Stevens' ratio scale
S. S. Stevens included "ratio scales" in his hierarchy of scales.
"A ratio scale is an interval scale in which distances are stated
with respect to a rational zero rather than with respect to, for
example, the mean" (Nunnally, 1967, p.14). A rational zero is a
location on an interval scale deliberately chosen for reasons
other than the current data. The distinctive feature of a ratio
scale is that it has a an origin defined by a dominating theory
(Stevens, 1959, p.25). Time measured from the "Big Bang" is on a
ratio scale. Height is on a ratio scale when measured from sea-
level, e.g., "one mountain is twice as high as another", but on
an interval scale when measured from the mean of a sample of
mountains.
When the "average" of a set of ratio scale numbers is intended to summarize their ratio, not interval, property, their ratio relationship with the scale's particular origin must be maintained. This is done by using the geometric, rather than the arithmetic, mean as the "average" for the ratios (Stevens, p. 27). To obtain the "average" of a set of ratio scale numbers, the logarithm of each number is calculated. The arithmetic mean of these logarithms is computed and this number is exponentiated to yield the geometric mean of the original ratio numbers.
Counts - A Special Case
"The numerosity of collections of objects [i.e., counts] ...
belongs to the class I have called ratio scales" ( Stevens, p.
20). Accordingly, in situations where it is important to
maintain the notion that a count of 0 means "none at all", rather
than "none extra", and a count 1 of means "only one object",
rather than "one more to go with those we already have", then
ratio scale arithmetic applies. To apply the usual interval
statistics to such counts requires the logarithms of the counts
to be obtained. This concept underlies one derivation of the
Rasch model (RMT 3:2 p.62).
Rasch's Scale of Ratios
Georg Rasch's use of the term "Ratio Scale" differs from
Stevens'. Rasch (1992, p. 69) specifies a scale of ratios,
analogous to the Richter scale for earthquakes. As originally
defined, zero on the Richter scale was intended to mean "no earth
movement". Then scale values are defined such that an earthquake
of "2" on the Richter scale is 10 times more severe than a "1". Richter-type scales are also called "logarithmic scales".
Rasch also conceptualized degrees of difficulty and ability in ratio units when he wrote the multiplicative form of his model (p. 107):
t/(1-t) = z/d
But, whenever convenient, Rasch switched to an additive, i.e., interval, form by taking logarithms (p. 119):
log_{e}(t/(1-t)) = log_{e}(z) - log_{e}(d)
In general, any interval scale can be transformed to a Stevens ratio scale by choosing an origin with some meaning. Further, any interval scale can be converted to a Rasch scale of ratios ("logarithmic scale") by choosing any origin and exponentiating. Consequently, the distinctions between these scale types have no mathematical importance.
Based on an AERA Division D Internet conversation between Bill Koch, Matt Schulz, Ray Wright, Richard Smith, Steve Lang et al.
Nunnally J.C. (1967) Psychometric Theory. New York: McGraw-Hill.
Rasch G. (1992) Probabilistic Models for Some Intelligence and Attainment Tests. Chi.:MESA Press.
Stevens S.S. (1959) Measurement, Psychophysics and Utility, Chap. 2, in C.W. Churchman & P. Ratoosh (Eds.), Measurement: Definitions and Theories. New York: John Wiley
What is a ratio scale? Koch W, Schulz EM, Wright R, Smith RM, Lang S. … Rasch Measurement Transactions, 1996, 9:4 p.457
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