Brereton Chemometrics

•The value is always greater than 0.

•The lower the value, the higher is the confidence in the prediction. A value of 1 indicates very poor prediction. A value of 0 indicates perfect prediction and will not be achieved.

•If there are P coefficients in the model, the sum of the values for leverage at each experimental point adds up to P . Hence the sum of the values of leverage for the 12 experiments in Table 2.14 is equal to 6.

In the design in Table 2.14, the leverage is lowest at the centre, as expected. However, the value of leverage for the first four points in slightly lower than that for the second four points. As discussed in Section 2.4, this design is a form of central composite design, with the points 1–4 corresponding to a factorial design and points 5–8 to a star design. Because the leverage, or confidence, in the model differs, the design is said to be nonrotatable, which means that the confidence is not solely a function of distance from the centre of experimentation. How to determine rotatability of designs is discussed in Section 2.4.3.

Leverage can also be converted to equation form simply by substituting the algebraic expression for the coefficients in the equation

h = d.(D .D)−1.d

where, in the case of Table 2.14,

d = ( 1 x1 x2 x12 x22 x1x2 )

to give an equation, in this example, of the form

h = 0.248 − 0.116(x12 + x22) + 0.132(x14 + x24) + 0.316x12x22

The equation can be obtained by summing the appropriate terms in the matrix (D .D)−1. This is illustrated graphically in Figure 2.16. Label each row and column by the corresponding terms in the model, and then find the combinations of terms in the matrix

Figure 2.16

Method of calculating equation for leverage term x12x22

that result in the coefficients of the leverage equation, for x12x22 there are three such combinations so that the term 0.316 = 0.250 + 0.033 + 0.033.

This equation can also be visualised graphically and used to predict the confidence of any point, not just where experiments were performed. Leverage can also be used to predict the confidence in the prediction under any conditions, which is given by

y± = s [F(1,N −P )(1 + h)]

√

where s is the root mean square residual error given by Sresid /(N − P ) as determined in Section 2.2.2, the F -statistic as introduced in Section 2.2.4.4, which can be obtained at any desired level of confidence but most usually at 95 % limits and is one-sided. Note that this equation technically refers to the confidence in the individual prediction and there is a slightly different equation for the mean response after replicates have been averaged given by

See Section 2.2.1 for definitions of N and P ; M is the number of replicates taken at a given point, for example if we repeat the experiment at 10 mM five times, M = 5. If repeated only once, then this equation is the same as the first one.

Although the details of these equations may seem esoteric, there are two important considerations:

•the shape of the confidence bands depends entirely on leverage;

•the size depends on experimental error.

These and most other equations developed by statisticians assume that the experimental error is the same over the entire response surface: there is no satisfactory agreement for how to incorporate heteroscedastic errors. Note that there are several different equations in the literature according to the specific aims of the confidence interval calculations, but for brevity we introduce only two which can be generally applied to most situations.

Table 2.15 Leverage for three possible single variable designs using a two parameter linear model.

Factorial designs are some of the simplest, often used for screening or when there are a large number of possible factors. As will be seen, they have limitations, but are the easiest to understand. Many designs are presented as a set of rules which provide the experimenter with a list of conditions, and below we will present the rules for many of the most common methods.

2.3.1 Full Factorial Designs

Full factorial designs at two levels are mainly used for screening, that is, to determine the influence of a number of effects on a response, and to eliminate those that are not significant, the next stage being to undertake a more detailed study. Sometimes, where detailed predictions are not required, the information from factorial designs is adequate, at least in situations where the aim is fairly qualitative (e.g. to improve the yield of a reaction rather than obtain a highly accurate rate dependence that is then interpreted in fundamental molecular terms).

Consider a chemical reaction, the performance of which is known to depend on pH and temperature, including their interaction. A set of experiments can be proposed to study these two factors, each at two levels, using a two level, two factor experimental design. The number of experiments is given by N = lk , where l is the number of levels (=2), and k the number of factors (=2), so in this case N = 4. For three factors, the number of experiments will equal 8, and so on, provided that the design is performed at two levels only. The following stages are used to construct the design and interpret the results.

1.The first step is to choose a high and low level for each factor, for example, 30◦ and 60◦, and pH 4 and 6.

2.The next step is to use a standard design. The value of each factor is usually coded (see Section 2.2.4.1) as − (low) or + (high). Note that some authors use −1 and +1 or even 1 and 2 for low and high. When reading different texts, do not get confused: always first understand what notation has been employed. There is no universally agreed convention for coding; however, design matrices that are symmetric around 0 are almost always easier to handle computationally. There are four possible unique

sets of experimental conditions which can be represented as a table analogous to four binary numbers, 00 (−−), 01 (−+), 10 (+−) and 11 (++), which relate to a set of physical conditions.

3.Next, perform the experiments and obtain the response. Table 2.16 illustrates the coded and true set of experimental conditions plus the response, which might, for example be the percentage of a by-product, the lower the better. Something immediately appears strange from these results. Although it is obvious that the higher

Table 2.16 Coding of a simple two factor, two level design and the response.

the temperature, the higher is the percentage by-product, there does not seem to be any consistent trend as far as pH is concerned. Provided that the experimental results were recorded correctly, this suggests that there must be an interaction between temperature and pH. At a lower temperature, the percentage decreases with increase in pH, but the opposite is observed at a higher temperature. How can we interpret this?

4.The next step, of course, is to analyse the data, by setting up a design matrix (see Section 2.2.3). We know that an interaction term must be taken into account,

and set up a design matrix as given in Table 2.17 based on a model of the form y = b0 + b1x1 + b2x2 + b11x1x2. This can be expressed either as a function of the true or coded concentrations, but, as discussed in Section 2.2.4.1, is probably best as coded values. Note that four possible coefficients that can be obtained from the four experiments. Note also that each of the columns in Table 2.17 is different. This is an important and crucial property and allows each of the four possible terms to be distinguished uniquely from one another, and is called orthogonality. Observe, also, that squared terms are impossible because four experiments can be used to obtain only a maximum of four terms, and also the experiments are performed at only two levels; ways of introducing such terms will be described in Section 2.4.

5.Calculate the coefficients. It is not necessary to employ specialist statistical software for this. In matrix terms, the response can be given by y = D.b, where b is a vector

of the four coefficients and D is presented in Table 2.17. Simply use the matrix inverse so that b = D−1.y. Note that there are no replicates and the model will exactly fit the data. The parameters are listed below.

•For raw values: intercept = 10

temperature coefficient = 0.2 pH coefficient = −2.5 interaction term = 0.05

•For coded values:

intercept = 17.5

temperature coefficient = 6.75 pH coefficient = −0.25 interaction term = 0.75

6.Finally, interpret the coefficients. Note that for the raw values, it appears that pH is much more influential than temperature, and also that the interaction is very small. In addition, the intercept term is not the average of the four readings. The

reason why this happens is that the intercept is the predicted response at pH 0 and 0 ◦C, conditions unlikely to be reached experimentally. The interaction term

appears very small, because units used for temperature correspond to a range of 30 ◦C as opposed to a pH range of 2. A better measure of significance comes from

Table 2.18 Four factor, two level full factorial design.

(a) Coded experimental conditions

(b) Design matrix

x0 x1 x2 x3 x4 x1x2 x1x3 x1x4 x2x3 x2x4 x3x4 x1x2x3 x1x2x4 x1x3x4 x2x3x4 x1x2x3x4

where there are N experiments. This a general feature of all saturated two level designs, and relates to an interesting classical approach for determining the size of each effect. Using modern matrix notation, the simplest method is simply to calculate

b = D−1.y

but many classical texts use an algorithm involving multiplying the response by each column (of coded coefficients) and dividing by the number of experiments to obtain the value of the size of each factor. For the example in Table 2.16, the value of the effect due to temperature can be given by

b1 = (−1 × 12 − 1 × 10 + 1 × 24 + 1 × 25)/4 = 6.75

Figure 2.21

Graph of levels of one term against another in the design in Table 2.18

identical with that obtained using simply matrix manipulations. Such method for determining the size of the effects was extremely useful to classical statisticians prior to matrix oriented software, and still widely used, but is limited only to certain very specific designs. It is also important to recognise that some texts divide the expression above by N /2 rather than N , making the classical numerical value of the effects equal to twice those obtained by regression. So long as all the effects are on the same scale, it does not matter which method is employed when comparing the size of each factor.