222
might lead to the addition of a third reference property, corresponding to such an
“unclear” situation.
7.3.5.3 Local scale construction and application
The reference properties are still empirical entities, being patterns which may be
read on a sheet of paper. A map X i
*
→ x i from reference properties X i
*
to values x i is
then defined (see Fig. 7.6). Typically, for each true/false item the patterns corresponding to the correct response are scored as 1 and the patterns corresponding to
the incorrect response are scored as 0, thus mapping responses into the informational sphere. Once the local scale has been constructed, whenever one of its reference properties X i
*
is selected by a reader, the corresponding value x i , which is an
informational entity, is recognized by applying the mapping (i.e., the reader’s
response is “scored”). This is the conclusion of the pre-measurement, as depicted in
Fig. 7.7.
7.3.5.4 Interlude: reality check
In fact, the theory and practice of RCA are not sufficiently advanced that such oneitem instruments could dependably provide information of sufficiently high quality
for most intended purposes. A more realistic situation is that test developers would
create a set of K items I m = {I mk }, where k = 1, …, K, that are, as above, designed to
interact with the RCA of a reader.
21
Then, once the reader has responded to all items
in the test, the transduced property is the vector of the responses of the reader to the
set of items, X m = (X m1 , …, X mK ) (i.e., a vector composed of responses like the one in
Fig. 7.17). These items may be designed so that some responses are indicative of
more comprehension (i.e., higher Θ), and other responses are indicative of less.
Thus, the response to each item can be matched using a reference property,
22
as in
Fig. 7.18, and then mapped to a local value, which in the example of true/false items
will be 1 for the correct response and 0 for the incorrect response. The local values
for the RCA of reader a can then be gathered into a vector x i = (x i1 , …, x iK ) of 1s and
0s of length K, which might be then summarized by taking the sum of the K values,
as before, referred to as “sum scores” (or “total scores”), as indeed has been done in
psychology and education for over a century within the framework of classical test
theory (e.g., Nunally & Bernstein, 1994: pp. 215–247, 308–310).
21 This is typical of all passive instruments, which are activated by the object under measurement
instead of by an external source. Also in this regard a RCA test is analogous to an alcohol thermometer, in which the upper surface of alcohol in the tube changes its position only because of the
energy transferred from the object whose temperature is under measurement.
22 Note that this will result in a matrix of local references X ik
*
.
7 Modeling measurement and its quality
might lead to the addition of a third reference property, corresponding to such an
“unclear” situation.
7.3.5.3 Local scale construction and application
The reference properties are still empirical entities, being patterns which may be
read on a sheet of paper. A map X i
*
→ x i from reference properties X i
*
to values x i is
then defined (see Fig. 7.6). Typically, for each true/false item the patterns corresponding to the correct response are scored as 1 and the patterns corresponding to
the incorrect response are scored as 0, thus mapping responses into the informational sphere. Once the local scale has been constructed, whenever one of its reference properties X i
*
is selected by a reader, the corresponding value x i , which is an
informational entity, is recognized by applying the mapping (i.e., the reader’s
response is “scored”). This is the conclusion of the pre-measurement, as depicted in
Fig. 7.7.
7.3.5.4 Interlude: reality check
In fact, the theory and practice of RCA are not sufficiently advanced that such oneitem instruments could dependably provide information of sufficiently high quality
for most intended purposes. A more realistic situation is that test developers would
create a set of K items I m = {I mk }, where k = 1, …, K, that are, as above, designed to
interact with the RCA of a reader.
21
Then, once the reader has responded to all items
in the test, the transduced property is the vector of the responses of the reader to the
set of items, X m = (X m1 , …, X mK ) (i.e., a vector composed of responses like the one in
Fig. 7.17). These items may be designed so that some responses are indicative of
more comprehension (i.e., higher Θ), and other responses are indicative of less.
Thus, the response to each item can be matched using a reference property,
22
as in
Fig. 7.18, and then mapped to a local value, which in the example of true/false items
will be 1 for the correct response and 0 for the incorrect response. The local values
for the RCA of reader a can then be gathered into a vector x i = (x i1 , …, x iK ) of 1s and
0s of length K, which might be then summarized by taking the sum of the K values,
as before, referred to as “sum scores” (or “total scores”), as indeed has been done in
psychology and education for over a century within the framework of classical test
theory (e.g., Nunally & Bernstein, 1994: pp. 215–247, 308–310).
21 This is typical of all passive instruments, which are activated by the object under measurement
instead of by an external source. Also in this regard a RCA test is analogous to an alcohol thermometer, in which the upper surface of alcohol in the tube changes its position only because of the
energy transferred from the object whose temperature is under measurement.
22 Note that this will result in a matrix of local references X ik
*
.
7 Modeling measurement and its quality