Pairwise Comparisons

Operations on Scale Values

The application of elementary statistics—such as standard deviation—to scale values requires the availability of the operations of addition and multiplication as well as order and the limit operation of calculus. Psychological variables to which addition, multiplication, order, and the limit operation are applicable must be modeled in the same manner—and for the same mathematical reasons—as such familiar physical variables as time, position of points on a straight line, potential energy, and temperature on the Fahrenheit or Celsius (but not Kelvin) scales.

Reference Objects

The building blocks for such scales require three or four rather than two objects. Consider, for example, the statement that the temperature of a certain object is 100 degrees on the Fahrenheit scale. As can be seen in Figure 1, this statement involves three empirical objects, three mathematical objects, and three correspondences: The empirical objects are freezing water, boiling water, and the object under measurement; the mathematical objects are the numbers 32, 100, and 212; and the correspondences are the assignments of the temperatures—{freezing water, 32}, {object under measurement, 100}, and {boiling water, 212}. It should be noted that this statement requires two empirical reference objects (freezing water and boiling water) and two corresponding mathematical reference objects (the numbers 32 and 212).

Removing one of these empirical reference objects and its corresponding mathematical reference object results in an ordinal pairwise comparison where neither differences nor ratios are defined. If both empirical reference objects are removed and the numerical ones are not, the statement “on a scale of 32 to 212, an object scores 100” is obtained. Statements of this form, such as the common phrase “on a scale of 1 to 10, an object scores 7,” have no mathematical meaning. No mathematical operations or statistics are applicable to numbers produced from such statements.

Figure 1 Example of Scale Values and Reference Objects

Ratios

Ratios of the type T1/T2 have become defined for temperature only after it has been established that temperature has an absolute zero. Conversely, for variables where the existence of an absolute zero has not been established, such ratios are undefined. For example, for time, the ratio t1/t2 where t1 and t2 are two points in time is undefined, whereas the ratio of two time differences, (i.e., time periods or time intervals) (Δt)1/(Δt)2, is well-defined. It follows that the ratio v1/v2 is undefined for any psychological variable because the existence of an absolute zero has not been established for such variables. In particular, decision methodologies such as the Analytic Hierarchy Process that depend on data in the form of ratios of preferences are not valid methodologies. In general, a ratio of differences depends on four points, but this ratio may depend on three variables rather than four when two of the four points are identical. Although the number of variables in this expression can be reduced from four to three, it cannot be further reduced to a pairwise comparison.

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