and the design is symmetric around this. We will illustrate the design for three factors, which can be represented by points on a cube, each axis corresponding to a factor. A central composite design is constructed as several superimposed designs.

•The smallest possible fractional factorial three factor design consists of four experiments, used to estimate the three linear terms and the intercept. Such as design will not provide estimates of the interactions, replicates or squared terms.

•Extending this to eight experiments provides estimates of all interaction terms. When

represented by a cube, these experiments are placed on the eight corners, and are consist of a full factorial design. All possible combinations of +1 and −1 for the three factors are observed.

•Another type of design, often designated a star design, can be employed to estimate

the squared terms. In order to do this, at least three levels are required for each factor, often denoted by +a, 0 and −a, with level 0 being in the centre. The reason

for this is that there must be at least three points to fit a quadratic. Points where one factor is at level +a are called axial points. Each axial point involves setting one factor at level ±a and the remaining factors at level 0. One simple design sets a equal to 1, although, as we will see below, this value of the axial point is not always recommended. For three factors, a star design consists of the centre point, and six in the centre (or above) each of the six faces of the cube.

•Finally it is often useful to be able estimate the experimental error (as discussed in Section 2.2.2), and one method is to perform extra replicates (typically five) in the centre. Obviously other approaches to replication are possible, but it is usual to replicate in the centre and assume that the error is the same throughout the response surface. If there are any overriding reasons to assume that heteroscedasticity of errors has an important role, replication could be performed at the star or factorial points. However, much of experimental design is based on classical statistics where there is no real detailed information about error distributions over an experimental domain, or at least obtaining such information would be unnecessarily laborious.

•Performing a full factorial design, a star design and five replicates, results in 20

experiments. This design is a type of central composite design. When the axial or star points are situated at a = ±1, the design is sometimes called a face centred cube design (see Table 2.31).

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	Table 2.31 Construction of a
	central composite design.
	Fractional factorial
1		1	1
1		−1	−1
	−1	−1	1
	−1	1	−1
	Full factorial
1		1	1
1		1	−1
1		−1	1
1		−1	−1
	−1	1	1
	−1	1	−1
	−1	−1	1
	−1	−1	−1
	Star		−1
0		0
0		0	1
0		1	0
0		−1	0
1		0	0
	−1	0	0
0		0	0
	Replication in centre
0		0	0
0		0	0
0		0	0
0		0	0
0		0	0
	Central composite
1		1	1
1		1	−1
1		−1	1
1		−1	−1
	−1	1	1
	−1	1	−1
	−1	−1	1
	−1	−1	−1
0		0	−1
0		0	1
0		1	0
0		−1	0
1		0	0
	−1	0	0
0		0	0
0		0	0
0		0	0
0		0	0
0		0	0
0		0	0