Complete factorial experiment

The experimental research of a process with k factors and one response can be carried out considering all the combinations of the k factors with each factor at both levels. Thus, before starting the experimental research, we have a plan of the experiments which, for the mentioned conditions, is recognized as a complete factorial experiment (CFE) or plan. The levels of each of the various factors establish the frontiers of the process-investigated domain. 

High-level FFEs such as, for example 1/4 or 1/8 from complete factorial experiments (CFEs) can be used for complex processes, especially if the effect on the response of some factors is the objective of the research. It is not difficult to decide that, if we have a problem with the k factors where the p linear effects compensate the effects of interaction, then the 2 p FFE can be used without any restriction. The plan 2 p FFE keeps the advantages of the CFE 2 plan, then ... 

One problem with a complete factorial experiment such as this is that the number of experiments required rises rapidly with the number of factors for k factors at two levels with two replicates for each combination of levels, 2 experiments are necessary, e.g. for five factors, 64 experiments. When there are more than three factors some economy is possible by assuming that three-way and higher-order interactions are negligible. The sums of squares corresponding to these interactions can then be combined to give an estimate of the residual sum of squares, and replicate... 

The Greco-Latin square is a fiactional, four-level, four-factor factorial experiment composed of a total of 16 sets of conditions. The complete factorial experiment requires 256 test conditions. It is a very efficient experiment when the data are commensurate with a model containing no interaction terms. Unfortunately, in corrosion experiments it is a... 

Crystal Structures.—Crystallization, a pre-requisite for diffraction studies, is a notoriously and unpredictably difficult exercise with new proteins. Carter and Carter have introduced a method which searches a large number of experimental variables (e.g., buffer components and pH) that can influence rates of crystallization. Random combinations of the variables are used to attempt crystallization and the resulting precipitated protein scored on an arbitrary crystallinity scale. After applying the appropriate statistic, a complete factorial experiment is set up using the conditions which promoted crystallization in the first experiment. 

If a complete factorial design has three levels (low, middle and high) for each of three factors (A = temperature B = pH and C = reaction time), it is said to be a 3 X 3 X 3 or 3 CFD and 27 runs are required, each one corresponding to a particular combination of the factor levels. Furthermore, if we replicate each run k times, then the number of experiments needed is k x 3. ... 

This model allows us to estimate a response inside the experimental domain defined by the levels of the factors and so we can search for a maximum, a minimum or a zone of interest of the response. There are two main disadvantages of the complete factorial designs. First, when many factors were defined or when each factor has many levels, a large number of experiments is required. Remember the expression number of experiments = replicates x Oevels) " (e.g. with 2 replicates, 3 levels for each factor and 3 factors we would need 2 x 3 = 54 experiments). The second disadvantage is the need to use ANOVA and the least-squares method to analyse the responses, two techniques involving no simple calculi. Of course, this is not a problem if proper statistical software is available, but it may be cumbersome otherwise. 

In a factorial experiment, a fixed number of levels are selected for each of a number of variables. For a full factorial, experiments that consist of all possible combinations that can be formed from the different factors and their levels are then performed. This approach allows the investigator to study several factors and examine their interactions simultaneously. The object is to obtain a broad picture of the effects of the selected experimental variables and detect major trends that can determine more promising directions for further experimentation. Advantages of a factorial design over single-factor experiments are (1) more than one factor can be varied at a time to allow the examination of interaction effects and (2) the use of all experimental runs in evaluating an effect increases the efficiency of the experiment and provides more complete information. 

Complete factorial designs 2k require a minimum of 2k experiments, which... 

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-... 

TABLE 3 Experimental Matrix for Investigating the Influence of Three Factors on the Ti02 Photocatalyzed Oxidative Degradation of Waste Water Pollutants in a Pilot Reactor Complete Factorial Design 23, Control Experiments 1 (Natural and Coded Variables are Indicated), and Values of the Experimental Response Y... 

Although this direct method is more adequate for the given example, because the number of the values that are not available are smaller than the sum of rows and columns, the constant method has also been demonstrated for the case of comparison. It should be noted that both methods are generally used in two-way classification such as designs of completely randomized blocks, Latin squares, factorial experiments, etc. Once the values that are not available are estimated, the averages of individual blocks and factor levels are calculated and calculations by analysis of variance done. The degree of freedom is thereby counted only with respect to the number of experimental values. Results of analysis of variance for this example are... 

Based on previous testing of the research subject, the design of the full factorial experiment 23 with one replication to determine experimental error has been chosen. To eliminate the influence of systematic error in doing the experiment, the sequence of doing design point-trials, in accord with theory of design of experiments, has been completely random. The outcomes are given in Table 2.107. 

The experiment has been done through the matrix of full factorial experiment 24, as shown in Table 2.109. Each trial has been done only once, with no replications. The sequence of doing trials has been completely random. 

To obtain the mathematical model of the process, 1/4-replica of a full factorial experiment of type 2s has been realized. Design points-trials have been done in a completely random order. The Table 2.129 shows conditions and outcomes of doing a 26 2 fractional factorial experiment. 

To control the complex process of cooking by the sulfate process cellulose, by means of a computer, it is necessary to have a mathematical model of the process. To obtain this model for the process of cooking by the sulfate process cellulose from a mixture of soft and hardwood and deciduous trees, we have used a fractional factorial experiment. It included these seven factors xi consumption of active lye, % Na20 on completely dry wood X2 cooking temperature, °C ... 

There are several ways to study the effect of various experimental factors on an analytical method. The classical method of studying one variable at a time while holding others constant is extremely inefficient. Other approaches such as regression analysis and complete factorial designs involve a large number of experiments and are also inefficient. For example, a factorial design of seven factors at two levels requires 2 or 128 experiments. Therefore, alternate approaches which reduce the experimental work are very attractive. 

Sample size and treatment choice are key design questions for general multifactor experiments. Authors have proposed the use of standard factorial experiments in completely randomized designs, block designs, or Latin squares (see, for example, Chapter 6 and Churchill, 2003). However, the unusual distribution of gene expression data makes one question the relevance of standard orthogonal factorial experiments in this context. 

If the experimental runs are completely randomized, then randomization theory (see Hinkelmann and Kempthorne, 1994) tells us that least squares gives us unbiased estimators of any pre-chosen set of n — 1 linearly independent contrasts among the n combinations of factor levels (treatments). In most factorial experiments the pre-chosen treatment contrasts would be main effects and, perhaps, interactions. However, in supersaturated designs there is no rational basis for choosing a set of n — 1 contrasts before the analysis. Any model selection method will lead to selection biases, perhaps large biases, in the estimators of effects. If a2 is assumed known, then we can test the null hypothesis that all n treatment populations have equal means. This would not be of great interest, because even if this null hypothesis were true it would not imply that all main effects are zero, only that a particular set of n - 1 linear combinations of treatment means are zero. Of course, in practice, a2 is not known. 

A complete factorial design will span the experimental space very efficiently and contains all comers of the (hyper)cube defined by the (—) and ( + ) settings of the variable. A fractional factorial design will distribute the experimental point so that as much as possible of the variation of the experimental space is covered. One example is shown in Fig. 2 where four experiments in a fractional design... 

Fractional factorial designs are constructed from the model matrix X of a 2k p complete factorial design using the orthogonal columns of X to define the variable settings in 2k p experiments. 

It is well known that the domains of the great curvature of the response surface are characterized by non-linear variable relationships. The most frequently used state of these relationships corresponds to a two-degree polynomial. Thus, to express the response surface using a two-degree polynomial, we must have an experimental plan which considers one factor and a minimum of three different values. A complete factorial 3 experiment requires a great number of experiments (N = 3 k = 3 N = 27 k = 4 N = 81). It is obvious that the reduction of the number of experiments is a major need here. We can consequently reduce the number of experiments if we accept the use of a composition plan (sequential... 

In this chapter, it is discussed how to select a subset of experiments from a complete factorial design in such a way that it will be possible to estimate the desired parameters through a limited number of experimental runs. We shall see that it is very easy to construct designs which are 1/2, l/4, 1/8, 1/16,... 1/2P fractions of a complete factorial design. This will give a total of 2 P experimental runs, where k is the number of variables, and p is the size of the fraction. 

This is one half-fraction of a complete three-variable factorial design and it is seen that the experiments correspond to Exp no 5, 2, 3, 8 in the complete factorial design. Fig. 6.1 shows how the experiments of the half-fraction are distributed in the space spanned by the three variables. 

One solution would be to run the experiments in three blocks where each block is a complete factorial design for each reagent. This would make it possible to estimate all main effects and interaction effects between the experimental variables, and also the effects caused by a change of reagent. This would give a total of 24 runs. The effects of the reagents is measured by the difference of the estimated effects of the variables obtained in the blocks. 

A complete factorial design for each reagent would provide rather detailed infonnation on main effects and interaction effects of the experimental variables. It would also provide infonnation on to how a change of reagent modifies these effects. In a screening situation, this may not be necessary. With complete factorial designs, the number of experiments is still rather high. 

It is impossible to evaluate all possible combinations of substrates, reagents, and solvents by experiments. It is quite cumbersome, even to run a complete factorial design with selected substrates, reagents and solvents, as was described in the examples above. To achieve a more manageable number of test systems, it is possible to use the principles of fractional factorial designs to select test systems by their principal properties To illustrate this, we shall once more make use of the Willgerodt-Kindler reaction. 

---