All-possible-combinations experiment

For example, consider a two-factor design with each factor at two levels. This is also a form of all-possible-combinations experiment. One item we note here is that there is more than one way to describe the form of an experiment, and we include a short digression here to explicate this multiplicity of ways of describing an experiment. In this particular case, we have two factors, each at two levels. We can describe it as a listing of values corresponding to each experiment (Table 10-1). 

Table 10-1 All-possible-combinations experiment organized as a list of values...

Table 10-2 All-possible-combinations experiment organized as a table where the body of the table contains the experiment number corresponding to each set of experimental conditions...

While all-possible-combinations is certainly one way to achieve this balance, the advantage of statistical deigns comes from the fact that clever ways have been devised to achieve balance while needing far fewer experiments than the all-possible-combinations approach would require (Table 8-1). 

As an illustration of this, let us consider the three aforementioned variables temperature, pressure, and concentration of reactant. An all-possible-combinations design would require eight experiments, with the following set of conditions in each experiment (where H and L represent the high and the low temperatures, pressures, etc.) ... 

Table 8-1 An all-possible-combinations design of three factors, needing eight experiments and sets of conditions...

There are several ways we can expand a design such as this we can increase the number of factors, the number of levels of each factor, or we can do both, of course. There are other differences than can be superimposed over the basic idea of the simple, all-possible combinations of factors, such as to consider the effect of whether we can control the levels of the factors (if we can then do things that are not possible to do if we cannot control the levels of the factors), whether the levels correspond to physical characteristics that can be evaluated and the values described have real physical meaning (temperature, for example, has real physical meaning, while catalyst type does not, even though different catalysts in an experiment may all have different degrees of effectiveness, and reproducibly so). 

In Chapter 15, which was based on reference [1] we began our discussions of factorial designs. If we expand the basic rc-factor two-level experiment by increasing the number of factors, maintaining the restriction of allowing each to assume only two values, then the number of experiments required is 2", where n is the number of factors. Even for experiments that are easy to perform, this number quickly gets out of hand if eight different factors are of interest, the number of experiments needed to determine the effect of all possible combinations is 256, and this number increases exponentially. 

A full factorial design contains all possible combinations (L ) between the different factors f and their levels L, with L = 2 for two-level designs. It allows estimating all main and interaction effects between the factors. A FF design will only perform a fraction of the full factorial. A two-level FF design 2 examines factors, each at two levels, in 2 experiments, with 1/2"... 

Factorial experiment an experiment designed to examine the effect of two or more factors, each applied at least at two levels of operation. The full factorial investigates all possible combinations of these factors at the indicated levels ... 

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. 

Recent experiments have shown that hK3 can be activated by hK15 [161]. hK4 has also recently been shown to activate hK3 and does so much more efficiently compared to hK2 [106]. hK5 is predicted to be able to activate hK7 in the skin [103]. The activation of hK3 by hK2 is also possible. Although Takayama et al., reported the ability of hK2 to activate hK3 [162], Denmeade et al. reported the opposite [105] and hypothesized that additional proteases may be required. It will be interesting to study all possible combinations of interactions among kallikreins, especially those... 

Very often it is sufficient to start with plans where only the lowest and the highest levels of each factor are considered. In this case we have two levels and n factors, which gives rise to a complete design of type 2". If all possible combinations of the two factors in two levels each are to be performed, a total of k = 2" experiments results. 

Our experiment will then investigate all possible combinations of these two factors, i.e. 5 x 2 = 10 combinations. When we use all combinations of two (or more) factors, it is called a full factorial experiment . We used five replicates of each combination and the results are shown in Table 13.7. 

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. 

The stability of the /3-barrel itself was demonstrated in engineering experiments with OmpA. The four external loops of OmpA were replaced by shortcuts in all possible combinations (Koebnik, 1999). The resulting deletion mutants lost their biological functions in bacterial / -conjugation and as bacteriophage receptors, but kept the transmembrane /1-barrel as demonstrated by their resistance to proteolysis and thermal denaturation. The results confirm the expectation that the large external loops do not contribute to /1-barrel folding and stability. 

A full factorial design is the experimental set-up that contains all possible combinations of variables and levels. The number of experiments (A) in a two-level full factorial design is 2 with / the number of factors considered. The design is also called a 2 design. 

Therefore, after defining the model, the number of experiments one is willing to perform is selected, e.g. eight (the minimal number for Eq. (6.2) is six). The D-optimal design is the one that for all possible combinations of eight experiments from the total number of feasible points (grid points) yields a maximal value of the determinant of the matrix X, with... 

Analysis of variance is most useful when applied to a set of experiments planned with statistical evaluation in mind. Factorial experiments, in which several factors are changed in all possible combinations in a single integrated experiment, allow the estimation of interaction effects, the simultaneous effects of two or more variables. Such interactions may be extremely important yet they may escape detection com-... 

Once identified all possible combination of the variables could of course be tested, however, this is typically prohibitively resource intensive to be practical. DoE is a class of statistically ba.sed methods to select combinations of variables that may be tested to yield the same information using a reduced number of experiments. DoE is often used in a totally empirical manner however, the knowledge of potentially critical variables can greatly simplify the process. DoEs may take the form of simple experiments which bracket the extremes of variable combinations, the so called extreme vertices method, or may be more intricate. Commonly, a partial matrix type screening DoE will be executed to zero in on the ranges in variables that show the maximum impact and/or that most closely bracket the desired responses from the process. This is followed by a more focused matrix DoE to determine optimum ranges of operations. 

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