Brereton Chemometrics

EXPERIMENTAL DESIGN	59

An important advantage of two level factorial designs is that some factors can be ‘categorical’ in nature, that is, they do not need to refer to a quantitative parameter. One factor may be whether a reaction mixture is stirred (+ level) or not (− level), and another whether it is carried out under nitrogen or not. Thus these designs can be used to ask qualitative questions. The values of the b parameters relate directly to the significance or importance of these factors and their interactions.

Two level factorial designs can be used very effectively for screening, but also have pitfalls.

•They only provide an approximation within the experimental range. Note that for the model above it is possible to obtain nonsensical predictions of negative percentage yields outside the experimental region.

•They cannot take quadratic terms into account, as the experiments are performed only at two levels.

•There is no replicate information.

•If all possible interaction terms are taken into account no error can be estimated, the F -test and t-test not being applicable. However, if it is known that some interactions are unlikely or irrelevant, it is possible to model only the most important factors. For example, in the case of the design in Table 2.18, it might be decided to model only the intercept, four single factor and six two factor interaction terms, making 11 terms in total, and to ignore the higher order interactions. Hence,

•N = 16;

•P = 11;

•(N − P ) = 5 terms remain to determine the fit to the model.

Some valuable information about the importance of each term can be obtained under such circumstances. Note, however, the design matrix is no longer square, and it is not possible to use the simple approaches above to calculate the effects, regression using the pseudo-inverse being necessary.

However, two level factorial designs remain popular largely because they are extremely easy to set up and understand; also, calculation of the coefficients is very straightforward. One of the problems is that once there are a significant number of factors involved, it is necessary to perform a large number of experiments: for six factors, 64 experiments are required. The ‘extra experiments’ really only provide information about the higher order interactions. It is debatable whether a six factor, or even four factor, interaction is really relevant or even observable. For example, if an extraction procedure is to be studied as a function of (a) whether an enzyme is present or not, (b) incubation time, (c) incubation temperature, (d) type of filter, (e) pH and

(f) concentration, what meaning will be attached to the higher order interactions, and even if they are present can they be measured with any form of confidence? And is it economically practicable or sensible to spend such a huge effort studying these interactions? Information such as squared terms is not available, so detailed models of the extraction behaviour are not available either, nor is any replicate information being gathered. Two improvements are as follows. If it is desired to reduce the number experiments by neglecting some of the higher order terms, use designs discussed in Sections 2.3.2 and 2.3.3. If it is desired to study squared or higher order terms whilst reducing the number of experiments, use the designs discussed in Section 2.4.

Sometimes it is not sufficient to study an experiment at two levels. For example, is it really sufficient to use only two temperatures? A more detailed model will be

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+1

0

−1
−1	0	+1

+1.5

0.5

−0.5

−1.5
−1.5	−0.5	+0.5	+1.5

Figure 2.22

Three and four level full factorial designs

obtained using three temperatures; in addition, such designs either will allow the use of squared terms or, if only linear terms used in the model, some degrees of freedom will be available to assess goodness-of-fit. Three and four level designs for two factors are presented in Figure 2.22 with the values of the coded experimental conditions in Table 2.20. Note that the levels are coded to be symmetrical around 0, and so that each level differs by one from the next. These designs are called multilevel factorial designs. The number of experiments can become very large if there are several factors, for example, a five factor design at three levels involves 35 or 243 experiments. In Section 2.3.4 we will discuss how to reduce the size safely and in a systematic manner.

2.3.2 Fractional Factorial Designs

A weakness of full factorial designs is the large number of experiments that must be performed. For example, for a 10 factor design at two levels, 1024 experiments are

EXPERIMENTAL DESIGN		61
	Table 2.20	Full factorial
	designs corresponding to
	Figure 2.22.
	(a) Three levels
	−1	−1
	−1	0
	−1	+1
0		−1
0		0
0		+1
+1		−1
+1		0
+1		+1
	(b) Four levels
	−1.5	−1.5
	−1.5	−0.5
	−1.5	+0.5
	−1.5	+1.5
	−0.5	−1.5
	−0.5	−0.5
	−0.5	+0.5
	−0.5	+1.5
+0.5		−1.5
+0.5		−0.5
+0.5		+0.5
+0.5		+1.5
+1.5		−1.5
+1.5		−0.5
+1.5		+0.5
	+1.5	+1.5

required, which may be impracticable. These extra experiments do not always result in useful or interesting extra information and so are wasteful of time and resources. Especially in the case of screening, where a large number of factors may be of potential interest, it is inefficient to run so many experiments in the first instance. There are numerous tricks to reduce the number of experiments.

Consider a three factor, two level design. Eight experiments are listed in Table 2.21, the conditions being coded as usual. Figure 2.23 is a symbolic representation of the experiments, often presented on the corners of a cube, whose axes correspond to each factor. The design matrix for all the possible coefficients can be set up as is also illustrated in Table 2.21 and consists of eight possible columns, equal to the number of experiments. Some columns represent interactions, such as the three factor interaction, that are not very likely. At first screening we may primarily wish to say whether the three factors have any real influence on the response, not to study the model in detail. In a more complex situation, we may wish to screen 10 possible factors, and reducing the number of factors to be studied further to three or four makes the next stage of experimentation easier.

How can we reduce the number of experiments safely and systematically? Two level fractional factorial designs are used to reduce the number of experiments by 1/2, 1/4, 1/8 and so on. Can we halve the number of experiments? At first glance, a simple

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Table 2.21 Full factorial design for three factors together with the design matrix.

Experiment	Factor 1	Factor 2	Factor 3	Design matrix
No.
				x0 x1 x2 x3 x1x2 x1x3 x2x3	x1x2x3
1	+	+	+	+ + + + + + +	+
2	+	+	−	+ + + − + − −	−
3	+	−	+	+ + − + − + −	−
4	+	−	−	+ + − − − − +	+
5	−	+	+	+ − + + − − +	−
6	−	+	−	+ − + − − + −	+
7	−	−	+	+ − − + + − −	+
8	−	−	−	+ − − − + + +	−

Factor 2

5	1
6	2
Factor 3
7	3
8	4
Factor 1

Figure 2.23

Representation of a three factor, two level design

approach might be to take the first four experiments of Table 2.21. However, these would leave the level of the first factor at +1 throughout. A problem is that we now no longer study the variation of this factor, so we do not obtain any information on how factor 1 influences the response, and are studying the wrong type of variation, in fact such a design would remove all four terms from the model that include the first factor, leaving the intercept, two single factor terms and the interaction between factors 2 and 3, not the hoped for information, unless we know that factor 1 and its interactions are insignificant.

Can a subset of four experiments be selected that allows us to study all three factors? Rules have been developed to produce these fractional factorial designs obtained by taking the correct subset of the original experiments. Table 2.22 illustrates a possible fractional factorial design that enables all factors to be studied. There are a number of important features:

EXPERIMENTAL DESIGN						63
	Table 2.22 Fractional factorial design.
	Experiment	Factor 1	Factor 2	Factor 3	Matrix of effects
	No.
					x0 x1 x2 x3 x1x2 x1x3 x2x3	x1x2x3
1		+	+	+	+ + + + + + +	+
2		+	−	−	+ + − − − − +	+
3		−	−	+	+ − − + + − −	+
	4	−	+	−	+ − + − − + −	+

Factor 2

1

4

Factor 3

3

2

Factor 1

Figure 2.24

Fractional factorial design

•every column in the experimental matrix is different;

•in each column, there are an equal number of − and + levels;

•for each experiment at level + for factor 1, there are equal number of experiments for factors 2 and 3 which are at levels + and −, and the columns are orthogonal.

The properties of this design can be understood better by visualisation (Figure 2.24): half the experiments have been removed. For the remainder, each face of the cube now corresponds to two rather than four experiments, and every alternate corner corresponds to an experiment.

The matrix of effects in Table 2.22 is also interesting. Whereas the first four columns are all different, the last four each correspond to one of the first four columns. For example, the x1x2 column exactly equals the x3 column. What does this imply in reality? As the number of experiments is reduced, the amount of information is correspondingly reduced. Since only four experiments are now performed, it is only possible to measure four unique factors. The interaction between factors 1 and 2 is said to be confounded with factor 3. This might mean, for example, that, using this design the interaction between temperature and pH is indistinguishable from the influence of concentration alone. However, not all interactions will be significant, and the purpose of

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a preliminary experiment is often simply to sort out which main factors should be studied in detail later. When calculating the effects, it is important to use only four unique columns in the design matrix, rather than all eight columns, as otherwise the matrix will not have an inverse.

Note that two level fractional factorial designs only exist when the number of experiments equals a power of 2. In order to determine the minimum number of experiments do as follows:

•determine how many terms are interesting;

•then construct a design whose size is the next greatest power of 2.

Setting up a fractional factorial design and determining which terms are confounded is relatively straightforward and will be illustrated with reference to five factors.

A half factorial design involves reducing the experiments from 2k to 2k−1 or, in this case, from 32 to 16.

1.In most cases, the aim is to

•confound k factor interactions with the intercept;

•(k − 1) factor interactions with single factor interactions;

•up to (k − 1)/2 factor interactions with (k − 1)/2 + 1 factor interactions if the number of factors is odd, or k/2 factor interactions with themselves if the number of factors is even.

i.e. for five factors, confound 0 factor interactions (intercept) with 5, 1 factor interactions (pure variables) with 4, and 2 factor interactions with 3 factor interactions, and for six factors, confound 0 with 6, 1 with 5, 2 with 4 factor interactions, and 3 factor interactions with themselves.

2.Set up a k − 1 factor design for the first k − 1 factors, i.e. a 4 factor design consisting of 16 experiments.

3.Confound the kth (or final) factor with the product of the other factors by setting the final column as either − or + the product of the other factors. A simple notation is often used to analyse these designs, whereby the final column is given by k = +1 2 . . . (k − 1) or k = −1 2 . . . (k − 1). The case where 5 = +1 2 3 4 is illustrated in Table 2.23. This means that a four factor interaction (most unlikely to have any physical meaning) is confounded with the fifth factor. There are, in fact, only two different types of design with the properties of step 1 above. Each

design is denoted by how the intercept (I) is confounded, and it is easy to show that this design is of the type I = +1 2 3 4 5, the other possible design being of type I = −1 2 3 4 5. Table 2.23, therefore, is one possible half factorial design for five factors at two levels.

4.It is possible to work out which of the other terms are confounded with each other,

either by multiplying the columns of the design together or from first principles, as follows. Every column multiplied by itself will result in a column of + signs or I as the square of either −1 or +1 is always +1. Each term will be confounded with one other term in this particular design. To demonstrate which term 1 2 3 is confounded with, simply multiply 5 by 4 since 5 = 1 2 3 4, so 5 4 = 1 2 3 4 4 = 1 2 3 since 4 4 equals I. These interactions for the design of Table 2.23 are presented in Table 2.24.

5.In the case of negative numbers, ignore the negative sign. If two terms are correlated, it does not matter if the correlation coefficient is positive or negative, they cannot

EXPERIMENTAL DESIGN							65
	Table 2.23 Confounding factor 5 with the product of factors 1–4.
	Factor 1	Factor 2		Factor 3	Factor 4		Factor 5
1		2		3	4		+1 2 3 4
	−		−	−	−		+
	−		−	−	+		−
	−		−	+	−		−
	−		−	+	+		+
	−	+		−	−		−
	−	+		−	+		+
	−	+		+	−		+
	−	+		+	+		−
+			−	−	−		−
+			−	−	+		+
+			−	+	−		+
+			−	+	+		−
+		+		−	−		+
+		+		−	+		−
+		+		+	−		−
+		+		+	+		+
			Table 2.24 Confounding interac-
			tion terms in design of Table 2.23.
			I		+1 2 3 4 5
		1			+2 3 4 5
		2			+1 3 4 5
		3			+1 2 4 5
		4			+1 2 3 5
		5			+1 2 3 4
			1 2		+3 4 5
			1 3		+2 4 5
			1 4		+2 3 5
			1 5		+2 3 4
			2 3		+1 4 5
			2 4		+1 3 5
			2 5		+1 3 4
			3 4		+1 2 5
			3 5		+1 2 4
			4 5		+1 2 3

be distinguished. In practical terms, this implies that if one term increases the other decreases.

Note that it is possible to obtain other types of half factorials, but these may involve, for example, confounding single factor terms with two factor interactions.

A smaller factorial design can be constructed as follows.

1.For a 2−f fractional factorial, first set up the design consisting of 2k−f experiments for the first k − f factors, i.e. for a quarter (f = 2) of a five (=k) factorial experiment, set up a design consisting of eight experiments for the first three factors.

2.Determine the lowest order interaction that must be confounded. For a quarter of a five factorial design, second-order interactions must be confounded. Then, almost

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	Table 2.25 Quarter factorial design.
	Factor 1	Factor 2	Factor 3	Factor 4	Factor 5
	1	2	3	−1 2	1 2 3
	−	−	−	−	−
	−	−	+	−	+
	−	+	−	+	+
	−	+	+	+	−
+		−	−	+	+
+		−	+	+	−
+		+	−	−	−
	+	+	+	−	+

arbitrarily (unless there are good reasons for specific interactions to be confounded) set up the last two columns as products (times − or +) of combinations of the other columns, with the proviso that the products must include as least as many terms as the lowest order interaction to be confounded. Therefore, for our example, any two factor (or higher) interaction is entirely valid. In Table 2.25, a quarter factorial design where 4 = −1 2 and 5 = 1 2 3 is presented.

3.Confounding can be analysed as above, but now each term will be confounded with three other terms for a quarter factorial design (or seven other terms for an eighth factorial design).

In more complex situations, such as 10 factor experiments, it is unlikely that there will be any physical meaning attached to higher order interactions, or at least that these interactions are not measurable. Therefore, it is possible to select specific interactions that are unlikely to be of interest, and consciously reduce the experiments in a systematic manner by confounding these with lower order interactions.

There are obvious advantages in two level fractional factorial designs, but these do have some drawbacks:

•there are no quadratic terms, as the experiments are performed only at two levels;

•there are no replicates;

•the number of experiments must be a power of two.

Nevertheless, this approach is very popular in many exploratory situations and has the additional advantage that the data are easy to analyse. It is important to recognise, however, that experimental design has a long history, and a major influence on the minds of early experimentalists and statisticians has always been ease of calculation. Sometimes extra experiments are performed simply to produce a design that could be readily analysed using pencil and paper. It cannot be over-stressed that inverse matrices were very difficult to calculate manually, but modern computers now remove this difficulty.

2.3.3 Plackett–Burman and Taguchi Designs

Where the number of factors is fairly large, the constraint that the number of experiments must equal a power of 2 can be rather restrictive. Since the number of experiments must always exceed the number of factors, this would mean that 32 experiments are required for the study of 19 factors, and 64 experiments for the study of 43 factors. In order

EXPERIMENTAL DESIGN	67

Table 2.26 A Plackett–Burman design.

to overcome this problem and reduce the number of experiments, other approaches are needed.

Plackett and Burman published their classical paper in 1946, which has been much cited by chemists. Their work originated from the need for war-time testing of components in equipment manufacture. A large number of factors influenced the quality of these components and efficient procedures were required for screening. They proposed a number of two level factorial designs, where the number of experiments is a multiple of four. Hence designs exist for 4, 8, 12, 16, 20, 24, etc., experiments. The number of experiments exceeds the number of factors, k, by one.

One such design is given in Table 2.26 for 11 factors and 12 experiments and has various features.

•In the first row, all factors are at the same level.

•The first column from rows 2 to k is called a generator. The key to the design is that there are only certain allowed generators which can be obtained from tables. Note

that the number of factors will always be an odd number equal to k = 4m − 1 (or 11 in this case), where m is any integer. If the first row consists of −, the generator will consist of 2m (=6 in this case) experiments at the + level and 2m − 1 (=5 in this case) at the − level, the reverse being true if the first row is at the + level. In Table 2.26, the generator is + + − + + + + − − − +−.

•The next 4m − 2 (=10) columns are generated from the first column simply by shifting the down cells by one row. This is indicated by diagonal arrows in the table. Notice that experiment 1 is not included in this procedure.

•The level of factor j in experiment (or row) 2 equals to the level of this factor in

row k for factor j − 1. For example, the level of factor 2 in experiment 2 equals the level of factor 1 in experiment 12.

There are as many high as low levels of each factor over the 12 experiments, as would be expected. The most important property of the design, however, is orthogonality. Consider the relationship between factors 1 and 2.

•There are six instances in which factor 1 is at a high level and six at a low level.

•For each of the six instances at which factor 1 is at a high level, in three cases factor 2 is at a high level, and in the other three cases it is at a low level. A similar

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	Table 2.27 Generators for Plackett–Burman design; first row is at – level.
	Factors	Generator
7		+ + + − + − −
11		+ + − + + + − − − + −
15		+ + + + − + − + + − − + − − −
19		+ + − + + + + − + − + − − − − + +−
	23	+ + + + + − + − + + − − + + − − + − + − − − −

relationship exists where factor 1 is at a low level. This implies that the factors are orthogonal or uncorrelated, an important condition for a good design.

• Any combination of two factors is related in a similar way.

Only certain generators possess all these properties, so it is important to use only known generators.

Standard Plackett–Burman designs exist for 7, 11, 15, 19 and 23 factors; generators are given in Table 2.27. Note that for 7 and 15 factors it is also possible to use conventional fractional factorial designs as discussed in Section 2.3.2. However, in the old adage all roads lead to Rome, in fact fractional factorial and Plackett–Burman designs are equivalent, the difference simply being in the way the experiments and factors are organised in the data table. In reality, it should make no difference in which order the experiments are performed (in fact, it is best if the experiments are run in a randomised order) and the factors can be represented in any order along the rows. Table 2.28 shows that for 7 factors, a Plackett–Burman design is the same as a sixteenth factorial (=27−4 = 8 experiments), after rearranging the rows, as indicated by the arrows. The confounding of the factorial terms is also indicated. It does not really matter which approach is employed.

If the number of experimental factors is less that of a standard design (a multiple of 4 minus 1), the final factors are dummy ones. Hence if there are only 10 real factors, use an 11 factor design, the final factor being a dummy one: this may be a variable that has no effect on the experiment, such as the technician that handed out the glassware or the colour of laboratory furniture.

If the intercept term is included, the design matrix is a square matrix, so the coefficients for each factor are given by

b = D−1.y

Table 2.28 Equivalence of Plackett–Burman and fractional factorial design for seven factors.