Experiments at two levels

The tabular value of t at the 95% level of confidence and one degree of freedom is 12.71 (see Appendix B). Thus, and we cannot conclude (at the 95% level 

The sum of squares due to purely experimental uncertainty, already calculated for 

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Figure 8.2 Relationship of three experiments at two levels to the fitted model = Po +...

Sometimes it is not sufficient to study an experiment at two levels. For example, is it really sufficient to use only two temperatures A more detailed model will be... 

Procedures are available for 2 designs involving two levels of n factors where n can, in principle, be any large number. However, the required number of runs for large n may be prohibitive. For a five-factor experiment, 25 runs are required in a single block but block size must be held to a minimum to control known sources of error. For this reason, fractional factorials which utilize some integer fraction (a multiple of the number of levels) of the total factorial experiments are used. The five-factor experiment at two levels would involve a total of 32 experiments in the factorial design whereas a fractional factorial... 

Cropley made general recommendations to develop kinetic models for compUcated rate expressions. His approach includes first formulating a hyperbolic non-linear model in dimensionless form by linear statistical methods. This way, essential terms are identified and others are rejected, to reduce the number of unknown parameters. Only toward the end when model is reduced to the essential parts is non-linear estimation of parameters involved. His ten steps are summarized below. Their basis is a set of rate data measured in a recycle reactor using a sixteen experiment fractional factorial experimental design at two levels in five variables, with additional three repeated centerpoints. To these are added two outlier... 

A risk assessment analyses systems at two levels. The first level defines the functions the system must perform to respond successfully to an accident. The second level identifies the hardware for the systems use. The hardware identification (in the top event statement) describes minimum system operability and system boundaries (interfaces). Experience shows that the interfaces between a frontline system and its support systems are important to the system cs aluaiion and require a formal search to document the interactions. Such is facilitated by a failure modes and effect analysis (FMEA). Table S.4.4-2 is an example of an interaction FMEA for the interlace and support requirements for system operation. 

Because all the variables that influence the properties of the final product are known, one can use a statistical design (known as a one-half factorial) to optimize the properties of the GPC/SEC gels. Factorial experiments are described in detail by Hafner (10). For example, four variables at two levels can be examined in eight observations. From these observations the significance of each variable as related to the performance of the gel can be determined. An example of a one-half factorial experiment applied to the production of GPC/SEC gel is set up in Table 5.2. The four variables are the type of DVB, amount of dodecane, type of methocel, and rate of stirring. 

From his conductivity measurements on solutions, Arrhenius concluded that strong electrolytes are not exceptions. Instead they dissociate into ions. When z = 2, it meant that each solute species dissociated to give two ions. A compound with Z = 3 dissociated to give three ions. Moreover, interpreting the results of his experiments at varying levels of concentration, Arrhenius concluded that at sufficiently high dilution, every electrolyte becomes fully dissociated. 

On the other hand, some sensible reduction may be acceptable. In the spiroxamine example, an appropriate reduced validation protocol may be as follows a full set of recovery experiments at both levels performed with the intact spiroxamine (which has the longest reaction pathway to the common moiety) and separately with one primary metabolite. Such two complete validations should be an acceptable test of the working range of the common moiety method. 

In the case that interactions prove to be insignificant, it should be gone over to the ab model the estimations of which for the various variance components is more reliable than that of the 2ab model. A similar scheme can be used for three-way ANOVA when the factor c is varied at two levels. In the general, three-way analysis bases on block-designed experiments as shown in Fig. 5.1. 

Table 5.10. Plackett-Burman design matrix lor N 8 experiments and consequently m = 7 factors (including dummy variables) at two levels... 

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). 

In Chapter 8, we looked at some experiments that involved two parameters (factors), each at two levels. In Chapter 10, we briefly looked at a three-factor, two-level design, with attention to how it could be represented geometrically. The use of the term three factor, two level to describe the design means that each factor was present at two levels, that is, the corresponding parameters were each permitted to assume two values. 

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"... 

Full factorial designs allow the estimation of all main and interaction effects, which is not really necessary to evaluate robusmess. They can perfectly be applied when the number of examined factors is maximally four, considering the required number of experiments. In references 69 and 70, four and three factors were examined at two levels in 16 and 8 experiments, respectively. When the number of factors exceeds four, the number of experiments increases dramatically, and then the full factorial designs are not feasible anymore. 

Figure 5.1 Graph of the deterministic model = Po + Pi-tw fitted to the results of two experiments at different levels of the factor x,.

Let us use the matrix least squares method to obtain an algebraic expression for the estimate of Pq in the model y, = Po + r, (see Figure 5.2) with two experiments at two different levels of the factor x,. The initial X, B, R, and F arrays are given in Equation 5.27. Other matrices are... 

Another statistical model that might be fit to two experiments at two different levels is... 

Unfortunately, two experiments at two different levels of a single factor cannot provide an estimate of the purely experimental uncertainty. The difference in the two observed responses might be due to experimental uncertainty, or it might be caused by a sloping response surface, or it might be caused by both. For this particular experimental design, the effects are confused (or confounded) and there is no way to separate the relative importance of these two sources of variation. 

In a full factorial design all combinations between the different factors and the different levels are made. Suppose one has three factors (A,B,C) which will be tested at two levels (- and +). The possible combinations of these factor levels are shown in Table 3.5. Eight combinations can be made. In general, the total number of experiments in a two-level full factorial design is equal to 2 with /being the number of factors. The advantage of the full factorial design compared to the one-factor-at-a-time procedure is that not only the effect of the factors A, B and C (main effects) on the response can be calculated but also the interaction effects of the factors. The interaction effects that can be considered here are three two-factor interactions (AB,... 

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