Suppes and Zinnes - basic measurement theory.pdf - Ted Sider

BASIC MEASUREMENT THEORY
Theorem 34 (Uniqueness Theorem). If the successive interval system
il3 = (B, k,f) is an A.M. system, then the derived scale (iB, R, !") is an
interval scale in both the narrow and wide senses.
Adams and Messick also note that if the values of f.i-a and ua are fixed
for one stimulus or, alternatively, if one of the normal cumulatives associated
with the stimuli is fixed, then the derived numerical assignment
p is uniquely determined.
For completeness we should point out that the numbers t 0 , tb ... , tk
can also be thought of as the values of a derived numerical assignment for
the category boundaries; that is, separate representation and uniqueness
theorems can be established for the boundaries as well as for the stimuli,
but since the theorems for the boundaries are similar to those just described
for the stimuli we shall not pursue these details further.
5. THE PROBLEM OF MEANINGFULNESS
Closely related to the two fundamental issues of measurement-the
representation and uniqueness problems-is a third problem which we
shall term the meaningfulness problem. Although this problem is not
central to a theory of measurement, it is often involved with various
aspects of how measurements may be used, and as such it has engendered
considerable controversy. It therefore merits some discussion here,
although necessarily our treatment will be curtailed.
5.1 Definition
To begin with, it will be well to illustrate the basic source of the meaningfulness
problem with two of the statements, (4) and (5), given previously
in Sec. 1.2. For convenience they are repeated here.
4. The ratio of the maximum temperature today (tJ to the maximum
temperature yesterday (tn_ 1 ) is 1.10.
5. The ratio of the difference between today's and yesterday's maximum
temperature Ctn and tn-J to the difference between today's and
tomorrow's maximum temperature Ctn and tn+1) will be 0.95.
In Sec. 1.2 statement 4 was dismissed for having no clear empirical
meaning and (5), on the other hand, was said to be acceptable. Here we
wish to be completely explicit regarding the basis of this difference. Note
first that both statements are similar in at least one important respect:
neither statement specifies which numerical assignment from the set of
THE PROBLEM OF MEANINGFULNESS
admissible numerical assignments is to be used to determine the validity
or truth of the statement. Are the temperature measurements to be made
on the centigrade, Fahrenheit, or possibly the Kelvin scale? This question
is not answered by either statement.
The distinguishing feature of ( 4) is that this ambiguity in the statement
is critical. As an example, suppose by using the Fahrenheit scale we found
that tn = 110 and 1n_ 1 = 100. We would then conclude that (4) was a
true statement, the ratio of tn to tn_ 1 being 1.10. Note, however, that if
the temperature measurements had been made on the centigrade scale
we would have come to the opposite conclusion, since then we would
have found that tn = 43.3 and tn-l = 37.8 and a ratio equal to 1.15
rather than 1.10. In the case of (4) the selection of a particular numerical
assignment influences the conclusions we come to concerning the truth of
the statement.
In contrast, consider statement 5. The choice of a specific numerical
assignment is not critical. To illustrate this point assume that when we
measure temperature on a Fahrenheit scale we find the following readings:
tn_ 1 = 60, tn = 79.0, and tn+ 1 = 99.0. Then the ratio described in (5) is
equal to
60- 79 = -19 = 0.95
79- 99 -20
which would indicate that (5) was valid. Now, if, instead, the temperature
measurements had been made on the centigrade scale, we would have
found that tn_ 1 = 15.56, tn = 26.11, and tn+ 1 = 37.20. But, since the
ratio of these numbers is
15.56- 26.11 = -10.55 = 0.95,
26.11- 37.20 -11.09
we would also have come to the conclusion that (5) was true. In fact, it is
easy to verify that if we restrict ourselves to numerical assignments which
are linearly related to each other then we will always arrive at the same
conclusion concerning the validity of (5). If the statement is true for one
numerical assignment, then it will be true for all. And, furthermore, if
it is untrue for one numerical assignment, then it will be untrue for all.
For this reason we can say that the selection of a specific numerical
assignment to test the statement is not critical.
The absence of units in (4) and (5) is deliberate because the determination
of units and an appreciation of their empirical signiHcance comes after,
not before, the investigation of questions of meaningfulness. The character
of (4) well illustrates this point. If the Fahrenheit scale is specified in (4),
the result is an empirical statement that is unambiguously true or false, but