## Abstract

In this empirically driven paper we compare the performance of two techniques in the literature of poverty measurement with ordinal data: multidimensional poverty indices and first order dominance techniques (FOD). Combining multiple scenario simulated data with observed data from 48 Demographic and Health Surveys around the developing world, our empirical findings suggest that the FOD approach can be implemented as a useful robustness check for ordinal poverty indices like the multidimensional poverty index (MPI; the United Nations Development Program’s flagship poverty indicator) to distinguish between those country comparisons that are sensitive to alternative specifications of basic measurement assumptions and those which are not. To the extent that the FOD approach is able to uncover the socio-economic gradient that exists between countries, it can be proposed as a viable complement to the MPI with the advantage of not having to rely on many of the normatively binding assumptions that underpin the construction of the index.

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## Notes

According to the ‘union’ approach, an individual should be labeled as ‘poor’ if s/he is deprived in at least one dimension. At the other extreme, the ‘intersection’ approach states that an individual is ‘poor’ if s/he is deprived in all dimensions simultaneously. In between these extreme perspectives, Alkire and Foster (2011) proposed a counting approach based on Atkinson (2003) suggesting that an individual is ‘poor’ when s/he is deprived in an intermediate number of dimensions that has to be decided by the analyst. These well-known approaches can be seen as particular cases of the more general identification method using partially ordered sets suggested in Fattore (2016).

A partial order relation in a set

*X*is a binary relation satisfying*reflexivity*(\(x \le x\) for all \(x \in X\)),*antisymmetry*(if \(x \le y\) and \(y \le x\), then \(x = y\) for all \(x,y \in X\)) and*transitivity*(if \(x \le y\) and \(y \le z\), then \(x \le z\) for all \(x,y,z \in X\)) (see Davey and Priestley 2002).We say that \(x = \left( {x_{1} , \ldots ,x_{k} } \right)\) vector dominates \(y = \left( {y_{1} , \ldots ,y_{k} } \right)\) if \(x_{i} \ge y_{i}\) for all \(i = 1, \ldots ,k\).

The time needed to run these simulations increases rapidly with

*k*(for the case*k*= 10 the computation time for anIntel^{®}Xeon^{®}E5-1650 v3 3, 5 GHz, RAM 16 GB is about 24 h). For this reason, in our simulations we have not considered more than 10 variables.The choice of different values of

*α*and*β*for the poverty index does not substantially change our findings. The results are not shown here, but are available upon request.In this example, the number of comparable and incomparable pairs equal 17,605 and 2295 respectively (hence, the relative shares of comparable and incomparable pairs are 88.5 and 11.5%, see Fig. 2). This means that the areas under the scaled density functions \(f_{C,3}\) and \(f_{N,3}\) equal 17,605 and 2295, respectively.

In its original definition, the UNDP’s MPI does not have the sub-domains we have introduced here. We have introduced them to have a more gradual dimensional refinement that allows exploring in more detail the effects of increasing dimensionality on the occurrence of FOD relationships.

Choosing alternative values for \(\alpha ,\beta\) leads to results that are highly correlated with the ones presented here, so they will not be reported.

For the five and ten dimensional cases the corresponding tables are considerably larger, so they are not shown here (they are available from the authors upon request).

Iglesias et al. (2016) compare confirmatory factor analysis, the Alkire and Foster counting approach and the posetic approach in the context of contemporary Switzerland.

A

*linear extension of*is a partial order that (i) is complete over (i.e. all pairs of elements are comparable), and (ii) respects the order established by vector dominance.Yet, imposing some simplifying assumptions (e.g. the

*k*attributes can be completely ordered in terms of relevance) the computational time can be considerably reduced (see Fattore and Arcagni 2017).

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## Acknowledgements

Financial support from the Spanish Ministry of Economy and Competitiveness “Ramón y Cajal” Research Grant Program and research projects ERC-2014-StG-637768 and ECO2013-46516-C4-1-R is gratefully acknowledged.

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## Appendices

### Appendix 1: Establishing the Existence of FOD

The applied algorithm to establish the existence of FOD relationships is based on a slight rephrasing of Arndt et al. (2015). Let *A* and *B* be two populations characterized by probability mass functions *f* and *g* respectively. For outcomes s and s′ with s′ ≤ s, let *t*
_{s,s′} be the amount of probability mass transferred from outcome s to s′. Note that the first subscript denotes the source of the transfer whereas the second denotes the destination. Given the conditions outlined above, population *A* dominates population *B* if and only if there exists a feasible solution to the following linear problem

where *S* is the space of outcomes—an issue that is determined via the GAMS 23.0 software (see https://www.gams.com).

### Appendix 2: The Multidimensional Binary Case

To illustrate the FOD definitions we focus on the case of different binary indicators that is applied in the empirical sections of the paper. While our empirical analysis explores the existence of *FOD*
_{
k
} relationships between pairs of countries when *k* goes from 1 to 10, for simplicity we base our illustration on the 3-dimensional case. In this case, the space of outcomes *S* is a partially ordered set with 2^{3} = 8 elements (*S* = {(0,0,0), (1,0,0), (0,1,0), (0,0,1), (1,1,0), (1,0,1), (0,1,1), (1,1,1)}) and the partial order given by the usual vector dominance ≤. Here (0,0,0) denotes the outcome where someone is deprived in all three dimensions simultaneously, (1,1,0) means that someone is only deprived in the third dimension, and so on. We now show the eleven inequalities (denoted as *I*
_{1},…,*I*
_{11}) derived from condition (3) in the definition of FOD that must be satisfied to conclude that, given two probability mass functions on *S* (*f* and *g*), one of them dominates the other (*f* FOD_{3} *g*).

The FOD concept is illustrated in Fig. 7 using three hypothetical welfare indicators denoted as I, II and III. The entries in the cells represent the probabilities for the joint distribution of all three indicators. We consider three hypothetical distributions: *q*, *r* and *s*. As can be seen, the percentage of individuals that are deprived in all three indicators at the same time in *q*, *r* and *s* are 20, 30 and 10 respectively. We see that *q* does not FOD *r*, and vice versa. This is because although *r*(0,0,0) > *q*(0,0,0) giving 30 > 20 (e.g. condition (*I*
_{1}) is fulfilled), we also see that *r*(0,0,0) + *r*(1,0,0) + *r*(0,1,0) + *r*(1,1,0) < *q*(0,0,0) + *q*(1,0,0) + *q*(0,1,0) + *q*(1,1,0) giving 34 < 35, which is a violation of condition (*I*
_{5}). The lack of FOD is illustrative of the fact that *q* and *r* alter rank depending on evaluation criteria. For instance *q* is better than *r* if the criterion is minimization of the group with members who are simultaneously worse off in all dimensions (*q*(0,0,0) < *r*(0,0,0) giving 20 < 30). But *r* is better than *q* if the criterion is instead maximization of population shares characterized by good outcomes in the three dimensions separately; shares with good outcomes in dimensions I–III is 66, 67 and 66% for distribution *r*, compared with 65% for each dimension in distribution *q*.

Looking next at welfare distributions *r* and *s* we also see that none dominates the other, e.g. lack of FOD. Condition (*I*
_{1}) is fulfilled since *r*(0,0,0) > *s*(0,0,0) (giving 30 > 10), but the last condition (*I*
_{11}) is not fulfilled (37 < 40). Again the problem that arises is that domination depends on evaluation criteria. Distribution *s* is better than *q* if the criterion is maximization of population shares characterized by good outcomes in the three dimensions separately; shares with good outcomes in dimensions I–III is 75% in each in distribution *s*, while the population shares are 66, 67 and 66% for distribution *r*. On the other hand *r* is better than *s* if we want to maximize the group that simultaneously does well on all three criteria, e.g. *r*(1,1,1) > *s*(1,1,1) giving 63 > 60.

The remaining comparison is between distributions *q* and *s*. If we insert Fig. 7 probabilities in conditions (*I*
_{1})–(*I*
_{11}) we see all are met, and we can therefore conclude that *s* FOD *q*. To reach that conclusion we can also use the intuitive strategy where we move probability mass from better to worse to see if one distribution (the dominated one) can be generated from the other (the dominating one). In this case we just need to move 10% from the best outcome (1,1,1) to the worst outcome (0,0,0) in distribution *s*, which will result in distribution *q*.

### Appendix 3: Comparison with Other Related Methods

The techniques analyzed in this paper bear some resemblance with other approaches recently proposed in the literature of multidimensional poverty in ordinal settings. For clarification purposes it will be useful to highlight what they have in common and what are the key differences between them. One of these approaches suggests using the theory of partially ordered sets to measure multidimensional deprivation (e.g. Fattore 2016) and the other proposes robust multidimensional poverty comparison techniques applied to ordinal data (see Yalonetzky 2014).

As regards the former, Annoni et al. (2011), Fattore et al. (2011) or Fattore (2016) among others have made great strides to apply partial order theory to better capture multidimensional poverty in ordinal settings. Like in this paper, individuals’ achievements are assessed via \(k \in {\mathbb{N}}\) ordinal variables, so evaluations are also based on the structure of the poset \(\left( {S, \le } \right)\). In a nutshell, the approach can be summarized as follows (for technical details, see Fattore 2016). First, a decision maker must establish deprivation thresholds, that is: identify what combinations of achievements constitute the unambiguously/completely deprived profiles. Second, an identification and severity functions assign the deprivation intensity associated to each achievement profile. These functions satisfy two consistency conditions: (1) if \(p \in S\) and \(q \in \downarrow p\), then the intensity of poverty in *p* is lower than the one in *q*; (2) deprivation thresholds are assigned the maximum deprivation degree. Lastly, a population level deprivation indicator is obtained aggregating individual deprivation levels.

Despite the limited structure of the original poset \(\left( {S, \le } \right)\) where many pairs of achievement profiles are incomparable in terms of vector dominance, the approach successfully generates a *complete* order that—unlike the FOD approach—allows comparing *all* possible pairs of countries (with each country represented as a probability mass function over *S*). The only assumption that has to be made to reach such completeness—which does not seem to be particularly restrictive—is that all the elements of the set of linear extensions^{Footnote 12} of *S* (denoted as \(\varOmega \left( S \right)\)) are supposed to be equally important. Unfortunately, the huge size of \(\varOmega \left( S \right)\) when the number of achievement profiles in *S* gets larger severely limits the applicability of the approach: as of now, computations are feasible for posets with a few hundreds of elements^{Footnote 13} (Arcagni and Fattore 2014; Fattore 2016). In this regard, the applicability of the FOD approach is also restricted by computational considerations: in this paper we have dealt with up to *k* = 10 binary variables, thus resulting in posets with 2^{10} = 1024 achievement profiles; see appendices 1 and 2 and footnote #5).

Another conceptually related approach has been recently suggested by Yalonetzky (2014). In that paper, the author presents the conditions that must be satisfied if one aims to conclude that the poverty levels associated with a multivariate ordinal distribution (*A*) are unambiguously lower than those of another distribution (*B*) *for all possible poverty thresholds Z and all possible weighting vectors W*. Yet, the results are only valid (i) when the function to identify the poor corresponds to the extreme union or intersection approaches; and (ii) when several restrictions on the signs of the cross partial derivatives of the underlying social welfare function—some of which being particularly difficult to interpret—are satisfied. On the other hand, the FOD techniques make no assumption regarding (i) the choice of weights; and (ii) the behavior of the underlying social welfare function. The last point is particularly attractive because in many empirical applications it is unclear whether the different pairs of dimensions should be treated as complements or substitutes.

Summing up, both approaches are interesting on their own right and each of them has its own advantages and disadvantages. While Yalonetzky (2014) method allows dominance analysis to varying deprivation cutoffs and weights but imposes several conditions on the underlying social welfare function, the FOD approach does neither impose conditions on that function nor in the weights but is dependent on the choice of deprivation cutoffs. Since neither of the methods encompasses the other, they have the potential of being complementary tools that can be jointly applied in empirical analyses.

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Permanyer, I., Hussain, M.A. First Order Dominance Techniques and Multidimensional Poverty Indices: An Empirical Comparison of Different Approaches.
*Soc Indic Res* **137**, 867–893 (2018). https://doi.org/10.1007/s11205-017-1637-x

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DOI: https://doi.org/10.1007/s11205-017-1637-x

### Keywords

- Multidimensional poverty measurement
- Poverty index
- First order dominance
- Robustness
- Simulations