Beyond Stevens:
A revised approach to measurement
for geographic information

Presented at AUTO-CARTO 12
Charlotte, North Carolina; March 1995
by Nicholas Chrisman
University of Washington

Outline of Presentation

Measurement Theory

abbreviated intellectual history

Classical school

developed in physics (and related sciences) by late 19th century
numerical relationship between standard& object measured; inherent in object
Example: US Office of Weights and Measures | standard kilogram | Legal Weights and Measures in the State of Indiana
[still a tacit part of geographic databases?]


Bertand Russell; Campbell; Bridgeman; Stevens ...
properties not inherent in object; numbers derive from measurement operations
20th century dominated by representationalists.

Stevens in Historical Context

Measurement theorists concentrated on physics
'extensive' measures (eg. length) emphasized strict limits; special role for addition
'Operationism' enshrines the procedure;
result: no room for social sciences

Stevens: a psychophysicist
research on perceived loudness of sound
measurements rejected by operationists
General movement for quantitative approach to social sciences

Stevens' Scales of Measurement

Published 1946 in Science (without any references)
Adopts 'nominalist' form of representationalism:
Defines measurement as 'assignment of numbers to objects according to a rule'
Measurement scale defined by properties of numbers (not properties of the rules or objects)
Scales organized by transformations under which the meaning remains invariant.
1946: Nominal, Ordinal, Interval, Ratio
(log-interval added in 1959)
Links scales to permissible statistics

Stevens' Scales of Measurement

(from Table 1, Stevens 1946, page 678)

Scale    Empirical   Invariance            Permissible
         Operations  Group                 Statistics

NOMINAL  equality    any 1:1 substitution  Mode
                                           Number of cases
ORDINAL   , >       any monotonic         Median
                     function              Percentiles
INTERVAL equality of x' = ax + b           Mean, std. dev.
         intervals                         Correlations
RATIO    equality of x' = ax               Coefficient of 
         ratios                            variation

Nominal Measures: Based on sets

Venn diagram of set membership
Objects classified by shared attributes.
All members belong equally.
NOTE: Examples based on a footrace. "Objects" in this case are people.

[Critique of traditional categories]

Ordinal Measures

Order of arrival of contestants in footrace

         Women's race  Men's race
First    Jane          Tom
Second   Melissa       Dick
Third    Leila         Harry

Ordinal measures may be complete orderings, partial orderings or ranked categories
(High, Medium, Low)
(slight, medium, severe, very severe limitations)
Examples: Grading Sweet Potato crops in Indiana | Plant Hardiness in Texas

Interval Measures

Clock time of arrival at Finish Line of race Interval measures require a fixed distance, but the zero point is arbitrary.
Differences between two interval measures are ratio.
Interval measures are often raw results, with some additional relationships they become ratio.

Temperature Scales

a misleading example of interval measurement

A frequent example of interval measurement is temperature in degrees F, or C.
The 'interval' equation does apply:
degrees F = ( 5/9 degrees C ) + 32
BUT degrees K is not the 'true' ratio scale (extensive).

To combine regular extensive measures: ADD
To combine temperatures:
Two objects brought in contact reach a weighted equilibrium temperature.

Current Local Weather data: (Use BACK to return)

Ratio Measures

Elapsed time of race Ratio measures have:
a fixed zero that means 'no quantity';
a constant interval (distance on the scale).

May be rescaled because '1' is arbitrary.

Examples of ratio scales: SI units | Physical Reference Data (CODATA 86) | Bushels in Indiana

Extensive and Derived Scales

Stevens conflated two types of ratio measures.

Extensive: based on addition rule for combination
examples: distance, mass, time
Derived: created by division of extensive measures;
velocity = distance / time (speed of light meters per second) , Newtonian constant of gravitation, etc.

Different axioms apply
(recognized in thematic cartography practice)
Examples of Proportional Symbols and Choropleth
Extensive: proportional symbols
Derived: choropleth [See Figure 1-8 and 1-9 in Exploring GIS]

Stevens adds a Scale

In 1959, Stevens recognizes that the invariance rules could allow another scale on the same 'level' as interval.

The invariance function would have the form: x' = a times x to the b power

This 'log-interval' scale is mostly theoretical, though earthquake intensities on Richter and Mercali scales use this form.

None of the social science statistics textbooks take notice; cartographers still use the basic 4.

Beyond Ratio... Absolute (percent)

Ratio is not the 'highest' level of measurement, '1' need not be arbitrary.

Using Stevens' invariance scheme, an 'absolute' scale could not be rescaled by any multipier.
Both zero AND one are fixed.

Probability is an absolute scale.
Bayes' Law and other axioms depend on absolute scaling.
Just as a difference of intervals is ratio, some divisions of ratios produce absolute measures.

Counts: a misfit

Some measures are obtained by tabulating the total of some objects inside a larger collecting unit.

Are counts higher than ratio?
Are counts 'discrete' like categorical measures?
One answer: Counts are counts;
Extend the system to handle all the cases.

Cyclical measures

Stevens scales only accomodate open-ended linear scales.
Some measures are cyclical, with a fixed range.
Example: Angles

Additional axioms are required to handle the fact that 359 is the same distance from 0 as 1 is.

Multidimensional measures

Multidimensional scales have extra relationships.

In Stevens' scheme, an unbounded linear axis could not be fit into a fixed interval without losing information.

yet, two unbounded Cartesian axes can be transformed into radial coordinates with only one unbounded axis, and one angle.

Conversion from Cartesian coordiantes to Radial

Multidimensional Measurements

Physics Devices at Cornell <Dead? too bad>

Reconsidering Categories

'Nominal' measures depend upon Aristotle's view:
categories imply shared values of attributes.
Hence, test for equivalence only.

Two alternative approaches to categories:

Both provide a metric for membership beyond the simple Yes/No of traditional categories.


Matthews Datasets | Fish and Wildlife Service "Cowardin" manual for wetland classification (and its official hierarchy); National list of plants associated with wetlands (1996) with 5 categories of association <now dead> | FGDC Vegetation Standard <relocated?>|

Verdict on Stevens

The four levels are not adequate.

New Requirements:

Measurement involves more axioms than just those involved in the number line.

A larger framework for measurement

Stevens never questions the nature of the 'object'.
[in psychology, the 'case' is not a problem]

Standard model: case 'has' attributes
(not relationships that can be measured)
Geographical Matrix: places 'have' attributes
'Case' permits measurement as a form of control.

Geographic control is not completely routine and accepted.
Scheme based on Sinton: fixed, controlled, measured

Measurement Frameworks

based on relationships between geometry & attribute
[Lecture on Measurement Frameworks]

Object Control
	Isolated objects: 'features'; isolines
	Connected objects: network;  coverage	
Spatial Control
	Point-based: center point
	Area-based: (many rules)
Relationship Control
	Triangular Irregular Network (TIN)
Composite Frameworks (eg. Choropleth)

Object Frameworks (Vector)

basic rules: attribute serves as control, positions measured to suit

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Version of 1 October 2003 (updted for links only)