Factorial designs data analysis

Box, G., Hunter, W., and Hunter, S. (1978). Factorial Design at Two Levels. In Statistics for Experimenters, an Introduction to Design, Data Analysis and Model Building. John Wiley and Sons, New York, pp. 306-644. 

When a reaction has many participants, which may be the case even of apparently simple processes like pyrolysis of ethane or synthesis of methanol, a factorial or other experimental design can be made and the data subjected to a re.spon.se. suiface analysis (Davies, Design and Analysis of Industrial Experiments, Oliver Boyd, 1954). A quadratic of this type for the variables X, Xo, and X3 is... 

Because earlier experimental results and data analyses (3-10) had led us to anticipate the inadequacy of the simple approach considered above, we also planned and carried out (2) a second order factorial design of experiments and related data analysis. Mathematical analysis (of the results of 11 experiments) based on the second order model showed that all of these results could be represented satisfactorily by an equation of the form... 

Figure 34-1 A simple factorial design for collaborative data collection. Each sample analyzed (in this hypothetical case n = 6) requires multiple labs, or operators, using both methods of analysis and replicating each measurements a number of times (r = 5) for this hypothetical case.

ANOVA in these chapters also, back when it was still called Statistics in Spectroscopy [16-19] although, to be sure, our discussions were at a fairly elementary level. The experiment that Philip Brown did is eminently suitable for that type of computation. The experiment was formally a three-factor multilevel full-factorial design. Any nonlinearity in the data will show up in the analysis as what Statisticians call an interaction term, which can even be tested for statistical significance. He then used the wavelengths of maximum linearity to perform calibrations for the various sugars. We will discuss the results below, since they are at the heart of what makes this paper important. 

This is a novel feature of factorial design when compared with the classical laboratory procedure which excludes indications of the interaction of the variable. The method of analysis of the data, due to Yates, which is commonly used to evaluate these effects, requires that the trials are conducted in the sequence shown above, and proceeds as follows. 

We will skip (1) and (2) above as methods not to be preferred as global analyses. Graphical displays have tremendous values as exploratory data analysis (EDA) techniques with the type of data one encounters in these studies. For formal analyses, one could weigh univariate repeated and other factorial designs against their true multivariate counterparts. 

Data analysis of factorial designs involves a comparison of the experimental responses at the high and low settings of each variable. The results can be plotted in several different ways to develop an understanding of the effect of changing two or more process variables at a time with regard to reaction yield and quality of the product. 

Table 14.4 shows a typical regression analysis output for the 2 factorial design in Table 14.3. Most of the output is self-explanatory. For the moment, however, note the regression analysis estimates for the parameters of the model given by Equation 14.5 and compare them to the estimates obtained in Equations 14.8-14.15 above. The mean is the same in both cases, but the other non-zero parameters (the factor effects and interactions) in the regression analysis are just half the values of the classical factor effects and interaction effects How can the same data set provide two different sets of values for these effects ...

The analysis of variance lends itself best to balanced factorial designs, whether complete, partially replicated, or otherwise modified. The concept of balance simplifies the calculations tremendously. There are ways of coping with missing data, unequal replication under various conditions, and even some lack of orthogonality in the design, but these methods seem to involve more calculation than the data may deserve. The analysis of variance is a procedure which makes it possible to compare the effects of the variables being studied, first independently of the effects of all other variables, and second in all possible combinations with one another. Sometimes the effect of a variable within a given level of another variable... 

Problems encountered in HPLC analysis most often stem from a lack of knowledge of the influence of the slight variation of the experimental parameters (pH, temperature, solvent composition, flow rate, etc.). The analyst has to set up the list of parameters and their possible interactions. There are hardware parameters (e.g., flow control, temperature control, lamp current) and software parameters used to interpret and report the results from stored data. The use of factorial designs is of great help. Software such as Validation Manager, from Merck, produces, in a table for each parameter and interaction, its percentage and confidence interval as well as information to help the analyst in concluding the study. 

Factorial design of experiments, combined with statistical methods of data analysis, offers wider and more differentiated information on the system, while conclusions are of greater usability. The results of all the eight runs in the analyzed example serve for determining the factor effects, with seven trials being independent possibilities of testing the effects and one serving for their comparison with the chosen fixed values. Three out of seven independently determined factor effects serve for... 

In the case of constraints on proportions of components the approach is known, simplex-centroid designs are constructed with coded or pseudocomponents [23]. Coded factors in this case are linear functions of real component proportions, and data analysis is not much more complicated in that case. If upper and lower constraints (bounds) are placed on some of the X resulting in a factor space whose shape is different from the simplex, then the formulas for estimating the model coefficients are not easily expressible. In the simplex-centroid x 23 full factorial design or simplex-lattice x 2n design [5], the number of points increases rapidly with increasing numbers of mixture components and/or process factors. In such situations, instead of full factorial we use fractional factorial experiments. The number of experimental trials required for studying the combined effects of the mixture com-... 

Liao, C. T. (2000). Identification of dispersion effects from unreplicated 2n k fractional factorial designs. Computational Statistics and Data Analysis, 33, 291-298. 

---