Editing Level of measurement (section)

== Debate on Stevens's typology ==
While Stevens's typology is widely adopted, it is still being challenged by other theoreticians, particularly in the cases of the nominal and ordinal types (Michell, 1986).<ref name = "Velleman and Wilkinson 1993">{{cite journal |last1=Velleman |first1=Paul F. |last2=Wilkinson |first2=Leland |journal=The American Statistician | title=Nominal, ordinal, interval, and ratio typologies are misleading |volume=47 | number=1 | year = 1993 |pages=65–72 |doi=10.2307/2684788 | jstor=2684788 }}</ref> Duncan (1986), for example, objected to the use of the word ''measurement'' in relation to the nominal type and Luce (1997) disagreed with Stevens's definition of measurement.

On the other hand, Stevens (1975) said of his own definition of measurement that "the assignment can be any consistent rule. The only rule not allowed would be random assignment, for randomness amounts in effect to a nonrule". Hand says, "Basic psychology texts often begin with Stevens's framework and the ideas are ubiquitous. Indeed, the essential soundness of his hierarchy has been established for representational measurement by mathematicians, determining the invariance properties of mappings from empirical systems to real number continua. Certainly the ideas have been revised, extended, and elaborated, but the remarkable thing is his insight given the relatively limited formal apparatus available to him and how many decades have passed since he coined them."<ref name = "Hand 2017">{{cite journal |last1=Hand |first1=David J.|journal=Measurement: Interdisciplinary Research and Perspectives | title=Measurement: A Very Short Introduction—Rejoinder to discussion |volume=15 | number=1 | year = 2017 |pages=37–50 |doi=10.1080/15366367.2017.1360022|hdl=10044/1/50223 |s2cid=148934577 |hdl-access=free }}</ref>

The use of the mean as a measure of the central tendency for the ordinal type is still debatable among those who accept Stevens's typology. Many behavioural scientists use the mean for ordinal data anyway. This is often justified on the basis that the ordinal type in behavioural science is in fact somewhere between the true ordinal and interval types; although the interval difference between two ordinal ranks is not constant, it is often of the same order of magnitude.

For example, applications of measurement models in educational contexts often indicate that total scores have a fairly linear relationship with measurements across the range of an assessment. Thus, some argue that so long as the unknown interval difference between ordinal scale ranks is not too variable, interval scale statistics such as means can meaningfully be used on ordinal scale variables. Statistical analysis software such as [[SPSS]] requires the user to select the appropriate measurement class for each variable. This ensures that subsequent user errors cannot inadvertently perform meaningless analyses (for example correlation analysis with a variable on a nominal level).

[[L. L. Thurstone]] made progress toward developing a justification for obtaining the interval type, based on the [[law of comparative judgment]]. A common application of the law is the [[analytic hierarchy process]]. Further progress was made by [[Georg Rasch]] (1960), who developed the probabilistic [[Rasch model]] that provides a theoretical basis and justification for obtaining interval-level measurements from counts of observations such as total scores on assessments.

===Other proposed typologies===
Typologies aside from Stevens's typology have been proposed. For instance, [[Frederick Mosteller|Mosteller]] and [[John Tukey|Tukey]] (1977) and Nelder (1990)<ref>Nelder, J. A. (1990). The knowledge needed to computerise the analysis and interpretation of statistical information. In ''Expert systems and artificial intelligence: the need for information about data''. Library Association Report, London, March, 23–27.</ref> described continuous counts, continuous ratios, count ratios, and categorical modes of data. See also Chrisman (1998), van den Berg (1991).<ref>van den Berg, G. (1991). ''Choosing an analysis method''. Leiden: DSWO Press</ref>

==== Mosteller and Tukey's typology (1977) ====
Mosteller and Tukey<ref name="Mosteller"/> noted that the four levels are not exhaustive and proposed seven instead:
# Names
# Grades (ordered labels like beginner, intermediate, advanced)
# Ranks (orders with 1 being the smallest or largest, 2 the next smallest or largest, and so on)
# Counted fractions (bound by 0 and 1)
# Counts (non-negative integers)
# Amounts (non-negative real numbers)
# Balances (any real number)

For example, percentages (a variation on fractions in the Mosteller–Tukey framework) do not fit well into Stevens's framework: No transformation is fully admissible.<ref name = "Velleman and Wilkinson 1993" />

==== Chrisman's typology (1998) ====
Nicholas R. Chrisman<ref name="Chrisman"/> introduced an expanded list of levels of measurement to account for various measurements that do not necessarily fit with the traditional notions of levels of measurement. Measurements bound to a range and repeating (like degrees in a circle, clock time, etc.), graded membership categories, and other types of measurement do not fit to Stevens's original work, leading to the introduction of six new levels of measurement, for a total of ten:
# Nominal
# Gradation of membership
# Ordinal
# Interval
# Log-interval
# Extensive ratio
# Cyclical ratio
# Derived ratio
# Counts
# Absolute

While some claim that the extended levels of measurement are rarely used outside of academic geography,<ref name = "Wolman 2006">{{cite journal |title=Measurement and meaningfulness in conservation science |journal=Conservation Biology |volume=20 |issue=6 |pages=1626–1634 |year=2006 |last=Wolman |first=Abel G |s2cid=21372776 |doi=10.1111/j.1523-1739.2006.00531.x|pmid=17181798 |bibcode=2006ConBi..20.1626W }}</ref> graded membership is central to [[fuzzy set theory]], while absolute measurements include probabilities and the plausibility and ignorance in [[Dempster–Shafer theory]]. Cyclical ratio measurements include angles and times. Counts appear to be ratio measurements, but the scale is not arbitrary and fractional counts are commonly meaningless. Log-interval measurements are commonly displayed in stock market graphics. All these types of measurements are commonly used outside academic geography, and do not fit well to Stevens's original work.

=== Scale types and Stevens's "operational theory of measurement" ===
The theory of scale types is the intellectual handmaiden to Stevens's "operational theory of measurement", which was to become definitive within psychology and the [[behavioral sciences]],{{citation needed|date=July 2012}} despite Michell's characterization as its being quite at odds with measurement in the natural sciences (Michell, 1999). Essentially, the operational theory of measurement was a reaction to the conclusions of a committee established in 1932 by the [[British Association for the Advancement of Science]] to investigate the possibility of genuine scientific measurement in the psychological and behavioral sciences. This committee, which became known as the ''Ferguson committee'', published a Final Report (Ferguson, et al., 1940, p.&nbsp;245) in which Stevens's [[sone]] scale (Stevens & Davis, 1938) was an object of criticism:

{{blockquote | …any law purporting to express a quantitative relation between sensation intensity and stimulus intensity is not merely false but is in fact meaningless unless and until a meaning can be given to the concept of addition as applied to sensation.}}

That is, if Stevens's ''[[sone]]'' scale genuinely measured the intensity of auditory sensations, then evidence for such sensations as being quantitative attributes needed to be produced. The evidence needed was the presence of ''additive structure''—a concept comprehensively treated by the German mathematician [[Otto Hölder]] (Hölder, 1901). Given that the physicist and measurement theorist [[Norman Robert Campbell]] dominated the Ferguson committee's deliberations, the committee concluded that measurement in the social sciences was impossible due to the lack of [[concatenation (mathematics)|concatenation]] operations. This conclusion was later rendered false by the discovery of the [[theory of conjoint measurement]] by Debreu (1960) and independently by Luce & Tukey (1964). However, Stevens's reaction was not to conduct experiments to test for the presence of additive structure in sensations, but instead to render the conclusions of the Ferguson committee null and void by proposing a new theory of measurement:

{{blockquote|Paraphrasing N. R. Campbell (Final Report, p. 340), we may say that measurement, in the broadest sense, is defined as the assignment of numerals to objects and events according to rules (Stevens, 1946, p. 677).}}

Stevens was greatly influenced by the ideas of another Harvard academic,<ref>[[Percy Bridgman]] (1957) ''[[The Logic of Modern Physics]]''</ref> the [[Nobel Prize|Nobel laureate]] physicist [[Percy Bridgman]] (1927), whose doctrine of [[operationalism]] Stevens used to define measurement. In Stevens's definition, for example, it is the use of a tape measure that defines length (the object of measurement) as being measurable (and so by implication quantitative). Critics of operationalism object that it confuses the relations between two objects or events for properties of one of those of objects or events (Moyer, 1981a, b; Rogers, 1989).<ref>{{cite journal | last1 = Hardcastle | first1 = G. L. | year = 1995 | title = S. S. Stevens and the origins of operationism | journal = Philosophy of Science | volume = 62 | issue = 3| pages = 404–424 | doi=10.1086/289875| s2cid = 170941474 }}</ref><ref> Michell, J. (1999). ''Measurement in Psychology &ndash; A critical history of a methodological concept''. Cambridge: Cambridge University Press.</ref>

The Canadian measurement theorist William Rozeboom was an early and trenchant critic of Stevens's theory of scale types.<ref>{{cite journal | last1 = Rozeboom | first1 = W. W. | year = 1966 | title = Scaling theory and the nature of measurement | journal = Synthese | volume = 16 | issue = 2| pages = 170–233 | doi=10.1007/bf00485356| s2cid = 46970420 }}</ref>

==== Same variable may be different scale type depending on context ====
Another issue is that the same variable may be a different scale type depending on how it is measured and on the goals of the analysis. For example, hair color is usually thought of as a nominal variable, since it has no apparent ordering.<ref>{{cite web|url=http://www.ats.ucla.edu/stat/mult_pkg/whatstat/nominal_ordinal_interval.htm|title=What is the difference between categorical, ordinal and interval variables?|work=Institute for Digital Research and Education|publisher=University of California, Los Angeles|archive-url=https://web.archive.org/web/20160125165359/http://www.ats.ucla.edu/stat/mult_pkg/whatstat/nominal_ordinal_interval.htm|access-date=7 February 2016|archive-date=2016-01-25}}</ref> However, it is possible to order colors (including hair colors) in various ways, including by hue; this is known as [[colorimetry]]. Hue is an interval level variable.