Page 43 - Contributed Paper Session (CPS) - Volume 4

CPS2111 Grant J. Cameron et al.
Axioms are rigorous properties for an index to satisfy, and they are more
formalized and generalized than the properties discussed earlier in Subsection
3.2.5. Axioms help in understanding what an index is actually measuring and
in deciding which index to use. Knowing which properties an index satisfies
can help in interpreting the empirical results obtained using that index; certain
forms of policy analysis become possible only when the index satisfies a given
property. Some axioms can be interpreted as “nuggets of policy” that specify
the kinds of changes that should leave the index value unchanged and those
that should alter it. Others break down the index value to help understand
how dimensions contribute to that value. Our proposed index satisfies three
axioms, which include symmetry, monotonicity, and subgroup
decomposability.
As noted in Foster et al (2013), axioms can be usefully grouped into three
categories: invariance axioms, which indicate what not to measure; dominance
axioms, which indicate what the index should measure; and subgroup axioms,
which break down or build up indices by variables or units of analysis. The
three axioms our proposed index of statistical capacity satisfies are closely
related with these three groups of axioms. In what follows, a generic index of
statistical capacity over profiles = (1,… , ) will be denoted by F.
Symmetry: in other measurement environments where the number of
people, dimensions, or other factors may differ across comparisons, invariance
axioms are often used to ensure consistency. In the present context, the index
F is being applied to one country’s data with a fixed number of dimensions
and dichotomous variables, so properties of this sort are not needed. A
second common form of invariance axiom is anonymity or symmetry whereby
the index value is unaffected when variable levels are switched. In the present
context, where the variables have a structure as represented by hierarchical
tree T and partition P, universal symmetry is not appropriate. Motivated by
Basu and Foster (1998), one might consider a weaker form of symmetry that
is contingent on variables being “similarly placed” in the variable structure.
We say that profile b is obtained from profile a by a basic switch if = ′ and
′ = for some ≠ ′ in the same basic group, while " = " for all other v".
In other words, the only difference between b and a is that two variable values
in the same basic group have been switched. A statistical capacity measure F
satisfies basic symmetry if () = () whenever is obtained from by a basic
switch. Notice that any nested counting index satisfies basic symmetry
because it has the same weight on every variable in the same basic group.
Monotonicity: the main axiom for F is an intuitive dominance axiom
requiring the index value to reflect improvements in variables. We say that
profile b is obtained from profile a by an improvement if ≥ for all v, and
≠ or, in other words, if profile vector dominates profile . A statistical
capacity index F satisfies monotonicity if () > () whenever is obtained

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