Factor experiments

The type of experiment which we have been discussing up to now has had only one independent variable or factor. In much industrial work, of course, there are a relatively large number of independent variables which we wish to investigate, and we therefore require a further set of techniques. 

The technique of factorial experimentation was developed in the science of agriculture in the twenties and thirties largely at the Rothamsted Experiment Station by R. A. Fisher and his colleagues. 

Fisher s approach to experimentation differs in two fundamental aspects from the classical one-vaiiable-at-a-time ideology. Firstly, he stresses the importance of obtaining an accurate estimate of the magnitude of the error variation, rather than its minimisation. The accurate estimate of the error variation is necessary in order to apply an exact test of significance. 

Secondly, Fisher emphasises the advantages to be gained from including in the same experiment as many as possible of the fectors whose effects are to be determined. The advantages of this are— 

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Because the preceding factor experiment suggests a, b, c, and abc as independent variables, cf. bottom row in Table 3.2, the data table would take on the form ... 

As the effect of aeration can still be predicted with less certainty than the stress caused by single-phase currents, due to the many influencing factors, experiments to investigate particle stress are particularly important. 

The table is not exhaustive, although it does include a majority of experimental designs that are used. One-at-a-time designs are the usual non-statistical type of experiments that are often carried out by scientists in all disciplines. Not included explicitly, however, are experimental designs that are generated from combinations of listed items. For example, a multi-factor experiment may have several levels of some of the factors but only two levels of other factors. 

Because the interactions measured in Mossbauer experiments are products of atomic and nuclear factors, experiments on iodine isotopes have yielded values of the change of nuclear radius between the ground state and the excited state, AR/R, quadrupole moment values Q, and magnetic moment values, fi, as well as electric field gradients and internal magnetic fields. 

Empirical Modeling. The effect of process variables on the rate of depKJsition and properties of electrolessly depKJsited metals is usually studied by one-factor-at-a-time experiments (one-factor experiments are discussed further later in the book). In these experiments the effect of a single variable (factor), such as Xj, in the multivariable process with the response y, y = fixi, %2, X3,. .., x ), is studied by varying the value (level) of this variable while holding the values of the other independent variable fixed, y Any prediction (extrapolation) of the effect of a single variable on... 

Figure 8.14. Rate of electroless cobalt deposition as a function of pH at 30 and 100 g/L citrate one-factor experiments. (From Ref. 65, with permission from the Electrochemical Society.)...

Although the influences of some factors are self-evident, if experimental data are sparse, it can be difficult to determine all the significant factors. Experience will help provide a list of candidates, and the suite of factors to be investigated will be constrained by the time and effort available. At first, the linear coefficients of the model will give most information about the... 

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. 

Domain of factors is marked O . The figure clearly shows that intervals of factor variations are part of the domain of factors when the optimization problem is being solved. This is necessary in order to realize movement towards optimum in this domain. The experiment domain is in the same figure marked by letter E". In studies with an objective of approximation or interpolation, that is mathematical modeling, the factor-variation intervals cover the whole of the domain of factors. For a two-factor experiment the upper level of factors X and X2 corresponds to values Xlmax,-and X2max, while the lower levels have values Xlmin, X2min. Domain of factors O is in that case called intcrpolational, and E the domain of extreme experiment. 

A one-factor experiment was done, which showed that, in the assumed experimental region, the quantity of extracted material and its saturation with mercury are of no importance. As in this experiment, a caustic with constant mercury content of 12x10"4% was used, this factor is also excluded from considerations. After this selection the following factors have remained ... 

With this backing one can attempt to extend the treatment to the even lighter isotopes 192>190Pb, where only the 12+ lifetime could be determined, as a PAD measurement of the quadrupole moment would need excessive statistics. The time spectrum obtained with the conventional pulsed beam technique for 192Pb (Fig. 4a) in parallel to the g-factor experiment to be... 

To compare our theoretical result with the current experimental data, we refer to the overview article [25] given elsewhere in this book. The results are in reasonable agreement and thus the gj factor experiment on hydrogenlike carbon in combination with our theoretical results forms one of the most stringent QED tests for systems with Z > 1. 

Comparison of theory to precision experiment often involves some other experimental data from different fields. In particular, the g factor experiment [1] deals with a comparison of two frequencies the Larmor spin precession frequency... 

Single factor experiments showed that Cu mixed analyte standards give a 25% higher response than single analyte standards for analysis in a lean flame (see Table XIV). 

The optimization of the atomic absorption method of determining metals in particulates found in the air of workplace is described. The Plackett-Burman Youden-Steiner balanced incomplete block designs as well as single-factor experiments were utilized with ten metals Be, Cd, Co, Cr, Cu, Mn, Mo, Ni, Pb, and Pd. Of the parameters tested, perchloric acid digestion, flame-stoichiometry, and the composition of the calibration standards were the most significant. Perchloric acid affected the recoveries of chromium. This was attributed to the formation of volatile chromylchloride. Flame-related phenomena and interelemental effects were brought under control using lanthanum flame buffer. 

Like the single-factor experiment (Table 13.1), this experiment is also balanced — there were equal numbers of replicates for each combination of factors. 

In more complex situations, such as 10 factor experiments, it is unlikely that there will be any physical meaning attached to higher order interactions, or at least that diese interactions are not measurable. Therefore, it is possible to select specific interactions that are unlikely to be of interest, and consciously reduce the experiments in a systematic manner by confounding these with lower order interactions. 

The main steps are as follows, exemplified by a two factor experiment. 

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... 

Equipment Count Factor - Experience has shown that in-house engineering is less sensitive to low equipment counts. The following correction factors are suggested ... 

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. 

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