Experimental design qualitative variables

In this chapter we discuss the multifactor concepts of confounding and randomization. The ideas underlying these concepts are then used to develop experimental designs for discrete or qualitative variables. 

Many papers have been published in which an experimental design is used to optimize tablet formulations. Both process variables as well as compositional variables (quantitative and qualitative) were used as independent (adjustable) variables. The studied dependent variables (factors to be optimized) are physical tablet properties directly after preparation such as weight variation, crushing strength, dissolution profile, disintegration time and friability. Doombos reviewed papers in which statistical methods were used to optimize tablet formulations [13]. 

Sometimes two of the modes of a three-way array are formed by a two-way ANOVA layout in qualitative variables and the three-way structure comes from measuring a continuous variable (e.g. a spectrum) in each cell of the ANOVA layout. In such cases each cell of the ANOVA has a multitude of responses that not even MANOVA can handle. When quantitative factors are used, one or two modes of the three-way array may result from an experimental design where the responses are noisy spectra that behave nonlinearly. Such data are treated in this section. 

The rate setting can be treated as either qualitative or quantitative, depending on the circumstances. It is this kind of variable that causes the most difficulties in formulating problems in experimental design Let us consider four possible circumstances in turn ... 

Experimental designs to establish the effect of the composition of a mixture on its properties differ from other designs in that the initial parameters are not usually independent of one another. The properties of a mixture of three diluents, lactose, calcium phosphate and microcrystalline cellulose, for example, may be defined in terms of the percentage of each component. If we know the concentration of the first two components we can automatically find the concentration of the third. This non-independence of the variables means that designs of the type outlined above are unsuitable for many problems involving mixtures. However the classical designs may be used in two circumstances - for the choice of excipients, where the factors are purely qualitative, and also in problems where the proportions of all but one of the excipients are constrained to be relatively small. 

Another approach is to treat the manufacturing scale as a normal qualitative variable, and optimize the process and/or formulation at each level. Alternatively it might also be treated as a quantitative discrete variable, as the volume of the apparatus, or possibly better, its logarithm. Literature examples of the use of experimental design in scale-up take this approach (14, 15, 16). 

Let U be the set of factors (the variables that are to be manipulated in the experimental design, with U = card(U) the number of elements) and V be the set measurement variables (in the following only the term variable will be used). With our mechanical system, the set U is heterogeneous (there are quantitative or qualitative scales) and U = 21 (maximum configuration). The induced problem is How to cover a heterogeneous experimental field with U = 21 factors plus their interactions in a minimum of tests. 

A company producing brake pads was interested to study the effect of seven qualitative variables, having 4, 2, 4, 4, 2, 2 and 2 levels each [7]. The total number of possible combinations is 1024 (2 4 ), while the number of coefficients in the model is 14. A D-Optimal Design detected the experimental matrix reported in Table 16, obtained by just 16 experiments. 

Note that 3 and X4 are qualitative factors and their values are limited to 1. An optimization method that could distinguish between discrete and continuous variables and which would thus only search the space of permitted values of the qualitative, or discrete quantitative variables, would be preferred. It can also be seen that the maximum desirability, on the edge of the square defined by X X2] at 1, is actually just outside the cylindrical experimental domain. The optimum should, strictly speaking, be displaced very slightly to lie on the edge of the design space. 

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