The Three Factor Experiment

Let us consider a hypothetical experiment, in which we wish to investigate the effects of three independent variables P, Q and R (which may be temperatures, pressures, flow rates, concentrations, etc.) upon a dependent variable x (which may be yield, purity, etc.). 

Suppose that in the first instance it will be adequate to investigate them each at only two levels. Thus suppose the normal values for the process are Pi, Qj, and Ri, and we wish to find the effects of increasing them to Pa, Qa and Ra respectively. How would we carry out this experiment in the classical style  

We would first do an experimental control with values Pj Qi Rj. A value of X would be obtained which we denote by (Pj Qi Ri)i. 

Now a very important point is that each of these experiments would have to be repeated not less than once, for without not less than two observations it is quite impossible to make any estimate of experimental error. Without such an estimate it is quite impossible to say whether any apparent difference between, say, (Pi Qi Ri)x and (Pu Qi Ri)x is real or is due to the errors of measurement and sampling, etc. We therefore require our four experiments to be repeated once, making eight observations in all. Each level of each effect will be given by two observations, i.e. we will be comparing the mean of two observations with the mean of two observations. 

No information is given by this experiment as to any possible interactions between the factors. Thus for example, (Ri— R ) with P at Pj may well be different from (Ri — Rjx) with P at Pj, but our experiment is quite unable to detect such an effect. 

---

The computation of the three factor experiment is discussed in Chapter XI (d). 

It will be noted that the four factor experiment is even more efficient relative to the classical design than the three factor experiment, which achieved double the accuracy for the main effects. 

Here we have a three factor analysis with a significant first order interaction, W X B. Accordingly we need to break the three factor experiment down, either into W X H experiments for Bi and for Ba or into B X H experiments for Wi and for Wj. 

If very little is known about a system, the three factors are varied over large intervals this maximizes the chances that large effects will be found with a minimum of experiments, and that an optimal combination of factors is rapidly approached (for example, new analytical method to be created, no boundary conditions to hinder investigator). 

Thus, there are two limitations of the pycnometric technique mentioned possible adsorption of guest molecules and a molecular sieving effect. It is noteworthy that some PSs, e.g., with a core-shell structure, can include some void volume that can be inaccessible to the guest molecules. In this case, the measured excluded volume will be the sum of the true volume of the solid phase and the volume of inaccessible pores. One should not absolutely equalize the true density and the density measured by a pycnometric technique (the pycnometric density) because of the three factors mentioned earlier. Conventionally, presenting the results of measurements one should define the conditions of a pycnometric experiment (at least the type of guest and temperature). For example, the definition p shows that the density was measured at 298 K using helium as a probe gas. Unfortunately, use of He as a pycnometric fluid is not a panacea since adsorption of He cannot be absolutely excluded by some PSs (e.g., carbons) even at 293 K (see van der Plas in Ref. [2]). Nevertheless, in most practically important cases the values of the true and pycnometric densities are very close [2,7],... 

Figure 12.3 is a pseudo-three-dimensional representation of a portion of the three-dimensional experiment space associated with the system shown in Figure 12.1. The two-dimensional factor subspace is shaded with a one-unit grid. The factor domains are again 0 < Xj < +10 and 0 < < +10. The response axis ranges from 0...

The aim of any clinical trial is to have low risk of Type I and II errors and sufficient power to detect a difference between treatments, if it exists. Of the three factors in determining sample size, the power (probability of detecting a true difference) is arbitrarily chosen. The magnitude of the drug s effect can be estimated with more or less accuracy from previous experience with drugs of the same or similar action, and the variability of the measurements is often known from published experiments on the primary endpoint, with or without the drug. These data will, however, not be available for novel substances in a new class and frequently the sample size in the early phase of development has to be chosen on an arbitrary basis. 

The three-factor central composite may be represented as a cube, with a centre point, experiments at each vertex and at the ends of axes radiating out from the centre through the middle of each face. The vertices will be designated as +1 or -1 for each factor. It is suggested that six runs be made at the centre, This design can be represented as indicated below the diagram,... 

As I have shown, the response given by the model equation (3.5) has an error term that includes the lack of fit of the model and dispersion due to the measurement (repeatability). For the three-factor example discussed above, there are four estimates of each effect, and in general the number of estimates are equal to half the number of runs. The variance of these estimated effects gives some indication of how well the model and the measurement bear up when experiments are actually done, if this value can be compared with an expected variance due to measurement alone. There are two ways to estimate measurement repeatability. First, if there are repeated measurements, then the standard deviation of these replicates (s) is an estimate of the repeatability. For N/2 estimates of the factor effect, the standard deviation of the effect is... 

Second Experimental Matrix. For the technical reasons mentioned above, a complete factorial design 23 was chosen for investigating the effects of the three factors DOC0, Ti02 concentration, and temperature (Table 3). An additional experiment at the center of the experimental region (A = XA = X5 = 0 i.e., DOC0 = 2700 ppm, [Ti02] = 2.75 g L 1, tempera-... 

Figure 11.3 is a pseudo-three-dimensional representation of a portion of the three-dimensional experiment space associated with the system shown in Figure 11.1. The two-dimensional factor subspace is shaded with a one-unit grid. The factor domains are again 0 < xl < +10 and 0 < x2 < +10. The response axis ranges from 0 to +8. The location in factor space of the single experiment at xn = +3, x2l = +7 is shown as a point in the plane of factor space. The response (yn = + 4.00) associated with this experiment is shown as a point above the plane of factor space, and is connected to the factor space by a dotted vertical line.

Most experiments result in some sort of model, which is a mathematical way of relating an experimental response to the value or state of a number of factors. An example of a response is the yield of a synthetic reaction the factors may be the pH, temperature and catalyst concentration. An experimenter wishes to run a reaction under a given set of conditions and predict the yield. How many experiments should be performed in order to provide confident predictions of the yield at any combination of the three factors Five, ten, or twenty Obviously, the more experiments, the better are the predictions, but the greater the time, effort and expense. So there is a balance, and experimental design helps to guide the chemist as to how many and what type of experiments should be performed. 

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. 

It may happen, however, that this second order interaction is large. A good example of this was in the Section on a Four Factor Experiment, when the four factor experiment was broken down into two three factor experiments, and the S 3 had a large Residual almost certainly because it contained a large second order interaction. 

A further example is in the Five Factor experiment discussed in Chapter XI (f), where the W X T x D interaction was very large. If the experiment had been carried out as a three factor one on W, T, and D, this large W X T X D interaction would have been used as the residual, and no conclusions could have been drawn. Actually, of course, the experiment included two further factors, S and G, and fortunately neither of these interacted, so a reasonably small residual was obtained. 

Let us introduce an appropriate symbolism. Let the small letters p, q, r, etc., stand for the condition that the factor referred to is at its upper level. For example rs would mean that p and q are at their lower levels and r and s at their upper levels. Let the capital letter P denote the difference between the sum of all the upper levels of P and the sum of all the lower levels of P, and similarly for the other factors. Then for a three factor experiment,... 

Now consider that we have a three factor experiment, which wiU take 2 = 8 runs, and we need to put this experiment not into one block of 8 runs but into 2 blocks of 4 runs, the block size admitting only up to 4 runs. If we consider that we are mainly interested in the main effects and the first order interactions, and are willing to sacrifice the second order interaction to achieve this end, we can allocate the treatments to the two blocks as in Table 14.1. 

Three operating variables have been analyzed hydrogen partial pressure, temperature and reaction time. A factorial design of experiments was used to choose the conditions of each run. The levels of the three factors used during the runs are shown in Table I In the same table we can see the expressions employed to normalize the factors. 

The three-factor interaction effect ABC is computed as half the average of the difference of the responses in experiments where all three factor have been changed between the runs. These experiments are joined by the large diagonals of the cube ... 

In a Box-Behnken design, the experimental points lie on a hypersphere equidistant from the center point as exemplified for a three-factor design in Figure 4.13 and Table 4.11. In contrast to the central composite design, the factor levels have only to be adjusted at three levels. In addition, if two replications are again performed in the center of the three-factor design, the total number of experiments is 15 compared to 17 with the central composite design. 

Whenever interaction exists between factors, the experiment must be run according to the factorial design (FD) shown in Table 6 to ensure accurate and precise conclusion. Note that this experiment covers each of the 27 combinations of the levels of the three factors. 

---