# Beyond Stevens:A revised approach to measurementfor geographic information

## Outline of Presentation

• Background: Measurement Theory
• Classical versus Representationalist
• Applied to cartography
• Steven's Scales of Measurement
• What is missing
• Measurement Frameworks for Geographic Information

## 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?]

#### Representationalism

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)
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

## 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

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

## 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

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

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)

Extensive: proportional symbols
Derived: choropleth [See Figure 1-8 and 1-9 in Exploring GIS]

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.

• Counts are discrete, since only integers can result.
• There is a real zero, and 2x does mean twice the quantity.
• The unit is not arbitrary; 'one' cannot be reset.

Are counts higher than ratio?
Are counts 'discrete' like categorical measures?
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

• Fixed zero (no turn)

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.

## 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:

• Graded membership : some scale of partial connection (eg. probability or fuzzy set)
• Distance from prototype (not normalized); [object fits with closest exemplar]; Common in photointerpretation, land use, etc. (eg. "dominant" species)

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

### Examples

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:

• Nominal must be expanded
• Extensive and derived must be distinguished
• Higher levels recognized
• Multidimensional relationships handled

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

• Isolated Objects: each category taken as Yes/No
• 'Spatial Object' (or cartographic feature view): each entity (point,line,area) surrounded by the void
• Isoline: continuous attribute sliced, disjoint
• Connected Objects: multinomial (more than one) categories
• Network: object connectivity relationships
• Categorical Coverage: exhaustive classification divides space with contiguous boundaries

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Version of 1 October 2003 (updted for links only)