Qualitative variables

Their value does not refer to a real degree of the risk level variation. Therefore a substance in NFPA code 4 cannot be considered as presenting a risk that is twice as high as of a code 2 substence. This indicates the limits of physical factors, which are only a qualitative variable masked by an arbitrary increase of modalities that nevertheless improve the accuracy of classification. 

The simplest case of this parameter estimation problem results if all state variables jfj(t) and their derivatives xs(t) are measured directly. Then the estimation problem involves only r algebraic equations. On the other hand, if the derivatives are not available by direct measurement, we need to use the integrated forms, which again yield a system of algebraic equations. In a study of a chemical reaction, for example, y might be the conversion and the independent variables might be the time of reaction, temperature, and pressure. In addition to quantitative variables we could also include qualitative variables as the type of catalyst. 

Yeast and fruit are input variables in the wine-making process. In the case of yeast, the amount of a given strain could be varied, or the particular type of yeast could be varied. If the variation is of extent or quantity (e.g., the use of one ounce of yeast, or two ounces of yeast, or more) the variable is said to be a quantitative variable. If the variation is of type or quality (e.g., the use of Saccharomyces cerevisiae, or Saccharomyces ellipsoideus, or some other species) the variable is said to be a qualitative variable. Thus, yeast could be a qualitative variable (if the amount added is always the same, but the type of yeast is varied) or it could be a quantitative variable (if the type of yeast added is always the same, but the amount is varied). Similarly, fruit added in the wine-making process could be a qualitative variable or a quantitative variable. In the algebraic system, x is a quantitative variable. 

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. 

The randomized paired comparison design discussed in the previous section separates the effect of a qualitative factor, fruit, from the effect of a quantitative factor, temperature (see Section 1.2). The randomized complete block design discussed in this section allows us to investigate more than one purely qualitative variable and to estimate their quantitative effects. 

We will define an input to the system as a quantitative or qualitative variable (experimental condition) that might have an influence upon the system. We can measure such an influence as a change in one or more output variables named responses. ... 

To apply these multivariate techniques, we require a data matrix with the information corresponding to n observations of p quantitative variables X, X2,..Xp). We could, also, have some qualitative variables, coded numerically, to classify the observations into groups. From a geometric perspective, the n observations of the data matrix would correspond to n points of the Euclidean space of the p variables, and the Euclidean distance between observations would correspond to a measure of proximity (similarity). 

The mean (denoted Y and also referred to as the arithmetic mean) is the average value of the data. It is obtained from the sum of all the individual data values divided by the number of data values (in symbolic terms, X) T/ ). The mean is a good measure of the centre of symmetrical frequency distributions of qualitative variables. It uses all of the numerical values of the... 

Discrete (qualitative) variables, which describe non-continuous variation, e.g. type of catalyst (Pd on Carbon or Pt on Alumina), type of solvent (carbon tetrachloride or hexane), type of equipment (reactor A, reactor B). 

To explore an experimental procedure, the experimenter chooses a range of variation for aU the experimental variables considered (a) for all continuous (quantitative) variables, the upper and lower bounds for their variation are specified (b) for the discrete (qualitative) variables, types of equipment, types of catalysts, nature of solvents etc. are specified. Assume that each experimental variable defines a coordinate axis along which the settings of the variables can be marked. Assume... 

Altland, K., Goedde, H. W., Held, K., Jensen, M., MUnsch, H., and Solem, E., New biochemical and immunological data on quantitative and qualitative variability of human pseudocholinesterase. Humangenetik 14, 56-60 (1971). 

This method is very interesting for the study of the qualitative variability of water and wastewater. 

This point is very important mainly in view of the design of a measurement strategy for industrial wastewater quality control. Among the analytical techniques available for the study of the complexity and qualitative variability of the medium, UV spectrophotometry is well adapted for the quality variation control of industrial wastewater (see Chapter 2). 

There are also situations where two of the modes of the three-way array can be expressed as qualitative variables in a two-way ANOVA and where the response measured is some continuous (spectral, chromatogram, histogram or otherwise) variable. Calling the ANOVA factors treatment A and treatment B a (Treatment x Treatment x Variable) array is made, see Figure 10.5. An example for particle size distribution in peat slurries was explained in Chapters 7 and 8. The two-way ANOVA can become a three-way ANOVA in treatments A, B and C, leading to a four-way array etc. Having three factors and only one response per experiment will lead to a three-way structure and examples will be given on how these can be modeled efficiently even when the factors are continuous. 

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. 

Gueguen, J. and Barbot, J., Quantitative and qualitative variability of pea Pisum sativum L.) protein composition, J. Set Food Agric., 42, 209-224, 1988. 

When a factor can take only 2 levels, whether these are qualitative or quantitative they are usually represented as -1 and +1. In this way, setting up the mathematical models and interpretation of the coefficients becomes much easier. Once a factor takes more than 2 quantitative levels this is no longer possible. This is why we will begin with a different approach here that might appear unduly complicated, but has the advantage that it can be extended to the study of qualitative variables at 3 or more levels. 

We have taken this "round trip" in order to insist on the fact that there is a fundamental difference between the meaning of the expressions "x = +1" or "a , = -1" referring to the level of a qualitative variable (with only 2 possible states) and that of the same expression when x, is the coded value of a continuous quantitative variable that has undergone a linear transformation and can therefore take all values between -1 and +1. 

The quantitative variables, X Xj, Xj and X4 are set at their extremes, 1 in the coded variables. We might wish to validate the method using several operators, and perhaps 3 different sets of apparatus. However we saw in the introduction to this chapter that the problem is considerably simpler if we limit ourselves to 2 levels of each qualitative variable, which are then set at levels -1, and +1. The problem of more than 2 levels of a qualitative variable will be treated in the following section. 

It is to be noted that if the (qualitative) variable X, is the nature of a certain class of excipient (for example, disintegrants), one of its levels may be the absence of a disintegrant. 

In a screening experiment it is rare for all variables to be quantitative. If they are not all quantitative then the centre point can no longer exist. If only one variable F is qualitative it would be possible to take the centre-points for all quantitative variables and repeat them at each level of the qualitative variable, Fj, Fj, F3... This already requires a considerable number of experiments. With more than one qualitative variable it is rarely feasible. There is no entirely satisfactory solution - one might repeat a number of experiments selected randomly or the experimenter can choose the experimental conditions that he thinks the most interesting. 

This way of representing an interaction is common in the literature. It has the advantage that the effects and the existence of an interaction may be seen at a glance. It has one major drawback it is the suggestion that an interpolation is possible between the experimental points. For qualitative variables this does not present any danger as there is no meaning to an interpolation between levels "-1" and "+1". For quantitative variables, such an interpolation could be possible, but it is not recommended in a factor study. 

In fact, sodium and potassium bicarbonate are not 2 separate constituents, as the authors did not intend making tablets containing both salts at the same time. There was therefore only a single constituent, "bicarbonate". Its nature, whether sodium or potassium salt is therefore considered as an independent qualitative variable. 

The formulation has a significant effect on the response. The only factor influencing the response is the concentration of bile salt. However we also see that there is almost as much residual variation (residual sum of squares) as there is variation explained by the model (regression sum of squares). The systematic differences between subjects (block sum of squares) are of similar importance. Note that the block effects sum to zero (compare with the presence-absence model for qualitative variables in chapter 2). 

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