12
value x = 1, then the only conclusion that can be drawn in this case is that Θ[a]/k ≥ 1,
and therefore that Θ[a] ≥ k. Thus, this Boolean thermometer has taken the underlying algebraically rich scale of the quantity subject to measurement (the temperature
Θ) and rendered it as an ordinal quantity: that is, it has operated as a pass/fail classifier. This then is the link to the correct/incorrect nature of the RCA items, as described
in the previous section. Of course, the imperfection of this instrument is clear—it
does no more than divide up temperatures into two categories, above k and below (or
equal to) k. And this is, of course, also the reason why RCA tests always consist of
multiple RCA items: so that the RCA scale will be able to distinguish more categories.
Then, to accomplish this using Boolean thermometers, suppose that an array of
M calibrated Boolean thermometers are available, each of them with a different
constant k i , and sequenced so that k i < k i+1 . (This sequencing is an immediate byproduct of the calibration described in the previous paragraph.) Then, given an
object a whose temperature is to be measured, the measurement procedure would be
to apply the measurand Θ[a] to the Boolean thermometers in sequence until the jth
thermometer is determined such that:
• Θ[a] generates an elongation in all thermometers i, i < j, i.e., the indication value
x i = 1 is obtained, so that Θ[a] ≥ k i .
• Θ[a] does not generate an elongation in the jth thermometer, i.e., the indication
value x j = 0 is obtained, so that Θ[a] < k j .
(Hence if j = 1, i.e., no thermometers elongate, Θ[a] < k 1 , and if j = M + 1, i.e., all
thermometers elongate, Θ[a] ≥ k M .)
In the simplest case of a sequence of M = 2 Boolean thermometers, with constants k 1 and k 2 , k 1 < k 2 , three cases can then arise:
(a) x 1 = 0, i.e., the applied temperature does not elongate any Boolean thermometer: Θ[a] < k 1 .
(b) x 1 = 1 and x 2 = 0, i.e., the applied temperature elongates the first Boolean thermometer but not the second one: k 1 ≤ Θ[a] < k 2 .
(c) x 2 = 1, i.e., the applied temperature elongates both the Boolean thermometers:
Θ[a] ≥ k 2 .
Clearly, this procedure can be extended to any number of Boolean thermometers
that were pragmatically usable in a given context. And that is exactly the formal
foundation for one classical approach to measurement in the human sciences, called
Guttman scaling (1944). Under this approach, RCA “Guttman” items are seen as
being related to the underlying scale as is the Boolean thermometer in Eq. (1.2), and
a sequence of successively harder Guttman items are generated, so that they specify
an ordinal scale of readers.
This illustrates what one can do if one already has an algebraically rich measurand such as temperature. The real situation in the case of RCA is, of course, that this
is not readily available, so that one must, in some sense, reverse the logic that was
worked through here, to proceed from the Guttman items back to the underlying
scale. The problem is actually a little bit more complicated than that, as the drawback
to this formulation is that RCA (and other human science items) only seldom function so exactly as given in Eq. (1.2), but rather they function in a less reliable way,
1 Introduction
value x = 1, then the only conclusion that can be drawn in this case is that Θ[a]/k ≥ 1,
and therefore that Θ[a] ≥ k. Thus, this Boolean thermometer has taken the underlying algebraically rich scale of the quantity subject to measurement (the temperature
Θ) and rendered it as an ordinal quantity: that is, it has operated as a pass/fail classifier. This then is the link to the correct/incorrect nature of the RCA items, as described
in the previous section. Of course, the imperfection of this instrument is clear—it
does no more than divide up temperatures into two categories, above k and below (or
equal to) k. And this is, of course, also the reason why RCA tests always consist of
multiple RCA items: so that the RCA scale will be able to distinguish more categories.
Then, to accomplish this using Boolean thermometers, suppose that an array of
M calibrated Boolean thermometers are available, each of them with a different
constant k i , and sequenced so that k i < k i+1 . (This sequencing is an immediate byproduct of the calibration described in the previous paragraph.) Then, given an
object a whose temperature is to be measured, the measurement procedure would be
to apply the measurand Θ[a] to the Boolean thermometers in sequence until the jth
thermometer is determined such that:
• Θ[a] generates an elongation in all thermometers i, i < j, i.e., the indication value
x i = 1 is obtained, so that Θ[a] ≥ k i .
• Θ[a] does not generate an elongation in the jth thermometer, i.e., the indication
value x j = 0 is obtained, so that Θ[a] < k j .
(Hence if j = 1, i.e., no thermometers elongate, Θ[a] < k 1 , and if j = M + 1, i.e., all
thermometers elongate, Θ[a] ≥ k M .)
In the simplest case of a sequence of M = 2 Boolean thermometers, with constants k 1 and k 2 , k 1 < k 2 , three cases can then arise:
(a) x 1 = 0, i.e., the applied temperature does not elongate any Boolean thermometer: Θ[a] < k 1 .
(b) x 1 = 1 and x 2 = 0, i.e., the applied temperature elongates the first Boolean thermometer but not the second one: k 1 ≤ Θ[a] < k 2 .
(c) x 2 = 1, i.e., the applied temperature elongates both the Boolean thermometers:
Θ[a] ≥ k 2 .
Clearly, this procedure can be extended to any number of Boolean thermometers
that were pragmatically usable in a given context. And that is exactly the formal
foundation for one classical approach to measurement in the human sciences, called
Guttman scaling (1944). Under this approach, RCA “Guttman” items are seen as
being related to the underlying scale as is the Boolean thermometer in Eq. (1.2), and
a sequence of successively harder Guttman items are generated, so that they specify
an ordinal scale of readers.
This illustrates what one can do if one already has an algebraically rich measurand such as temperature. The real situation in the case of RCA is, of course, that this
is not readily available, so that one must, in some sense, reverse the logic that was
worked through here, to proceed from the Guttman items back to the underlying
scale. The problem is actually a little bit more complicated than that, as the drawback
to this formulation is that RCA (and other human science items) only seldom function so exactly as given in Eq. (1.2), but rather they function in a less reliable way,
1 Introduction