Guttman scales examples [easy] 2 of 4

["Robert J. Chassell" <bob@GNU.AI.MIT.EDU>]

This is the 2nd of 4 related messages.

Here are examples of different kinds of scale.  This message follows
after, and is best understood, within the context of another messages
I just posted, called `Guttman Scales'.

Categorical scales
------------------

When you use a categorical scale, you say that a proposition belongs
to the category of truthful propositions or to the category of false
propositions.  When you use such a scale, you are not saying how much
truth there is in a proposition, only that it is true, not false.  In
physics, for example, an entity either is or is not an electron.

Much logic is based on there being only two categories, true and false
One important reason for this is that it makes the mathematics
simpler.  Another is that parts of reality do seem to fit this, as
with electrons.

As a practical matter, other instances of reality really cannot
readily be assigned to one or other category, but need to be so
assigned for practical reasons.

Roy Rappaport, in _Ecology, Meaning, and Religion_, argues that a
major function of ritual is to impose unambiguous distinctions on
ambiguous differences.  For example, as he says, "...a young Tahitian
decides to have himself supercised at about the age of twelve.  He
thereby makes clear his transistion from the status of child to that
of {T'aura'area}.  The process of maturation is slow, continuous, and
obscure; the ritual summarizes the decision as a simple yes or no
signal."

In Western society, much of law has to do with imposing unambiguous
distinctions on ambiguous differences.


Linear ordering
---------------

A linear ordering provides more information than mere categorization.
A doctor often needs to know more than whether a patient is in pain or
not in pain.  A mild pain may signify a state quite different than a
strong pain.

Steven Belknap mentioned that he uses scales to help diagnose illness.
As Peter Schuerman so rightly says, you cannot say that a `level 8'
pain is twice as "bad" as a `level 4' pain.  The scale lacks the
mathematical properties of an Archimedean ordered field--you cannot
divide one level by another.  Indeed, such a scale also lacks the
properties of an ordered Abelian group--you cannot add levels.

The scale is a linear ordering.  As Schuerman says, the numbers can be
thought of as "awful", "bad", "fair", "good", "wonderful".  But
nonetheless, the results provide helpful information to Belknap and
the others around him.

With clever operations, you can sometimes convert such a scale to an
interval scale--not necessarily to a Fahrenheit-style interval scale,
but a scale in which there is one operation more or less parallel to
addition.  I'll discuss one such scale below, for dealing with
uncertainty.

Interval scales
---------------

The Fahrenheit and Celsius scales of temperature are examples of
interval scales.

In an interval scale, both the unit and the zero point of the measure
are arbitary, but the intervals are equal.  A Fahrenheit scale, for
example, contains 180 degrees between the temperatures of the freezing
and boiling points of water, and the zero point is 32 degrees below
freezing.

Such a scale provides more information than a linear ordering: not
merely that `Tuesday was warmer than Monday, and Monday was warmer
than Sunday', but that `Tuesday was 10 degrees warmer than Monday, and
Monday was 3 degrees warmer than Sunday'.

You can graph interval scales.

Of all the types of scale, interval scales are the most widely
misunderstood.  As far as I can see, this is because people take
Fahrenheit and Celsius temperature scales for granted, and
automatically avoid the problems they create by using an absolute
(Kelvin) scale as needed.

Contemporary environmental regulation illustrates the
misunderstandings that come from interval scales.

A regulator can say that it is better to preserve two acres of swamp
along the Mississippi River than one; likewise, he or she can also say
that it is better to save two owls than one.

But, no one can say that whether it is better to save one owl rather
than one acre of swamp.  This leads to many controversies.  Part of
the problem is that there is no way (instrinsic to ecology) to compare
owls to swamps.  If it were possible to compare the two, you could
draw up a graph trading off one or the other, and pick the best ratio
of spending.  But this cannot be done.  Yet funding is in money, which
fits into a ratio scale, and people expect that funding for owls
should be visibly comparable to funding for swamps (or tax cuts).

Further below, I will develop an example of an interval scale that has
an operation of addition/subtraction that does not work exactly like
conventional addition/subtraction.

Ratio scales
------------

A ratio scale permits intercomparison among different entities.  We
cannot say that George's euphoria is twice that of Steven, but we can
say that George's weight is twice that of Steven.  We can also say
that George is 10% taller than Steven.

Market prices are a ratio scale; these oranges cost twice as much as
those apples.

You can compare apples and oranges when you use a ratio scale, by
using weight, absolute temperature, price, or other measure that has a
ratio scale.

You cannot compare apples and oranges when you use some other metric.


    Robert J. Chassell               bob@gnu.ai.mit.edu
    25 Rattlesnake Mountain Road     bob@rattlesnake.com
    Stockbridge, MA 01262-0693 USA   (413) 298-4725