A method of scaling for a mixed set of discrete and continuous variables.

A method of equalized scaling is proposed for taxonomic or classificatory systems, designed so that problems involving a mixed set of variables (some quantitative and others multistate qualitative) can be handled automatically and so that variables of neither class will unduly dominate the classifying process. The method is formulated in terms of Euclidean distance in n-dimensional space but is easily adapted to a system using similarity coefficients. Other advantages of the method are discussed. The method of normalized scaling presented here is designed to equalize the potential weight or influence of each variable where the variable set contains a mixture of 1) the qualitative type of variable which can assume any one of two or more discrete states, and 2) the quantitative type of variable which can assume any value along a continuum. This method is intended particularly for use in classification systems (cluster analysis, Bonner, 1964; numerical taxonomy, Rohlf, 1963; Sneath and Sokal, 1963; Sokal, 1961; etc.) whether formulated in terms of Euclidean distances in n space or in terms of the similarity coefficient. The method is fundamentally nonparametric and offers certain advantages over scaling in units of sample deviation. One approach to the mixed variable problem is to treat a qualitative variable as quantitative by appropriate scaling. This presents little difficulty where the variable is truly binary (e.g., the variable sex) and one state is simply represented by 0 while the other is scaled as 1. However, the general problem often involves variables which can assume more than two discrete and mutually exclusive states which do not lend themselves to rank ordering or other numerical scaling techniques (e.g., the variable of hair color where the discrete states might be black, brown, red, blond, gray, and white). Another approach is to treat a quantitative variable as qualitative by defining a series of discrete intervals along its continuum. This results in a loss of any measure of the degree of difference between objects with respect to the quantitative variables. For example, all persons falling within a discrete interval of age, such as 60 to 69 years, would be treated as alike, but any person falling outside of this range would be treated as simply different, whether the age is 19 or 59. Another consequent of this approach is that the greater the number of discrete intervals, the fewer the number of objects (e.g., persons) that will be considered as alike and, hence, the less influence the variable is likely to have on the classifying process. (For systems employing the similarity coefficient, Rubin [1965, 1966] has devised a scheme which overcomes this latter difficulty. ) Because of these considerations, it is desirable that a system or classification treat each type of variable according to its own nature, yet not let either unduly dominate the classifying process. In developing a scaling method for this purpose, I considered first the nature of the contribution to Cartesian distance from a qualitative variable p which can assume k discrete and mutually exclusive states, i.e., can be coded as 1 for no more than one of these states, and 0 for all others. The squared distance between two objects i and j with respect to variable p will be