A Reduced Posetic Approach to the Measurement of Multidimensional Ordinal Deprivation

Abstract

In this paper, we discuss the existence of particular systems of generators for posets associated to multidimensional systems of ordinal indicators and derive a reduced posetic procedure for the measurement of multidimensional ordinal deprivation. The proposal is motivated by the need to lessen the computational complexity of the original posetic procedure described in Fattore (Soc Indic Res 128(2):835–858, 2015), so as to make it applicable to larger multi-indicator systems, particularly to those comprising many variables scored on “short” scales, as typical in deprivation studies. The reduced procedure computes identification and severity functions based only on so-called lexicographic linear extensions. These are a particular generating system for the basic achievement poset, naturally associated to rankings of deprivation attributes. After motivating this choice, both from an interpretative and a computational point of view, the paper provides some simulated examples, comparing the reduced and the non-reduced procedures.

Appendix

In this appendix, we show that representation (4) holds for a generic product order of k linear orders. The proof is simple, but rather technical and is given only for sake of completeness.

Let $\varPi$ be the product order of k chains, arising from k attributes $v_1,\ldots ,v_k$, scored on degrees $m_1,\dots ,m_h$, $h=1,\ldots ,k$ (in the following, for sake of simplicity, we assume degrees are actually coded as $0,1,2\ldots$). Observe that:

• 1. Partial orders of profiles built “conditioning” on each attribute separately are extensions of $\varPi$, since the new comparabilities added to $\varPi$ involve only profiles with conflicting scores (i.e. incomparable in $\varPi$). From this, we see that conditional linear orders are extensions of $\varPi$ as well.

• 2. Profiles having the same “sum” of scores are incomparable in $\varPi$, since they necessarily have conflicting scores. Moreover, if profile $\varvec{q}$ covers profile $\varvec{p}$ in $\varPi$, then $\varvec{p}$ and $\varvec{q}$ are identical on all of their components but one, otherwise other profiles would be in between the two. Let this component be the j-th. For the very same reason, it must be $q_j^{}=p_j^{}+1$. This implies that profiles on the same level from bottom, have the same sum of scores. Adding edges to the Hasse diagram of $\varPi$ in such a way that profiles on a level cover profiles on the level below thus produces a compensative extension of $\varPi$. Hence, also compensative linear orders are extensions of $\varPi$.

• 3. Consider poset $\varvec{2}^{4}_{}$. The following total order ”<” on the achievement profiles

  $$\begin{aligned}&0000<0001<0010<0011<0100<0101<0110<1000<\\<&0111<1001<1010<1011<1100<1101<1110<1111 \end{aligned}$$

  is a linear extension of $\varvec{2}^{4}_{}$, but it is neither a conditional, nor a compensative linear extension (i.e. it is “mixing”).

By construction, $\varPi$ is thus the intersection of conditional, compensative and mixing extensions. Since conditional linear extensions cannot be compensative and both, by construction, cannot be mixing, we see that the families of linear extensions on the right hand of formula (4) have no elements in common.