## What is a Ratio Scale?

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):

loge(t/(1-t)) = loge(z) - loge(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

Rasch Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
in Spanish: Análisis de Rasch para todos, Agustín Tristán Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez

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June 25 - July 23, 2021, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com