Marco teórico

The simplest example of the consideration of multiple experimental fac- tors would involve two factors. Taking the earlier example of a chemical synthesis, suppose that we were interested in the effect of two different reaction temperatures, T1 and T2, and two different solvents, S1 and S2, on the yield of the reaction. The minimum number of experiments required to give us information on both factors is three, one at T1S1 (y1), a second at T1S2 (y2) involving change in solvent, and a third at T2S1 (y3) involving a change in temperature (see Table 2.3). The effect of changing temperature is given by the difference in yields y3−y1 and the effect of changing solvent is given by y2−y1. Confirmation of these results could be obtained by duplication of the above requiring a total of six experiments.

This is a ‘one variable at a time’ approach since each factor is exam- ined separately. However, if a fourth experiment, T2S2 (y4), is added to Table 2.3 we now have two measures of the effect of changing each factor but only require four experiments. In addition to saving two exper- imental determinations, this approach allows the detection of interaction effects between the factors, such as the effect of changing temperature in solvent 2 (y4−y2) compared with solvent 1 (y3−y1). The factorial approach is not only more efficient in terms of the number of experi- ments required and the identification of interaction effects, it can also be useful in optimization. For example, having estimated the main effects and interaction terms of some experimental factors it may be possible to predict the likely combinations of these factors which will give an optimum response. One drawback to this procedure is that it may not always be possible to establish all possible combinations of treatments, resulting in an unbalanced design. Factorial designs also tend to involve a large number of experiments, the investigation of three factors at three levels, for example, requires 27 runs (3f_{where f is the number of factors)}

without replication of any of the combinations. However, it is possible to reduce the number of experiments required as will be shown later.

A nice example of the use of factorial design in chemical synthesis was published by Coleman and co-workers [1]. The reaction of 1,1,1-trichloro-3-methyl-3-phospholene (1) with methanol produces 1-methoxy-3-methyl-2-phospholene oxide (2) as shown in the reaction scheme. The experimental procedure involved the slow addition of a known quantity of methanol to a known quantity of 1 in dichloromethane held at subambient temperature. The mixture was then stirred until it reached ambient temperature and neutralized with aqueous sodium carbonate solution; the product was extracted with dichloromethane.

Scheme 2.1 .

The yield from this reaction was 25 % and could not significantly be improved by changing one variable (concentration, temperature, addi- tion time, etc.) at a time. Three variables were chosen for investigation by factorial design using two levels of each.

A: Addition temperature (−15 or 0◦C)

B: Concentration of 1 (50 or 100 g in 400 cm3_{dichloromethane)}

C: Addition time of methanol (one or four hours)

This led to eight different treatments (23_{), which resulted in several yields} above 25 % (as shown in Table 2.4), the largest being 42.5 %.

The effect on an experimental response due to a factor is called a main effect whereas the effect caused by one factor at each level of the other factor is called an interaction effect (two way). The larger the number of levels of the factors studied in a factorial design, the higher the order of the interaction effects that can be identified. In a three-level factorial design it is possible to detect quadratic effects although it is often difficult to interpret the information. Three-level factorial designs also require a considerable number of experiments (3f_{) as shown above.}

For this reason it is often found convenient to consider factors at just two levels, high/low or yes/no, to give 2f_{factorial designs.}

Table 2.4 Responses from full factorial design (reproduced from ref. [1] with permission of the Royal Society of Chemistry).

Order of Treatment

treatment combinationa _{Yield (%)}

3 − 24.8 6 a 42.5 1 b 39.0 7 ab 18.2 2 c 32.8 4 ac 33.0 8 bc 13.2 5 abc 24.3

a_{Where a lower-case letter is shown, this indicates that a} particular factor was used at its high level in that treatment, e.g. a means an addition temperature of 0◦C. When a letter is missing the factor was at its low level.

Another feature of these full factorial designs, full in the sense that all combinations of all levels of each factor are considered, is that inter- actions between multiple factors may be identified. In a factorial design with six factors at two levels (26_{= 64 experiments) there are six main ef-} fects (for the six factors), 15 two-factor interactions (two-way effects), 20 three-factor, 15 four-factor, 6 five-factor, and 1 six-factor interactions. Are these interactions all likely to be important? The answer, fortunately, is no. In general, main effects tend to be larger than two-factor interac- tions which in turn tend to be larger than three-factor interactions and so on. Because these higher order interaction terms tend not to be significant it is possible to devise smaller factorial designs which will still investigate the experimental factor space efficiently but which will require far fewer experiments. It is also often found that in factorial designs with many experimental factors, only a few factors are important. These smaller factorial designs are referred to as fractional factorial designs, where the fraction is defined as the ratio of the number of experimental runs needed in a full design. For example, the full factorial design for five factors at two levels requires 32 (25_{) runs: if this is investigated in 16} experiments it is a half-fraction factorial design. Fractional designs may also be designated as 2f−n_{where f is the number of factors as before and} n is the number of half-fractions, 25−1is a half-fraction factorial design in five factors, 26−2is a quarter-fraction design in six factors.

Of course, it is rare in life to get something for nothing and that prin- ciple applies to fractional factorial designs. Although a fractional design

allows one to investigate an experimental system with the expenditure of less effort, it is achieved at the expense of clarity in our ability to sep- arate main effects from interactions. The response obtained from certain treatments could be caused by the main effect of one factor or a two- (three-, four-, five-, etc.) factor interaction. These effects are said to be confounded; because they are indistinguishable from one another, they are also said to be aliases of one another. It is the choice of aliases which lies at the heart of successful fractional factorial design. As mentioned before, we might expect that main effects would be more significant than two-factor effects which will be more important than three-factor effects. The aim of fractional design is thus to alias main effects and two-factor effects with as high-order interaction terms as possible.

The phospholene oxide synthesis mentioned earlier provides a good example of the use of fractional factorial design. Having carried out the full factorial design in three factors (addition temperature, concentra- tion of phospholene, and addition time) further experiments were made to ‘fine-tune’ the response. These probing experiments involved small changes to one factor while the others were held constant in order to determine whether an optimum had been reached in the synthetic con- ditions. Figure 2.1 shows a response surface for the high addition time in which percentage yield is plotted against phospholene concentration and addition temperature. The response surface is quite complex and demonstrates that a maximum yield had not been achieved for the fac- tors examined in the first full factorial design. In fact the largest yield found in these probing experiments was 57 %, a reasonable increase over the highest yield of 42.5 % shown in Table 2.4.

Figure 2.1 Response surface for phospholene oxide synthesis (reproduced from ref. [1] with permission of the Royal Society of Chemistry).

Table 2.5 Responses from fractional factorial design (reproduced from ref. [1] with permission of the Royal Society of Chemistry).

Treatment

combinationa _{Yield Aliasing effect}

− 45.1

ad 60.2 A with BD

bde 62.5 B with CE+AD

abe 46.8 D with AB ce 77.8 C with BE acde 49.8 AC with DE bcd 53.6 E with BC abc 70.8 AE with CD a_{As explained in Table 2.4.}

The shape of the response surface suggests the involvement of other factors in the yield of this reaction and three more experimental variables were identified: concentration of methanol, stirring time, and tempera- ture. Fixing the concentration of phospholene at 25 g in 400 cm3 _of dichloromethane (a broad peak on the response surface) leaves five ex- perimental factors to consider, requiring a total of 32 (25_{) experiments to} investigate them. These experiments were split into four blocks of eight and hence each block is a quarter-fraction of 32 experiments. The results for the first block are shown in Table 2.5, the experimental factors being

A: Addition temperature (−10 or 0◦ C)

B: Addition time of methanol (15 or 30 minutes)

C: Concentration of methanol (136 or 272 cm3₎

D: Stirring time (0.5 or 2 hours)

E: Stirring temperature (addition temperature or ambient)

This particular block of eight runs was generated by aliasing D with AB and also E with BC, after carrying out full 23 _{experiments of A, B,} and C. As can be seen from Table 2.5, the best yield from this principal block of experiments, which contains variables and variable interac- tions expected to be important, was 78 %, a considerable improvement over the previously found best yield of 57 %. Having identified impor- tant factors, or combinations of factors with which they are aliased, it is possible to choose other treatment combinations which will clarify the situation. The best yield obtained for this synthesis was 90 % using treatment combination e (addition temperature−10 ◦C, addition time 15 mins, methanol concentration 136 cm3_{, stirring time 0.5 hours, stir-} ring temperature ambient).