## 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-Related Resources: Rasch Measurement YouTube Channel
Rasch Measurement Transactions & Rasch Measurement research papers - free An Introduction to the Rasch Model with Examples in R (eRm, etc.), Debelak, Strobl, Zeigenfuse Rasch Measurement Theory Analysis in R, Wind, Hua Applying the Rasch Model in Social Sciences Using R, Lamprianou El modelo métrico de Rasch: Fundamentación, implementación e interpretación de la medida en ciencias sociales (Spanish Edition), Manuel González-Montesinos M.
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Rasch Models for Measurement, David Andrich Constructing Measures, Mark Wilson Best Test Design - free, Wright & Stone
Rating Scale Analysis - free, Wright & Masters
Virtual Standard Setting: Setting Cut Scores, Charalambos Kollias Diseño de Mejores Pruebas - free, Spanish Best Test Design A Course in Rasch Measurement Theory, Andrich, Marais Rasch Models in Health, Christensen, Kreiner, Mesba Multivariate and Mixture Distribution Rasch Models, von Davier, Carstensen
Rasch Books and Publications: Winsteps and Facets
Applying the Rasch Model (Winsteps, Facets) 4th Ed., Bond, Yan, Heene Advances in Rasch Analyses in the Human Sciences (Winsteps, Facets) 1st Ed., Boone, Staver Advances in Applications of Rasch Measurement in Science Education, X. Liu & W. J. Boone Rasch Analysis in the Human Sciences (Winsteps) Boone, Staver, Yale Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes Rasch Models for Solving Measurement Problems (Facets), George Engelhard, Jr. & Jue Wang Statistical Analyses for Language Testers (Facets), Rita Green Invariant Measurement with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments (Facets), George Engelhard, Jr. & Stefanie Wind Aplicação do Modelo de Rasch (Português), de Bond, Trevor G., Fox, Christine M
Exploring Rating Scale Functioning for Survey Research (R, Facets), Stefanie Wind Rasch Measurement: Applications, Khine Winsteps Tutorials - free
Facets Tutorials - free
Many-Facet Rasch Measurement (Facets) - free, J.M. Linacre Fairness, Justice and Language Assessment (Winsteps, Facets), McNamara, Knoch, Fan

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