168
The actual numbers obtained for θ n and δ i are termed “logits” (i.e., log of the
odds, or log-odds units),
27
and are typically used to generate property values in ways
similar to the way that is done for temperature units: the logits are on an interval
scale, so what is needed are two fixed and socially accessible points. One standard
way is to assign two relatively extreme values: for example, one might decide that
for a given population of readers, say State X for year 20YZ, the 100.0 point level
might be the mean of the logits for readers in Grade 1, while a higher value, say
500.0, would be chosen as the mean for students in Grade 12: this would be suitable,
for example, for a reading test used in a longitudinal context (for an expanded discussion, see, e.g., Briggs, 2019).
A second but similar way, perhaps more suitable for test applications focused on
particular grades, would be to allocate a value, say 500 as the mean for readers in
Grade 6, say, and 100 as the standard deviation for the same students. Some applications also use the raw logits from their analyses—this effectively embeds the interpretation of the units in a given sample, which may be acceptable in some research
and development situations, but would be difficult to justify in a broadly used application. There is also a more traditional set of practices that use (a) an ordinal
approach to classifying readers into a sequence of reading performance categories,
and (b) a “norm-referenced” approach that carries out similar techniques to those
just described, but uses the raw scores from the reading tests rather than estimates
from a psychometric model.
The conclusion is then that values of RCA are individual abilities identified as
elements in a log-odds (interval) scale based on ratios of probabilities (see Freund,
2019, for a discussion of these types of scales).
6.4 The epistemic role of Basic Evaluation Equations
The conclusion reached in the previous section has an important implication for an
ontology of quantities, and properties more generally (as developed further in the
following section). A Basic Evaluation Equation such as Q[a] ≈ x q ref reports not
just an attribution or a representation, but the claim of an indistinguishability, and in
the form Q[a] = x q ref the claim of an equality, of individual quantities: if it is true,
it informs us that two individual quantities that were known according to different
criteria are in fact one and the same. In detail:
• Before the relation is evaluated, we are able to identify an individual quantity q
as a quantity of an object a, Q[a], and a set of individual quantities q x , each as a
value x q ref , for a given q ref and a number x varying in a given set; q and each q x
are quantities of the same kind.
27 Equation (6.2) has no closed-form solution for θ and δ; hence we are not providing equations for
them.
6 Values, scales, and the existence of properties
The actual numbers obtained for θ n and δ i are termed “logits” (i.e., log of the
odds, or log-odds units),
27
and are typically used to generate property values in ways
similar to the way that is done for temperature units: the logits are on an interval
scale, so what is needed are two fixed and socially accessible points. One standard
way is to assign two relatively extreme values: for example, one might decide that
for a given population of readers, say State X for year 20YZ, the 100.0 point level
might be the mean of the logits for readers in Grade 1, while a higher value, say
500.0, would be chosen as the mean for students in Grade 12: this would be suitable,
for example, for a reading test used in a longitudinal context (for an expanded discussion, see, e.g., Briggs, 2019).
A second but similar way, perhaps more suitable for test applications focused on
particular grades, would be to allocate a value, say 500 as the mean for readers in
Grade 6, say, and 100 as the standard deviation for the same students. Some applications also use the raw logits from their analyses—this effectively embeds the interpretation of the units in a given sample, which may be acceptable in some research
and development situations, but would be difficult to justify in a broadly used application. There is also a more traditional set of practices that use (a) an ordinal
approach to classifying readers into a sequence of reading performance categories,
and (b) a “norm-referenced” approach that carries out similar techniques to those
just described, but uses the raw scores from the reading tests rather than estimates
from a psychometric model.
The conclusion is then that values of RCA are individual abilities identified as
elements in a log-odds (interval) scale based on ratios of probabilities (see Freund,
2019, for a discussion of these types of scales).
6.4 The epistemic role of Basic Evaluation Equations
The conclusion reached in the previous section has an important implication for an
ontology of quantities, and properties more generally (as developed further in the
following section). A Basic Evaluation Equation such as Q[a] ≈ x q ref reports not
just an attribution or a representation, but the claim of an indistinguishability, and in
the form Q[a] = x q ref the claim of an equality, of individual quantities: if it is true,
it informs us that two individual quantities that were known according to different
criteria are in fact one and the same. In detail:
• Before the relation is evaluated, we are able to identify an individual quantity q
as a quantity of an object a, Q[a], and a set of individual quantities q x , each as a
value x q ref , for a given q ref and a number x varying in a given set; q and each q x
are quantities of the same kind.
27 Equation (6.2) has no closed-form solution for θ and δ; hence we are not providing equations for
them.
6 Values, scales, and the existence of properties