Measuring the depth of multidimensional poverty with ordinal data

Measuring the depth of m ultidimensional p o v ert y with ordinal data F ernando Flores T a v ares a a Dep artment of Ec onomics and Statistics, University of Siena, Italy, Email: fernando.tavar es@unisi.it Abstract This pap er proposes a p ositional p o vert y gap measure of m ultidimensional p o v erty within the Alkire–F oster coun ting framew ork. The measure captures the depth of depriv ations even when indicators are ordinal, unlik e the stan- dard p o vert y gap, whic h requires cardinal v ariables. The prop osed metho d dra ws on the fuzzy set literature and in tro duces a distribution-based measure of depriv ation depth using the empirical cumulativ e distribution of eac h in- dicator, with the most deprived group as the benchmark. F or each deprived individual, the metho d assigns a score based on the individual’s relativ e p osition in the distribution. Depth is thus expressed as a difference in distri- butional positions, motiv ating the lab el positional p o vert y gap. The paper demonstrates that this measure preserv es the iden tification and aggregation structure of t he coun ting approac h and satisfies its axiomatic prop erties when the reference distribution remains fixed ov er time. The framework remains flexible b ecause it accommo dates differen t iden tification rules, depriv ation cutoffs, and v ariable types. Overall, it offers a simple, meaningful, and the- oretically grounded wa y to incorp orate depth into multidimensional p o vert y measuremen t with ordinal data. Keywor ds: Multidimensional p o v erty, p ositional p o vert y gap, ordinal v ariables, Alkire-F oster metho d, fuzzy-set approac h JEL Classific ation: I32, C43, O15, D63 Marc h 17, 2026 1. In tro duction This pap er introduces a p ositional p o vert y gap measure for multidimen- sional pov ert y within the Alkire–F oster coun ting framew ork, extending it to capture depriv ation depth when indicators are ordinal. The con tribu- tion bridges the Alkire–F oster (AF) metho d ( Alkire and F oster , 2011a ) and the T otally F uzzy and Relativ e (TFR) approach ( Cheli and Lemmi , 1995 ). It pro vides a coherent framework for incorp orating depth in to multidimen- sional p o vert y measurement while preserving the identification and aggrega- tion structure of the coun ting approach. In the unidimensional con text, depth is t ypically measured through the p o v erty gap, the normalized shortfall from the p o v erty line, first prop osed b y Sen ( 1973 ) and generalized by F oster et al. ( 1984 ) in to a class of measures. Alkire and F oster ( 2011a ) extended the F oster–Greer–Thorb eck e (FGT) class of measures to a m ultidimensional setting, capturing the extent, breadth, and depth of m ultidimensional p o v erty . The latter is measured through the multi- dimensional analogue of the FGT p o vert y gap, whic h captures the normalized shortfall of an individual’s ac hiev emen t from the depriv ation cutoff. Ho w- ev er, this measure requires cardinal v ariables, since ordinal v ariables do not pro vide meaningful information ab out the magnitude of differences b et ween categories ( Alkire and F oster , 2011b ). In practice, empirical applications of the AF framew ork rarely implement the p o v erty gap comp onen t b ecause most p o v erty indicators are ordinal ( D’Attoma and Matteucci , 2024 ). As a result, the depth of m ultidimensional p o vert y remains largely understudied. T o fill this gap, I build on the TFR fuzzy-set approach and introduce a relativ e measure of depriv ation depth based on the empirical cumulativ e distribution of each indicator. F or each depriv ed individual, the metho d assigns a score that reflects the share of the p opulation with higher outcomes, normalized by the corresp onding share for the most deprived p erson. Depth is thus expressed as a difference in distributional p ositions: it increases as the individual mov es closer to the w orst observed outcome. This framing motiv ates the lab el p ositional pov ert y gap. T o analyze the b eha vior of incorp orating a p ositional p o v erty gap within the AF class of measures, I formalize and verify its prop erties. I sho w that holding the distributional reference fixed ensures it satisfies the main desir- able axioms, including p o vert y fo cus, monotonicity , subgroup consistency , and decomp osabilit y , yielding predictable b eha vior across p o vert y compar- isons ( Alkire et al. , 2015 ). 2 I then illustrate the metho ds with t wo applications. Using data from Brazil, I sho w that individuals depriv ed in few indicators can still exhibit high p ositional p ov ert y gap within those indicators, a pattern that incidence and in tensity alone miss. With data from Bangladesh, I compare the p ositional p o v erty gap measure with the AF p ov ert y gap using cardinal indicators: the t wo rank individuals similarly and their differences reveal complemen tary asp ects of p ov ert y depth. The con tributions of this pap er are practical and p olicy-relev ant. A first con tribution is to op en depth measurement to a wider range of empirical settings. F or instance, it allows researchers to ask, for the first time in ordinal and multidimensional settings, how deeply deprived the p o or are, whether depth and intensit y p oin t to the same individuals and groups, and ho w the distribution of sev erity v aries within the p o or p opulation. The global MPI and most national m ultidimensional p ov ert y indices rely hea vily on ordinal indicators. The prop osed measure is directly applicable to these existing framew orks without requiring new data collection or changes to indicator selection. Depth measurement, in turn, has a direct p olicy application. By iden- tifying individuals with few but deep depriv ations, the p ositional p ov ert y gap measure can help direct resources to ward those whom standard mea- sures ma y giv e lo w er priorit y—not b ecause they are not p o or, but b ecause their p o v erty is deep rather than broad. Moreov er, by incorp orating a rel- ativ e component to the class of measures, the p ositional p o vert y gap index accoun ts for the so cial context of depriv ation, complementing the absolute measures of incidence and in tensity . The pap er proceeds as follows. Sections 2 and 3 review the background literature and introduce the notation. Section 4 discusses the construction of indicators and the conditions required for ordinally consistent measurement. Section 5 presen ts the p o vert y iden tification approach. Sections 6 and 7 define the in tensity and p ositional p o v erty gap comp onen ts of the index. Section 8 combines them into the o verall m ultidimensional p o vert y index. Section 9 establishes the axiomatic prop erties of the measure. Section 10 presents t wo empirical illustrations using data from Brazil and Bangladesh. Section 11 concludes. 3 2. Bac kground The AF metho d is the most used approach in the literature on multidi- mensional p o v erty ( D’Attoma and Matteucci , 2024 ). This approac h offers an in tuitive, simple, and flexible w ay to capture o verlapping depriv ations ( Alkire and F oster , 2011b ). These asp ects reflect its success in reaching b e- y ond academia, as 49 countries w orldwide ha ve used or are using the metho d to build official national or lo cal multidimensional p o v erty indexes ( Multidi- mensional Po v ert y P eer Netw ork , 2025 ), achieving its practical ob jectiv e of informing action to reduce p o v erty . Despite its success, the approac h is not imm une to criticism. Tw o limi- tations are particularly relev an t to the discussion in this study . First, most AF applications cannot measure depth. Second, the additive aggregation structure do es not accoun t for cross-dimensional complementarit y , the com- p ounding effect of exp eriencing multiple depriv ations simultaneously . P at- tanaik and Xu ( 2018 ) elab orate b oth p oin ts: under AF’s dual-cutoff rule, a p erson with small depriv ations in k dimensions is classified as po or while someone with larger depriv ations in k − 1 dimensions is not; and the ad- ditiv e structure cannot capture the in teraction of disadv an tages, conflicting with the Stiglitz-Sen-Fitoussi principle that “multiple disadv an tages far ex- ceed the sum of individual effects” ( Stiglitz et al. , 2009 ). Their prop osed solution, raising aggregated intensit y to a con vex p o wer, p enalises higher cu- m ulative depriv ation disprop ortionately , but remains neutral to how those depriv ations are spread across dimensions. Rippin ( 2010 ) addresses cross-dimensional complemen tarit y more directly , prop osing a class of measures that w eigh ts eac h individual’s con tribution b y an increasing con vex function of their total depriv ation count. Con vexit y implies that the marginal p o vert y impact of eac h additional depriv ation rises with the num ber already exp erienced. Datt ( 2019 ) extends this logic in to a t wo-parameter family , applying separate conv exit y parameters within eac h dimension, p enalising larger gaps more hea vily , and across dimensions, p e- nalising m ultiple depriv ations more than their sum. AF and P attanaik-Xu emerge as sp ecial cases of Datt’s family . How ev er, while these extensions usefully expand the scop e of the AF framew ork, they do not address the depth limitation. A parallel literature based on fuzzy-set approaches provides an alternative route to measuring depriv ation. Instead of assigning a binary p o or/non-po or status, these metho ds represent depriv ation as a matter of degree through 4 mem b ership functions that exploit the full v ariation a v ailable in ordinal indi- cators. A foundational con tribution is the distribution-based fuzzy approach of Cheli and Lemmi ( 1995 ), whose TFR metho d derives membership v al- ues purely from rank p ositions and therefore do es not require a cardinal in terpretation of the data. Betti and V erma ( 1999 ) prop osed an alternativ e form ulation based on the complement of the Lorenz curve computed from w eighted ac hiev ements, whic h instead requires cardinal measuremen t. Betti et al. ( 2006 ) sho wed how these t wo distribution-based membership functions can be com bined within a unified framew ork. Building on this work, Betti et al. ( 2015 ) prop osed a m ultidimensional implemen tation in whic h indica- tors are first transformed using the normalised TFR, enabling the consisten t application of the Lorenz-curv e complement across indicators. The depth index prop osed here builds on this transformation approach, adapting it to measure depriv ation depth. Despite the adv an tages, fuzzy approaches ha v e tw o limitations relev an t here. First, without a depriv ation threshold or censoring, all individuals receiv e non-zero scores, violating the depriv ation and p o vert y fo cus axioms. Second, the o verall fuzzy index yields a single aggregate measure without sep- arating p o v erty into incidence, in tensity , and depth, making it incompatible with the rep orting framework that has made AF attractiv e for p olicy . Tw o pap ers hav e attempted to bridge the tw o streams. Kobus ( 2017 ) dev elops a fuzzy coun terpart to the AF MPI linking union and dual-cutoff iden tification, and Gangopadh y ay et al. ( 2021 ) prop ose a comprehensiv e ax- iomatic framework combining absolute depriv ation with a distributional com- p onen t. Both preserve elements of the AF structure, but b oth retain a car- dinal depriv ation gap as the within-dimension depth score (in the tradition of Bourguignon and Chakrav arty , 2003 ), making them inapplicable when indicators are purely ordinal Prop osals ha v e also b een made to measure p o v erty appropriately when indicators are ordinal. A prominen t con tribution is Seth and Y alonetzky ( 2021 ), who dev elop a class of indices defined as w eigh ted sums of the p opu- lation shares in depriv ation categories b elow a cutoff, using ordering w eigh ts that assign larger weigh ts to more depriv ed categories and decrease mono- tonically as categories b ecome less deprived. Their prop osal, how ev er, ap- plies only to unidimensional settings. As the ordering w eights are defined o ver eac h indicator’s sp ecific set of depriv ation categories, a given weigh t v alue reflects different levels of depriv ation depending on the num ber of cat- egories and their definitions. As a result, the scale of depriv ation severit y 5 v aries across indicators. Com bining such weigh ts in a multidimensional ag- gregation w ould therefore require an additional normalization step to ensure comparabilit y across indicators, which is absent from their framework. This pap er bridges these differen t streams of m ultidimensional pov ert y . It k eeps the AF framew ork structure, incorporating the distribution-based score of the fuzzy tradition within it and formalising the score and its axiomatic prop erties for ordinal indicators in a m ultidimensional setting. In doing so, it pro vides a new in terpretation of the score as a p ositional analogue of the AF p o vert y gap. The o verall index factors into incidence, in tensity , and depth, a structure the fuzzy literature do es not pro vide. This m ultiplicative structure is also compatible with cross-dimensional complemen tarity exten- sions, such as those prop osed b y Rippin ( 2010 ), Pattanaik and Xu ( 2018 ), or Datt ( 2019 ), where simultaneous depriv ations are penalised more heavily . The following sections formalize the prop osed depth measure, establish its axiomatic prop erties, and illustrate its application. 3. Notation Let X N D denote the ac hievemen t matrix of size n × d , where ro ws index individuals i = 1 , 2 , . . . , n and columns index indicators j = 1 , 2 , . . . , d . The en try x ij represen ts the achiev emen t of individual i in indicator j . Eac h indicator j is asso ciated with a depriv ation cutoff z j . I define the depriv ation status of i in j as g 0 ij = 1 { x ij < z j } , (1) where 1 {·} is the indicator function that equals one if the statement is true and zero otherwise 1 . Indicators are assigned relative imp ortance through a w eight vector w = ( w 1 , . . . , w D ), with P D j =1 w j = 1. In addition, to capture depth I define the empirical cum ulative distribu- 1 The g 0 ij elemen ts form the depriv ation matrix g 0 and z j depriv ations cutoffs form the v ector z . Let g 0 = [ g 0 ij ] n × d =    g 0 11 · · · g 0 1 D . . . . . . g 0 N 1 · · · g 0 N D    z =    z 1 . . . z D    . 6 tion function (CDF) for eac h indicator j as F j ( x ) = 1 n n X i =1 1 { x ij ≤ x } , (2) whic h gives the prop ortion of individuals whose achiev emen t in j do es not exceed x . Its complement, 1 − F j ( x ij ), represents the share of the p opula- tion that has strictly higher ac hievemen ts than individual i in indicator j 2 . This transformation places all indicators on a common [0 , 1] scale, enabling meaningful comparisons across differen t types of v ariables. 4. Indicator dev elopmen t process Man y factors need consideration when building an indicator, suc h as indi- cator selection, unit of identification, applicable p opulation, and others (see Alkire et al. , 2015 ; T av ares and Betti , 2024 ). These factors dep end on the study’s ob jective and the researcher’s judgment v alues. In this subsection, I fo cus on desirable conditions to build ordered indicators, apply admissi- ble transformations, and ensure comparability , considering meaningfulness, uncertain ties, and data av ailability 3 . 4.1. Indic ators c onstruction One of the adv an tages of the prop osed metho d is to use the informa- tion av ailable in eac h v ariable to estimate their depriv ation depth (p ositional p o v erty gap), even when they are ordinal. Before estimating the index, a pre- vious step is to build indicators that establish a clear and complete ranking for eac h element based on data a v ailabilit y . Based on the v ariable t yp e, one can implemen t different pro cedures to build indicators, as outlined b elo w: Nominal variables. This v ariable type cannot b e ordered, so it is irrele- v an t for measuring p ov ert y . Ho wev er, they ma y b e a v ariable that explains p o v erty , so one can use them in subgroup analysis, conditional exp ectations, and other m ulti-v ariable models. Or dinal variables. Supp ose that, in a dataset, a ra w v ariable r = { 0 , . . . , R } represen ts an attribute that has R categories. T o build an indicator based 2 In the empirical application, a p opulation-w eighted version of F j is used. 3 This section draws partially from insights presented in Chapter 2 of Alkire et al. ( 2015 ). 7 on these elemen ts, they m ust b e classified into ordered ac hievemen ts such that eac h ascending v alue represents a strictly higher achiev emen t ( < ) than the previous v alue, and in a wa y that one can define a depriv ation cutoff z j and non-trivially split the set in to at least t w o parts. Let the ordered set of ac hievemen ts for indicator j be X j = { x j 0 , . . . , x j X } with X > 2, and for eac h x j e , e ∈ { 0 , . . . , X − 1 } , the relationship x j e < x j,e +1 holds. Additionally , for an individual i , the observ ed achiev emen t x ij ∈ X j is classified as deprived if x ij < z j and as non-deprived if x ij ≥ z j . When r has only t wo categories or can only b e partitioned into t w o ordered sets, then X = 2. Ho wev er, it is not alw ays p ossible to rank all elements in a set. Some elemen ts may b e incomparable in relation to each other, meaning that an individual may not ha ve a preference among them, or it is not p ossible to distinguish which one is more adequate with resp ect to the other. In this case, these elements ma y be assigned the same achiev emen t v alue x j e , forming a semiorder. Another p ossibilit y is that sets of elemen ts can only b e distin- guished b et ween inadequate and adequate, forming a weak order. Therefore, an achiev emen t x j e ma y include one or more categories of the ra w v ariable r . If it is not clear whether a category is adequate or not, a go od prac- tice is to make decisions based on the context of the coun try/region and the ob jective of the study , and run robustness analysis with differen t p ossible classifications. As mentioned previously , the depriv ation cutoff z j separates the ac hieve- men ts X j in to t wo sets: deprived and non-depriv ed. The p ositional p ov ert y gap measure, how ev er, focuses on the ordering within the deprived set, since it censors achiev emen ts that meet the adequacy threshold. Measuring p osi- tional pov ert y gap depth, therefore, requires distinguishing degrees of depri- v ation among those identified as deprived. As an illustration, consider an indicator of housing conditions, whic h usually ev aluates the material condition of houses’ flo or, walls, and ro of. T o simplify things, let us consider only the flo or. Materials should first b e classified into adequate and inadequate, based on depriv ation cutoff z j that splits these tw o groups. Then, inadequate materials should b e ranked from most inadequate to least inadequate. Ho wev er, some materials are not clearly differen tiable from others in terms of adequacy of housing conditions (e.g., sand and other natural materials). The adequacy ma y dep end on the climatic, geographical, and so cial con texts. Therefore, one p ossibilit y is to classify these materials with the same ac hievemen t v alue. T o measure the depth of depriv ations (p ositional p o v erty gap), it is nec- 8 essary to non-trivially divide the set of deprived achiev emen ts, x ij < z j , into at least t wo ordered subsets. T o increase the possibilities of building m ultiple categories, t w o or more ra w v ariables that represent the same attribute ma y b e com bined. F or example, a v ariable that indicates if a household has access to electricit y is usually binary , so the depriv ation set cannot b e divided in t wo or more ordered parts. But, if there is a v ariable of frequency of elec- tricit y p er week, these v ariables ma y b e com bined to pro duce more than tw o ordered depriv ation sets, as follo ws: 0 = no electricit y; 1 = electricit y once p er w eek; 2 = electricit y twice a week; and so on. Another p ossibilit y to build indicators with more than tw o depriv ation sets is to combine tw o or more raw v ariables that are related but do not represen t the same attribute. F or example, the v ariables of housing materials and electricit y may b e com bined to build an indicator of housing conditions with multiple ordered categories. How ever, the analysis should av oid such com binations when p ossible, b ecause they mak e it more difficult to identify the factors driving the depriv ation score and, consequen tly , to deriv e precise p olicy implications. Car dinal variables. Raw v ariables r that are cardinal (interv al scale or ratio scale) are originally in a ranked order. Therefore, one need only to define the depriv ation cutoff z j and, when necessary , c hange the direction of the v ariable from most depriv ed to non-deprived. 4.2. A dmissible tr ansformations After building indicators, the next step is to ensure that, when estimating the measures, only admissible transformations are applied, and admissible statistical and mathematical op erations in the indicators considering their t yp e (i.e., ordinal, and cardinal). In this w ay , the measures will maintain meaningfulness. F ollo wing Alkire et al. ( 2015 ), I rely on the classification of scales of measuremen t and their resp ective admissible transformations as defined by Stev ens ( 1946 ). This author classified the v ariables in to four types of scales: nominal, ordinal, interv al scale, and ratio scale. He classifies the first t wo as qualitativ e and the last tw o as cardinal. F or eac h type of v ariable, Stev ens listed the admissible mathematical operations and p ermissible statistics. Con- sidering the t yp es of v ariables relev ant to m ultidimensional pov ert y , T able A1 in the App endix A sho ws these admissible op erations and examples relev an t to p o v erty measurement. 9 As T able A1 demonstrates, for ordinal v ariables, no mathematical op er- ation and only some statistics are admissible. Using non-admissible math- ematical op erations suc h as subtraction in ordinal v ariables w ould assume that the differences b et w een the indicator’s v alues hav e a meaning. How ever, these differences are meaningless b ecause the interv als b et ween the v alues are not necessarily equal or consisten t. F or instance, the differen t categories of t yp es of sanitation facilities rank these types based on adequacy , but they only pro vide information on the order of adequacy and not on the magnitude of an impro vemen t from one category to another. The p ov erty gap is an example of measure that applies mathematical op- erations that are not admissible for ordinal indicators (i.e., subtraction and division). That is wh y the p o vert y gap can only b e used when all the indi- cators of a m ultidimensional index are cardinal. In contrast, the in tensity as estimated in the AF metho d use an amissible transformation: it dic hotomizes the indicators into depriv ed and non-depriv ed, whic h is a monotonic trans- formation. The prop osed p ositional p o v erty gap measure extends the AF coun ting framew ork while relying only on admissible transformations and statistics. T o meaningfully estimate the depth of depriv ations, the p ositional p o vert y gap measure relies on the frequency distribution, whic h is a statistic p ermissi- ble for ordinal and cardinal v ariables types (see T able A1 ) 4 . More specifically , it applies a cum ulative distribution function (CDF), whic h conv erts all v ari- ables in to the same unit: the share of individuals with higher achiev emen ts than those who ha ve ac hieved x ij . In this wa y , the resulting quantities p ossess ratio-scale prop erties, with a meaningful zero and in terpretable differences and ratios. 4.3. Comp ar ability acr oss indic ators and individuals Besides ensuring that the transformations are admissible, one should v erify whether these transformations create indicators that are comparable among eac h other and among individuals. Comparabilit y is a c hallenge b e- cause multidimensional indexes deal with man y v ariables that differ in terms of scale and unit. T o assure comparabilit y , the v ariables m ust be transformed so that they b ecome meaningfully comparable. 4 Sections 6 and 7 further elab orate on the construction of the measures. 10 In terms of comparabilit y across indicators, using ordinals without trans- formations w ould wrongly assume that differences among the v alues of an indicator ha ve the same scale as the differences of other indicators’ v alues. Therefore, ordinal v ariables require an admissible transformation, as the last section show ed, that con verts the v ariable into the same unit and scale while k eeping the ranking order of the original v ariable. Moreo ver, it is usually assumed that weigh ts correctly represent the relativ e imp ortance of each in- dicator and dimension, ensuring comparabilit y among them. Regarding the comparability across individuals, in addition to the pre- viously outlined requiremen ts, it is assumed that an individual’s level of depriv ation represents an equiv alen t state of p ov erty when compared to oth- ers exhibiting iden tical depriv ation levels. This is a basic assumption that is also presen t in monetary p o vert y measures ( Alkire et al. , 2015 ). In the case of positional p o vert y gap, it transforms the v ariables using the frequency distribution, which produces indicators that range b et ween 0 and 1. The common unit is then the share of individuals with higher ac hieve- men ts than those who ha ve achiev ed x ij . All these pro cedures establish comparabilit y across indicators and individuals. 5. P o vert y iden tification The identification of p o vert y is not trivial, esp ecially in a m ultidimen- sional context. The approac h to identifying depriv ations and p o vert y is not only a statistical matter, but it is a c hoice that has p otential p olicy implica- tions. In this section, I explain the main p o v erty iden tification metho ds and ho w the metho d incorp orates them. Before iden tifying the po or, for eac h individual i , the metho d coun ts the indicators j in whic h they are deprived, denoted g 0 ij . As Subsection 4.1 shows, for each j a cutoff z j is defined, suc h that g 0 ij = 1 if x ij < z j and g 0 ij = 0 otherwise. The next step is calculating the individual depriv ation score c i , whic h is the sum of weigh ted depriv ations: c i = d X j =1 w j g 0 ij . (3) where the w eights w j are p ositiv e and normalized to sum to one, P d j =1 w j = 1. 11 Giv en a p o v erty cutoff k , the iden tification function ρ i ( k ) assigns p o vert y status according to the individual depriv ation score: ρ i ( k ) = ( 1 if c i ≥ k , 0 if c i < k . (4) When the depriv ation status is censored to only consider p o or individuals, g 0 ij ( k ) = ρ i ( k ) g 0 ij . If k = min j w j , ρ i ( k ) corresp onds to the union approach, classifying p eople as p o or if they are deprived in at least one indicator with p ositiv e w eight. If k = 1, ρ i ( k ) corresp onds to the interse ction approac h, classifying as p o or only those deprived in all indicators. Finally , for in termediate v alues min j w j < k < 1, ρ i ( k ) corresp onds to the dual-cutoff approac h prop osed b y Alkire and F oster ( 2011a ), which allo ws flexible p o v erty lines b et ween the extremes. Considering the dual-cutoff approac h, Alkire and F oster ( 2011a , b ) classify the union and in tersection iden tification as extreme cases, and advocate for in termediate pov ert y lines according to the ob jectives and preferences of the user, whic h is a w ay of em b o dying normative judgments. They recognize that p o v erty line c hoices are p oten tially arbitrary and, to minimize this problem, recommend robustness analysis to test for differen t c hoices. Therefore, if users w ant to align the metho d with the dual cutoff approac h, they ma y set an in termediate p o vert y line such as min j w j < k < 1. In the fuzzy literature, pov ert y is treated as a v ague indicator, recognizing that there is no exact p o vert y line that can clearly classify p eople as p o or or non-p o or. T o reduce arbitrariness, the approach usually disp enses with the definition of depriv ation cutoffs and p o vert y lines by assigning pov ert y degrees to every unit in the sample, which is compatible with the union approac h. Therefore, if users w an t to align the metho d with the fuzzy ap- proac h, they can adopt the union iden tification rule ( k = min j w j ) and set the depriv ation cutoffs at the maxim um of each v ariable, z j = X . In practice, this sp ecification implies that no effectiv e depriv ation and pov ert y cutoffs are imp osed. 5 Therefore, one ma y view the prop osed approach here as a more general framew ork that encompasses b oth the coun ting and fuzzy approac hes. Ev en 5 Ho wev er, under this sp ecification, one cannot estimate p ov erty intensit y and may violate the fo cus axiom (see Section 9 ). 12 under the dual-cutoff iden tification, it treats pov ert y as a matter of degree rather than a strictly binary condition. Once the metho d iden tifies an in- dividual as p oor, the analysis can also assess the extent of that p erson’s depriv ation. In this sense, the p o vert y line separates the non-p oor from the p oor, while allowing those iden tified as p o or to exp erience p o vert y to differen t degrees. 6. In tensit y F ollo wing Alkire and F oster ( 2011a ), the measure of intensit y represen ts the breadth of multidimensional pov ert y based on the num b er of weigh ted depriv ations, c i . The individual in tensity is defined as A i = d X j =1 w j g 0 ij ( k ) = c i ( k ) , (5) whic h censors the v alues of non-p oor. Since the w eights w j are normalized to sum to one, A i ranges b etw een zero and one, with greater v alues indicating stronger depriv ation breadth. Finally , the aggregated intensit y is measured by the av erage share of w eighted depriv ations among the p oor: A = 1 q q X i =1 d X j =1 w j g 0 ij ( k ) = 1 q q X i =1 A i , (6) where q is the n um b er of po or individuals. 6 In this w a y , in tensity captures the breadth of multidimensional p o vert y by iden tifying individuals with many depriv ations o ccurring sim ultaneously . 7. P ositional P o vert y Gap The p ositional pov erty gap measures the depth of multidimensional pov ert y . The prop osed approac h builds on the cumulativ e distribution function (CDF), dra wing on the in tuition of fuzzy p ov erty measures ( Cheli and Lemmi , 1995 ; 6 Throughout, q > 0 is as sumed when computing A and S ; b y conv ention b oth equal zero when q = 0. 13 Betti et al. , 2006 ), but reinterprets it here as a positional p o v erty gap mea- sure. This subsection formalizes the measure and clarifies its in terpretation as a depth index. Consider an individual i deprived in indicator j . The depth of this depri- v ation is measured b y comparing the individual’s achiev ement x ij with the w orst observed achiev ement in the same indicator, relativ e to the empirical distribution. Sp ecifically , the p ositional depth sc or e is defined as s ij = 1 − F j ( x ij ) 1 − F j (min( x j )) , (7) where F j ( x ij ) ∈ [0 , 1] is the individual’s p ositional rank in the distribution of indicator j , 1 denotes the top p osition in that distribution, and min( x j ) denotes the low est achiev ement observ ed for j . This score has a direct in ter- pretation: it measures the prop ortion of individuals who are b etter off than i , expressed relativ e to the p osition of the most depriv ed. The score s ij also admits a natural interpretation as a normalized p o- sitional depth. In the AF framework, the p o vert y gap measures depth as ( z j − x ij ) /z j , that is, the normalized shortfall of individual i ’s ac hievemen t from the depriv ation threshold. As discussed in previous sections, this for- m ulation relies on a cardinal notion of distance. F or ordinal indicators, such cardinal distances are not defined. Instead, depth can b e captured from differences of distribution p ositions: the quan tit y 1 − F j ( x ij ) represen ts the individual’s p osition relative to the top of the distribution, namely the share of the p opulation with higher outcomes. The term 1 − F j (min x j ) represents the same quan tity for the most deprived individual. The ratio s ij therefore measures the individual’s relative distance from the top, normalized b y the maxim um suc h distance. The reference point thus shifts from the depriv ation threshold in the AF p o v erty gap to the top of the distribution. This shift follo ws from ordinalit y: the cutoff z j has no cardinal in terpretation, whereas the top of the distribution is alw ays iden tifiable from rankings alone. By con- struction, the most deprived individual receives s ij = 1, and non-depriv ed individuals are censored to zero. In the case of binary indicators, all depriv ed individuals tak e the v alue x ij = min x j and therefore receiv e s ij = 1. After calculating the scores, I then censor them to ensure that they only apply to depriv ed indicators. I define the censored p ositional depth score of eac h depriv ed indicator as g 1 ij = g 0 ij s ij ( z ), replacing eac h item in the gap matrix with the indicator’s p ositional depth score and replacing with zero when the indicator x ij ≥ z j . I then apply the pov ert y iden tification function 14 suc h that g 1 ij ( k ) = g 1 ij ρ i ( k ), further replacing elements with zero when an individual is not p oor. The p ositional depth scores are inv ariant to c hanges in the depriv ation cutoff. Because the scores s ij are computed from the distribution of x ij b e- fore applying censoring, they do not dep end on the choice of z j . When the cutoff changes, only iden tification changes: individuals who mo ve abov e the threshold are censored to zero, while the scores of those who remain depriv ed sta y the same. The cutoff thus gov erns identificat ion, whereas the p ositional depth score reflects only the individual’s position in the distribution, inde- p enden tly of where the threshold is set. A t the individual lev el, p ositional pov ert y gap is defined as the weigh ted a verage of censored p ositional depth scores, normalized b y the individual’s censored depriv ation score. F ormally , S i = P d j =1 w j g 1 ij ( k ) P d j =1 w j g 0 ij ( k ) . (8) By con v ention, S i = 0 whenever P j w j g 0 ij ( k ) = 0, that is, for all non-p oor individuals. The aggregate p ositional p ov erty gap used in the index is then: S = q X i =1 d X j =1 w j g 1 ij ( k ) q X i =1 d X j =1 w j g 0 ij ( k ) . (9) In words, S is the av erage depth of a typic al deprivation episo de among the p oor (equiv alen tly , a depriv ation-weigh ted a verage of individual p ositional p o v erty gap v alues). The reference p opulation used to compute F j is an analytical choice op en to the researcher, and eac h option em b eds a different normative assump- tion. A national sample captures relative depriv ation within a country at a giv en p oint in time. A baseline y ear fixes the reference distribution, so that c hanges in measured depth reflect genuine c hanges in individuals’ p ositions relativ e to a fixed standard, rather than relativ e to a shifting distribution. P o oled cross-sections place all y ears on a common scale, 62suited to retro- sp ectiv e comparisons across perio ds. This flexibilit y mirrors the role of the 15 p o v erty cutoff k in AF: it is a normativ e decision, not a technical default, and should b e stated and justified alongside the choice of dimensions, weigh ts, and thresholds. 8. Ov erall m ultidimensional pov erty index This section integrates the t wo preceding comp onen ts, intensit y A i and the individual p ositional p o v erty gap S i , into a single p o vert y degree at the individual level. Aggregating these degrees across the p opulation yields the o verall index P , whic h captures incidence, breadth, and depth of multidi- mensional p o v erty in a single summary measure. 8.1. Individual p overty de gr e es As men tioned previously , m ultidimensional p o vert y is in terpreted in terms of degrees. The measure must reflect in termediate situations at the individual lev el, rather than an “all-or-nothing” pov ert y condition. T o ac hieve this, I define pov ert y degrees as the multiplication of the individual in tensit y and p ositional p ov erty gap measures. In formal terms, the p o vert y degree of individual i is giv en by P i = A i · S i . (10) Substituting the definitions of A i and S i , one obtains P i = d X j =1 w j g 0 ij ( k ) ! · P d j =1 w j g 1 ij ( k ) P d j =1 w j g 0 ij ( k ) ! = d X j =1 w j g 1 ij ( k ) . (11) Th us, the p o vert y degree of a multidimensionally p oor individual corre- sp onds to the w eighted a verage of the censored positional depth scores across the indicators in which the individual is depriv ed. Higher v alues indicate greater p ov erty lev els, with zero indicating non-p o vert y , one indicating total p o v erty , and v alues b et w een zero and one indicating in termediate p o vert y situations. Therefore, the measure follows the fuzzy logic, and P i can b e in terpreted as a membership function. 8.2. A ggr e gate d multidimensional p overty The incidence is the most used p o vert y measure b oth in academic researc h and official national statistics. This measure is b oth simple and intuitiv e, widely recognized b y people ev erywhere. Therefore, it is important to include the incidence as a comp onen t of the index while adjusting for its limitations. 16 The incidence of m ultidimensional p o vert y is the p ercen tage of p eople who are iden tified as p o or giv en a p o vert y line, k . It can be represented as follo ws: H = 1 n n X i =1 ρ i ( k ) = q n , (12) where ρ i ( k ) is the p o vert y identific ation function, n total p opulation, and q n umber of po or. H therefore measures the prop ortion of p eople who are iden tified as multidimensionally p o or. As one can see in Equation 12 , H is obtained only as an aggregated measure, as it is not p ossible to hav e individual levels. That is why it ma y b e preferable to use H as a separate measure from the p o v erty degrees, whic h is defined as an individual measure, and adjust it to b etter represen t the differen t asp ects of p o v erty 7 . I then define the aggregated multidimensional p o v erty , or adjusted p osi- tional p o v erty gap, as the pro duct of the three censored a verages: P = H · A · S = 1 n n X i =1 P i , (13) where A and S are the a v erage intensit y and p ositional p ov erty gap among the p oor, resp ectiv ely . Hence the adjusted p ositional p o vert y gap index P equals the p opulation mean of individual p ov erty degrees. The p ositional depth scores s ij can b e raised to a p o wer α > 1 to place greater weigh t on the most deeply deprived. One can then use an alternative notation and define a generalized index : P α = 1 n P i P j w j g α ij ( k ), whic h is analogous to the AF’s generalized family M α = 1 n P i P j w j g α ij ( k ). When α = 1, P is the p ositional p ov erty gap index, analogue of the AF adjusted p o v erty gap M 1 = H · A · G , where G is the normalized p ov erty gap (see Alkire et al. , 2015 ). Note that when all indicators are binary , P reco vers the AF’s M 0 . 7 F or simplicity , users can also employ H as a standalone pov erty indicator and inter- pret it in conjunction with the p o vert y degree, revealing that X % of the population is exp eriencing p ov erty but at v arying degrees. 17 9. Prop erties Axiomatic approaches to multidimensional p o vert y sp ecify prop erties de- signed to guarantee w ell-b eha v ed p o vert y indices. Most prop erties are temp o- ral, as they specify ho w the index should resp ond when recomputed for a new time p oin t in a panel. In this section, I discuss and verify whether standard axioms used in the multidimensional pov ert y literature ( Alkire and F oster , 2011a ; Alkire et al. , 2015 ) apply to the prop osed metho d. Here, I fo cus on the axioms for which the justification differs from that in the AF framew ork due to the p ositional p o v erty gap component. These axioms are: depriv ation and p o v erty fo cus, monotonicit y , subgroup consistency and decomp osabilit y , and w eak transfer. The remaining prop erties -symmetry , replication inv ari- ance, and weak rearrangemen t- are briefly discussed, as their verifications are direct and the same as for the AF’s measures. Moreo ver, to analyse the inno v ative part of the index, I fo cus on P (or, equiv alen tly , P α with α ≥ 1), distinguishing b et w een individual or aggregated lev els when necessary . This fo cus is due to the individual and aggregated in tensity here b eing the same as in the AF framew ork, and their prop erties are discussed elsewhere (see Alkire et al. ( 2015 )). I consider t wo p ossible implemen tations: (i) anchor e d CDFs , where eac h F j is estimated once either on a baseline sample (then used to score sub- sequen t years) or on a p ooled sample con taining all years under analysis (one-time estimation for the full set), and is then held fixed for ev aluation; (ii) in-sample CDFs , where F j is computed from the curren t sample, without a baseline reference, whic h means that the CDF is recalculated at eac h time p eriod. T able 1 summarises the prop erties and whether they hold or not, considering the t wo p ossible implemen tations. In what follows, I presen t a formal and an intui tive presentation of the prop erties. App endix B sho ws the verifications and pro ofs. Deprivation F o cus. Let X ′ N D b e obtained from X N D b y c hanging a single cell ( i, d ) suc h that x ′ id > x id ≥ z d (an impro v ement in a non-depriv ed indi- cator), and x ′ nj = x nj for all ( n, j )  = ( i, d ). A p o vert y measure P satisfies depriv ation fo cus if P ( X ′ N D ; z , k ) = P ( X N D ; z , k ) . This axiom guarantees that non-deprived indicators do not influence the index. P satisfies this axiom. This result follo ws from censoring the v alues of 18 T able 1: Axioms summary Prop ert y Anc hored CDF In-sample CDF Symmetry , replication, b ounds ✓ ✓ Ordinal in v ariance (admissible transf.) ✓ ✓ Depriv ation fo cus ✓ ✓ P ov erty focus ✓ ⋄ 1 Own-p erson monotonicity ✓ ✓ Aggregate monotonicit y ✓ ✗ Dimensional monotonicit y ✓ ✗ Decomp osabilit y & subgroup consistency ✓ ✗ W eak rearrangemen t ✓ ✗ W eak transfer ✗ ✗ Notes: ✓ satisfied; ⋄ conditional; ✗ fails. 1. Satisfied only when the p o vert y iden tification is by union ( k = min j w j ). the p ositional depth score for nondepriv ed indicators. The formal verification is in the App endix B . Poverty F o cus. Let X ′ N D b e obtained from X N D b y changing the ac hieve- men ts of a single p erson i so that ρ i ( k ) = ρ ′ i ( k ) = 0 (the p erson remains non- p oor), with x ′ nj = x nj for all ( n, j )  = ( i, · ). A p o v erty measure P satisfies Poverty F o cus if P ( X ′ N D ; z , k ) = P ( X N D ; z , k ) . This axiom states that changes in the ac hiev ements of the non-p o or do not affect measured p o vert y . P satisfies this prop ert y when the CDFs are anchor e d and also under in-sample CDFs if the identification is b y union ( k = min j w j ), b ecause non-p o or then hav e no depriv ed cells. The axiom ma y fail with in-sample CDFs and in termediate k , but only in a sp ecific case: an y c hange b y a non-p oor p erson (who has at least one depriv ation) to one of their depriv ed indicators ( x id < z d ), alters the empirical CDF for indicator d and thereby affects the p ositional depth scores of other individuals, so P ( X ′ N D ; z , k )  = P ( X N D ; z , k ). Appendix B presen ts the formal verification for the anc hored and union cases. Monotonicity. Let X ′ N D b e obtained from X N D b y worsening p erson i in indicator d : either (a) x ′ id < x id < z d (still deprived), or (b) x ′ id < z d ≤ x id (crosses into depriv ation). Monotonicity requires P ( X ′ ; z , k ) ≥ P ( X ; z , k ), with strict inequality whenev er the p erson is (or b ecomes) p o or and the 19 w orsening is strict in a deprived indicator. F or own-person monotonicit y , P i will alwa ys increase when a depriv ation gets worse. This is b ecause the p ositional p o v erty gap is like a “p ercen t- b ehind” score that is dep enden t on the p osition of the distribution. If x ij drops , i will ha v e a low er ranking. That can only mak e the p ercen t of p eople ahead of i stay the same or go up. App endix App endix B formally shows the v erification of this prop ert y . F or aggregated monotonicity , violation or not dep ends on whether one uses anchor e d CDFs or in-sample CDFs . F or the later, monotonicity will not hold. Because F j is a relativ e measure, when it is recomputed from the sample, a c hange in x id shifts all p ositional depth scores in column d : ∆ P = 1 n w d h ( s ′ id − s id ) ρ i ( k ) g 0 id + X n  = i  s ′ nd − s nd  ρ n ( k ) g 0 nd i . The second (externality) term can b e negative and dominate the first; hence ∆ P can b e < 0 ev en though individual i worsens in a deprived indicator. Therefore aggregate monotonicit y ne e d not hold with in-sample CDFs. 8 F or the anchor e d CDFs monotonicit y will hold, as c hanges in x id will not shift any other cells’ depth terms. F ormal v erification is in App endix App endix B . Sub gr oup Consistency and De c omp osability. Let the p opulation b e par- titioned in to m ≥ 2 m utually exclusive and collectiv ely exhaustive subgroups ℓ 1 , . . . , ℓ m with sizes n ℓ (so P m ℓ =1 n ℓ = n ), and let X ℓ denote the achiev ement matrix restricted to subgroup ℓ . Fix cutoffs ( z , k ), weigh ts w , and ev alu- ate the total p opulation and all subgroups using the same reference CDFs (anc hored). Then: Sub gr oup Consistency. If X ′ is obtained from X b y c hanging outcomes only in subgroup ℓ ′ so that P ( X ′ ℓ ′ ; z , k ) < P ( X ℓ ′ ; z , k ) and P ( X ′ ℓ ; z , k ) = 8 With in-sample CDFs, aggregate monotonicit y can fail via tw o c hannels, even if the p oor set is fixed. (i) Denominator (minimum) effe ct: if the minim um m d c hanges or the shar e at the minimum changes, the common normaliser D IS d = 1 − F IS d ( m d ) shifts b y ∆ D IS d = ± 1 / N (use the weigh ted analogue with surv ey weigh ts), whic h rescales all p ositional depth scores in column d : ∆ s IS nd = − (1 − F IS d ( x nd )) ∆ D IS d / ( D IS d ) 2 , ev en if ranks are unchanged. (ii) Pe er-r e distribution effe ct: with m d fixed, moving one observ ation from a to b > a (b oth b elo w z d ) changes the empirical CDF by ∆ F IS d ( x ) = − 1 / N for p eers with x ∈ [ a, b ) (and +1 / N if b < a ), shifting their numerators and hence ∆ s IS nd = − ∆ F IS d ( x nd ) /D IS d . Either effect can outw eigh the w orsened person’s increase, so the aggregate change can b e negative. 20 P ( X ℓ ; z , k ) for all ℓ  = ℓ ′ , while subgroup sizes remain the same, then P ( X ′ ; z , k ) < P ( X ; z , k ) , and the inequalit y reverses if subgroup ℓ ’ worsens. Population Sub gr oup De c omp osability. The ov erall index equals the p opulation- weighte d me an of subgroup indices—each subgroup con tributes in prop ortion to its p opulation share: P ( X ; z , k ) = m X ℓ =1  n ℓ n  P ( X ℓ ; z , k ) . This iden tity holds whether one computes the subgroups first or the total first, provided all are ev aluated against the same reference CDFs and fixed ( z , k , w ). F ormal v erification for both subgroup consistency and decompos- abilit y are av ailable at App endix B . We ak tr ansfer. An equalizing sw ap within one indicator, taking a lit- tle achiev emen t from a less-depriv ed p o or p erson and giving it to a more- depriv ed p o or p erson, keeping their a verage fixed, should not raise measured p o v erty . How ever, the prop osed measures do not satisfy this prop ert y , b e- cause they use a rank–based p enalty s ij ∝ 1 − F j ( x ij ). Its curv ature dep ends on the shap e of F j : if the density rises o ver the deprived range, 1 − F j ( · ) is c onc ave , so an equalising (Pigou–Dalton) transfer b et w een t wo p oor in in- dicator j can incr e ase the sum of p enalties. Hence, unlike AF gaps (which are conv ex for α ≥ 1), weak transfer is not distribution-free for CDF–based depth; with in-sample CDFs it can fail even more often due to denominator (minim um) rescaling and p eer-redistribution effects. The w eak transfer in the AF-framew ork holds only when α ≥ 1, whic h is rarely used. In practice, therefore, most studies do not comply with this prop ert y and do not rely on them. The metho dology also satisfies the following properties. Symmetry. Relabelling individuals or indicators (with weigh ts aligned) lea ves P unchanged. R eplic ation invarianc e. Duplicating the p opulation do es not change P since it is a (w eighted) mean of cell contributions. Bounds/normalisation. P ∈ [0 , 1], equals 0 when no one is p oor, and reac hes 1 only at maximal depriv ation depth and breadth by construction of g α ij ( k ) and normalised s ij . 21 Or dinal invarianc e. Monotone reco dings of ordinal indicators do not affect P b ecause s ij dep ends only on ranks via F j . We ak r e arr angement (holds under anchor e d CDFs). Within a p erson, re- placing a more deeply deprived indicator b y a less deeply deprived indicator, k eeping the n umber of deprived indicators unchanged, do es not increase P , b ecause anc hored s ij resp ects the indicator ordering. With in-sample CDFs, this ma y fail due to the same externalities noted ab o ve. Theorem 1. Given weights w , cutoffs ( z , k ) , and P α with α ≥ 1 , the metho d- olo gy with p ositional depth sc or es satisfies: symmetry , replication in v ariance , b ounds/normalisation , ordinal inv ariance , and depriv ation fo cus (anchor e d or in-sample); pov ert y fo cus under anc hored CDFs for any k , and under in-sample CDFs only under union identific ation; individual-lev el monotonic- it y (anchor e d or in-sample); aggregated-lev el and dimensional monotonicity under anc hored CDFs; subgroup decomp osabilit y and subgroup consistency under anc hored CDFs; and weak rearrangement under anchored CDFs. Pro of. The pro of for Theorem 1 is in the App endix Appendix B . When using the anchored CDFs for temp oral analysis, one p ossibility is to use a h ybrid implementation: the depth comp onen t S is computed with anchor e d CDFs F j , whereas the incidence H and breadth A are computed for eac h time p erio d. Anchoring remov es cross-person externalities in S , while H and A reflect pure contemporaneous depriv ations. F or purely cross-sectional analysis (a single y ear), the distinction anchored vs. in-sample across time is inapplicable. If subgroup decomp osability is desired within that year, ho wev er, the CDF should b e estimated once on the full-year sample and used as a common reference across subgroups. 10. Illustrativ e application This section presen ts tw o illustrations of the metho d. The first uses Brazilian data and rep orts results b y p o v erty line and by subgroups. It aims to demonstrate the added v alue of the p ositional p o vert y gap measure, S . The second illustration uses data from Bangladesh and relies exclusiv ely on cardinal v ariables. It rep orts results by p o vert y line and compares the rank distributions of the p ositional p o vert y gap and AF’s pov ert y gap. The 22 aim is to analyze the asso ciation b et w een the t wo measures and show that, although they are highly correlated, they remain complemen tary measures. 10.1. Br azil The data for this illustration comes from the Brazilian Con tinuous Na- tional Household Sample Survey (PNAD Con t ´ ın ua) 2023, fourth quarter, with 473,206 observ ations ( Instituto Brasileiro de Geografia e Estat ´ ıstica , 2024a ). F or this exercise, I created an index with four dimensions, eac h one ha ving one v ariable: (1) education, measured by a cardinal indicator of the deviation in y ears of schooling from an age-sp ecific reference lev el; (2) fo o d insecurit y measured by an ordinal indicator following Brazilian Institute of Geograph y and Statistics’s official classification ( Instituto Brasileiro de Ge- ografia e Estat ´ ıstica , 2024b ); (3) assets cardinally measured b y the num b er of household assets (i.e., tv, radio, tablet, in ternet access, cellphone); and (4) an ordinal indicator for income depriv ation (i.e., extreme deprived, de- priv ed, at-risk of depriv ation, and not-deprived). F or simplicity , the w eights are assumed to b e equal among the dimensions. Because not all measures are cardinal, it is not possible to calculate the p o v erty gap, so T able 2 sho ws the results for H , H · A , and P . The results are presen ted for three p o vert y lines: deprived in one indicator out of four k = 0 . 25, whic h is equiv alen t to the union approac h; depriv ed in t wo indicators k = 0 . 50, representing an in termediate level; depriv ed in three indicators k = 0 . 75; and deprived in all for indicators k = 1 . 00, which corresponds to the in tersection approach. T able 2 shows that for all measures the v alues decrease as the num b er of indicators increases in the p o vert y line. When k = 0 . 50, the share of m ultidimensionally p oor p eople equals H = 25 . 6%, and the adjusted equals A = 0 . 58, whic h means that, on a verage, p oor p eople are deprived in ab out 58% of the indicators. The p ositional p o v erty gap index equals S = 0 . 87, in- dicating that, on a verage across po or individuals and indicators, depriv ations reac h ab out 87% of the maxim um depriv ation observ ed in the p opulation, where the maxim um corresp onds to the most deprived category . These last t wo indices yield the H · A and P = H · A · S . Figure 1 depicts the v alues of P for the four p ossible p o vert y lines and b y color/ethnicit y . The results show that p o vert y dominance among the subgroups is robust, with Indigenous with the highest v alues, follo wed by Blac k and P ardo, and White and Asian. The curves show a tendency of 23 T able 2: Results by p o vert y cutoff ( k ), Brazil, PNADC 2023 Measure k = 0 . 25 k = 0 . 50 k = 0 . 75 k = 1 . 00 H 0.565 0.256 0.074 0.008 H · A 0.226 0.149 0.057 0.008 P = H · A · S 0.194 0.129 0.050 0.007 Note: k = 0 . 25 is the union approac h; k = 1 . 00 is the intersection. con vergence, suc h that when k = 1 . 00, the three subgroups ha ve more similar m ultidimensional p o vert y v alues. Figure 1: P by p o vert y line k Figure 2 compares the individual positional depth scores with depriv ation scores, sho wing how the p ositional p o vert y gap v aries within each p ossible in tensity group. F or A i = 0 . 25 (i.e., depriv ed in one indicator), the S i v aries from less than 0 . 75 to 1 . 00. This means that ev en if an individual has only one depriv ation, this depriv ation can b e very deep. This situation would not b e captured b y measuring only prev alence and in tensity . Moreov er, the higher is A i , the S i v alues also tend to b e greater. This subsection demonstrates that the new p ositional p o vert y gap mea- sure satisfies the dominance prop ert y and captures p olicy-relev ant situations 24 Figure 2: Individual p ositional depth scores by depriv ation intensit y Notes: Each p oint represents one individual classified as p oor under the union cutoff ( k = 0 . 25). Diamonds indicate the survey-w eighted mean S i within each intensit y group. n ≈ 267 , 409. 25 that standard approac hes ma y o verlook. Individuals with only a few but v ery deep depriv ations may not be identified as po or under con v entional multidi- mensional pov ert y measures. Ev en when they are iden tified, their scores fail to reflect the depth of their condition, as they would receiv e the same score as individuals who exp erience the same n um b er of depriv ations but at lo wer lev els of p ositional p o v erty gap. 10.2. Bangladesh This illustration uses data from UNICEF’s Multiple Indicator Cluster Surv ey (MICS) 2019, round 6, for Bangladesh, which includes information on children under fiv e y ears old and their mothers or primary caregiv ers (n=22,108) ( Bangladesh Bureau of Statistics and UNICEF , 2019 ). The aim here is to compare S with the AF’s p ov erty gap, G , so I built an index with three dimensions containing only cardinal indicators. The first dimension is education, measured by the mother’s y ears of education. The second dimen- sion is health, represen ted b y the Bo dy Mass Index-for-age z-score. The third dimension is Assets, measured b y the num b er of household assets. Here, I also set all w eights equally . T able 3 rep orts the main results b y p o v erty line, including the p o vert y index with p ositional p o vert y gap, P , and the AF’s adjusted p o vert y gap, P AF . The incidence ranges from 30 . 3% to 0 . 4%. So increasing the p ov erty cutoff from k = 0 . 67 to the intersection threshold ( k = 1) sharply reduces all p o vert y measures. Both P and P AF decline fast as the cutoff increases, with v ery similar prop ortional reductions (around 90%), with P decreasing marginally faster than P AF . T able 3: Results by p o vert y cutoff ( k ), Bangladesh, MICS 2019 Measure k = 0 . 33 k = 0 . 67 k = 1 . 00 H 0.303 0.057 0.004 H · A 0.121 0.039 0.004 P = H · A · S 0.116 0.038 0.003 P AF = H · A · G 0.065 0.021 0.002 Notes: k = 0 . 33 is the union approac h; k = 1 . 00 is the intersection. P AF is computed follo wing Alkire et al. ( 2015 ); G denotes the a verage normalised p o vert y gap. The t wo indices measure differen t concepts. Although b oth reflect the depth of p o v erty , they do so from distinct p ersp ectiv es. Therefore, their 26 magnitudes are not directly comparable. F or this reason, I compare only their rankings to assess whether the rank classifications pro duced by the t wo measures are similar. If they are, the p ositional p o vert y gap measure prop osed in this paper could serve as a v alid alternativ e to the AF p ov erty gap in the absence of cardinal indicators. That said, to further assess the relationship b et w een P and P AF , T able 4 rep orts Pearson, Sp earman, and Kendall τ b correlations. The co efficients indicate a strong p ositive asso ciation. Pearson’s co efficien t (0.880) captures linear dependence in lev els and sho ws substan tial co v ariance betw een the t wo measures. Sp earman’s rank correlation (0.899), sligh tly higher, reflects an ev en stronger monotonic asso ciation, indicating that individuals with higher AF gaps almost systematically rank higher in p ositional p o vert y gap, ev en if the relationship is not perfectly linear. Kendall’s τ b (0.772) is smaller in mag- nitude b ecause it is based on pairwise concordance and is more conserv ativ e, particularly in the presence of ties. T able 4: Correlations b et ween AF’s p ov erty gap and p ositional p o vert y gap Metho d Estimate SE N P earson 0.880 0.003 6859 Sp earman 0.899 0.003 6859 Kendall τ b 0.772 0.009 6859 Notes: P earson’s and Sp earman’s correla- tions are design-based and computed using 500 b ootstrap replicate weigh ts. Kendall’s τ b correlations are unw eighted. Figure 3 illustrates the measures’ rank relationship visually . Graph (a) plots the percentile rank of the p ositional p o vert y gap ( S i ) in relation to the p o v erty gap ( G i ) for p o or individuals ( k = 1 / 3). The diagonal dotted line represen ts p erfect concordance in rank. The outcomes confirm the high rank correlation b et w een the measures and show that higher depriv ation counts exhibit greater concordance. Graph (b) is a histogram sho wing the distribution of individual rank dif- ferences b et ween S i and G i . Eac h bar is the p ercen tage of p o or p eople that exhibit the difference ( S i − G i ) rep orted on the horizontal axis. P ositiv e v alues indicate that S i assigns a higher rank than G i , whereas negativ e v al- 27 ues indicate that G i assigns a higher rank than S i . This graph shows that roughly one in three p o or individuals ( ∼ 34%) get the same rank under b oth measures. The negativ e side is denser and more concen trated (with a clus- ter around − 0 . 15), whereas the p ositiv e side is sparser but more disp ersed (extending up to +0 . 40). As a result, the mean remains close to zero despite the visually asymmetric distribution: the relatively few individuals with pos- itiv e v alues exhibit larger magnitudes, whic h approximately offset the many individuals with negativ e v alues clustered around − 0 . 15. These outcomes rev eal a strong rank asso ciation b et w een the tw o mea- sures, indicating broad agreement on who is most and least deprived. At the same time, the asymmetry b et ween the t w o tails in the Bangladesh case rev eals informative differences in how eac h measure conceptualizes depriv a- tion. The negativ e cluster (around − 0 . 15) corresp onds to individuals with 3-4 y ears of education. Although they are closer to the depriv ation threshold (fiv e y ears, corresp onding to completion of primary education) in absolute terms than those with zero sc ho oling, the p ositional p o v erty gap measure S i assigns them p ositional p o vert y gap v alues similar to those with no school- ing. This reflects the distributional con text. In a so ciet y where educational attainmen t is concentrated at the extremes, with many individuals having no schooling and man y ha ving completed primary education, those with 3- 4 years of schooling remain b elo w the primary completion threshold and therefore likely exp erience disadv an tages in lab our market access and so cial mobilit y comparable to those faced by individuals with no formal education. In parallel, the AF p o vert y gap clearly distinguishes each education level from zero sc ho oling b y its absolute distance to the threshold. The p ositiv e tail, by contrast, is driv en by individuals with moderate n u- tritional depriv ation, b elo w the ZBMI cutoff: their absolute shortfall from the threshold is small, y et S i ranks them high b ecause the n utritional distri- bution is concen trated well ab o ve that cutoff, with most of the p opulation ac hieving substan tially b etter outcomes. The p ositional p ov erty gap is again capturing distributional con text: when nutritional depriv ation is rare, even mo derate shortfalls get high p ositional depth scores, p oten tially signaling marginalization and p ersisten t disadv antage within a context where most individuals are not deprived. Complementarily , the AF pov ert y gap, in con- trast, assigns these individuals low v alues b ecause their shortfall from the threshold is ”mo derate” in absolute terms. In b oth cases, the div ergence reflects a more general prop ert y: the p osi- 28 tional p o vert y gap is sensitive to the shap e of the ac hievemen t distribution, whereas the AF gap measures the absolute distance from the depriv ation cutoff. (a) Within-po or p ercen tile ranks: p ositional povert y gap vs. AF p o vert y gap ( n = 6 , 859). Dashed line: 45 ° reference (p erfect concordance). ρ S = 0 . 899 (design-based Sp earman, 500 b ootstrap replicates). Colours indicate num b er of active de- priv ations. (b) Distribution of individual rank differences in percentage p oints (positional p o vert y gap rank − AF p o vert y gap rank), survey-w eighted. P ositive v alues: CDF assigns a higher rank; negativ e: AF assigns a higher rank. V ertical line: zero difference. Figure 3: Rank concordance b etw een p ositional p o vert y gap and AF p o vert y gap among the p oor ( k = 1 / 3, union cutoff ), Bangladesh MICS 2019. Ov erall, this illustration sho ws that, on cardinal data where b oth mea- sures are computable, the p ositional p o vert y gap produces rankings with a high degree of association with the established AF gap. This pro vides confi- dence that the p ositional p o vert y gap captures meaningful depriv ation depth and extends this to settings where cardinal indicators are una v ailable. At the same time, it confirms that the measures giv e differen t p ersp ectiv es. Rather than capturing the same phenomenon with different precision, they em b o dy distinct conceptions of depth: absolute shortfall from a normativ e standard v ersus relative disadv an tage within the distribution of achiev ements. 11. Conclusion This pap er proposed a positional p o v erty gap measure that mak es depth measurable in ordinal settings, dra wing on the distributional logic of the TFR 29 approac h within the AF counting framework. The prop osed distribution- based measure has a natural in terpretation as depth, expressed as the nor- malized distance of an individual’s relativ e p osition from the top of the dis- tribution, or equiv alently as the share of the p opulation with higher ac hieve- men ts than the individual. It th us opens a new line of researc h, extending analysis in ordinal settings to incorp orate positional p o vert y gap alongside incidence and in tensity . Researc hers and policymakers can no w examine not only ho w man y di- mensions an individual or household exp eriences depriv ation in, but also ho w deep each depriv ation is. This p ersp ectiv e allo ws the identification of indi- viduals with few but deep depriv ations. It also reveals subgroups with low incidence but high p ositional p o vert y gap, meaning that although few in- dividuals are p oor, those who are p oor face deep disadv antages. Incidence and in tensit y alone do not capture these patterns and ma y therefore o verlook vulnerable groups in p olicy targeting. The empirical applications illustrate these additional insigh ts. The Brazil application rev eals substantial within-p ov erty heterogeneit y in p ositional p o v erty gap that the in tensity alone do es not detect. The Bangladesh illustration sho ws a high rank asso ciation b etw een the positional pov ert y gap and the AF p o v erty gap, while also demonstrating their complemen tarity: the p o v erty gap captures absolute need relative to a threshold, whereas the p ositional p o v erty gap captures context through distributional p osition. The prop osed measure satisfies desirable axiomatic prop erties when the reference distribution is kept fixed ov er time, confirming its internal con- sistency under w ell-defined conditions. When comparisons across time or coun tries are required, the choice of benchmark requires an explicit norma- tiv e decision. This should b e seen as a feature rather than a limitation, as it mak es transparen t an assumption that other approac hes often leav e implicit. Cross-sectional applications, whic h account for most national p o vert y assess- men ts, do not require suc h a decision. More broadly , introducing a positional p o v erty gap helps address an issue that has received little attention in multi- dimensional p o vert y measurement: capturing the depth of depriv ation when indicators are ordinal. The framework can b e easily implemen ted within ex- isting AF applications and pro vides a straigh tforward wa y to obtain richer p o v erty profiles that go b ey ond counting depriv ations. 30 References Alkire, S. and F oster, J. (2011a). Coun ting and m ultidimensional p ov erty measuremen t. Journal of Public Ec onomics , 95(7-8):476–487. Alkire, S. and F oster, J. (2011b). Understandings and misunderstandings of m ultidimensional pov ert y measurement. 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Gender differences in m ultidimensional p o v erty in Brazil: A fuzzy approach. So cial Indic ators R ese ar ch . 33 App endix A. Admissible transformations T able A1: Admissible transformations for v ariables that are relev ant for p o vert y measure- men t T yp e of v ariable Admissible transforma- tions Admissible mathe- matical op erations P ermissible statis- tics Examples rele- v ant for pov ert y measuremen t Qualitativ e: Ordinal Order- preserving transfor- mations (monotonic increasing functions). None F requency distribution, mo de, con tingency , cor- relation, median p er- cen tiles T yp e of sanitation facilit y , source of drinking w ater, co oking fuel, floor, w alls, and ro of; lev els of schooling. Cardinal: In terv al scale linear trans- formations Add, subtract F requency distribution, mo de, con tingency , correlation, median p ercen tiles, mean, standard deviation, rank-order, correlation, pro duct-momen t corre- lation. z-scores of n utri- tional indicators (eg., w eight for age); Bo dy Mass Index (BMI). Cardinal: Ratio scale linear and ratio transfor- mations Divide, multi- ply F requency distribution, mo de, con tingency , correlation, median p ercen tiles, mean, standard deviation, rank-order, correlation, pro duct-momen t corre- lation, co efficien t of v ariation Income, consump- tion exp enditure, n umber of deaths a mother exp e- rienced, y ears of sc ho oling, n umbers of b edro oms, n um- b er of assets (e.g., car, computer, fridge, and others). Source: Adapted from Stevens ( 1946 ) and Alkire et al. ( 2015 ). 34 App endix B. Axioms V erification and Pro ofs V erification Depriv ation F o cus. A t ( i, d ), g 0 id = 0 b oth b efore and after, hence g 1 id ( k ) = ρ i ( k ) s id g 0 id = 0 in b oth matrices. F or any deprived p eer ( n, d ) with x nd < z d , the CDF term F d ( x nd ) is the (w eighted) shar e of the reference distribution—empirical or anc hored—at or b elo w x nd . Since x id , x ′ id ≥ z d > x nd , this share is unc hanged, so s nd (and th us g 1 nd ( k )) is unc hanged. The normaliser 1 − F d (min x d ) is also unc hanged: with in-sample CDFs a change ab o ve z d cannot alter the minim um among deprived v alues (and if no one is depriv ed, the column contributes zero); with anc hored CDFs the reference minim um is fixed. Therefore P ( X ′ ) = P ( X ). V erification Po vert y F o cus. Anchor e d CDFs. F or the non-po or p er- son i ⋆ , ρ i ⋆ ( k ) = ρ ′ i ⋆ ( k ) = 0, so g 1 i ⋆ j ( k ) = ρ i ⋆ ( k ) s i ⋆ j g 0 i ⋆ j = 0 and g 1 ′ i ⋆ j ( k ) = ρ ′ i ⋆ ( k ) s ′ i ⋆ j g 0 ′ i ⋆ j = 0 for all j , regardless of x ′ i ⋆ j . F or every other ( n, j ) with n  = i ⋆ , anc horing k eeps s nj fixed; and b y h yp othesis x ′ nj = x nj , hence g 1 ′ nj ( k ) = g 1 nj ( k ). Therefore ev ery summand in P ( X ; z , k ) = (1 / N ) P i,j w j g 1 ij ( k ) is un- c hanged and P ( X ′ ) = P ( X ). In-sample CDFs under union identific ation. Here ρ i ⋆ ( k ) = 0 implies g 0 i ⋆ j = 0 for all j , i.e. x i ⋆ j ≥ z j in ev ery indicator, and the same holds after the change. F or an y indicator j and any depriv ed peer n with x nj < z j , the empirical CDF F IS j ( x nj ) equals the (weigh ted) share of observ ations ≤ x nj . Because x i ⋆ j , x ′ i ⋆ j ≥ z j > x nj , that share at x nj is unchanged, so the n umer- ators 1 − F IS j ( x nj ) are unc hanged for all deprived peers. The denominator 1 − F IS j ( m j ), with m j = min i x ij , is also unc hanged: if there exists at least one depriv ed observ ation in j , then m j < z j and altering a non-depriv ed v alue cannot change the minimum; if there are no deprived observ ations, then g 0 nj ( k ) ≡ 0 and the column con tributes zero b oth b efore and after. Hence all g 1 nj ( k ) are unc hanged for ev ery ( n, j ), and the terms for i ⋆ are zero b oth b efore and after. Consequen tly , P ( X ′ ) = P ( X ). V erification Monotonicity . Own-p erson. Consider a w orsening for p erson i in indicator d . Case (i): x ′ id < x id < z d (still depriv ed). Then g 0 id and ρ i ( k ) are un- c hanged, and only s id mo ves. Since F d is nondecreasing, x ′ id < x id ⇒ F d ( x ′ id ) ≤ F d ( x id ), hence s ′ id ≥ s id with strict inequality for a strict w ors- ening. Therefore ∆ P i = w d  s ′ id − s id  ≥ 0 ( > 0 if x ′ id < x id ) . 35 Case (ii): x ′ id < z d ≤ x id (crosses into depriv ation). Here g 0 id switc hes from 0 to 1, so a new non-negativ e term app ears: ∆ P i = w d s ′ id ρ ′ i ( k ) ≥ 0 , whic h is strictly p ositiv e whenever i is (or b ecomes) po or so that ρ ′ i ( k ) = 1 (and equals 0 if ρ ′ i ( k ) = 0). Hence dimensional monotonicit y also holds. In all cases, a w orsening cannot reduce P i . A ggr e gate (anchor e d CDFs). Let X ′ b e obtained by worsening cell ( i, d ). With anc hored { F j } , s nj is unc hanged for all ( n, j )  = ( i, d ), so only g 1 id ( k ) = ρ i ( k ) g 0 id s id can c hange. Consider tw o parallel cases. Case (i): x ′ id < x id < z d (still depriv ed). If i is p o or b oth b efore and after, ρ i ( k ) = 1 and ∆ P = P ( X ′ ; z , k ) − P ( X ; z , k ) = 1 n w d  s ′ id − s id  > 0 . If i is non-po or b oth b efore and after, then ρ i ( k ) = ρ ′ i ( k ) = 0 and ∆ P = 0 (b y p o vert y fo cus). Case (ii): x ′ id < z d ≤ x id (crosses into depriv ation). A new term en ters for p erson i : ∆ P = 1 n w d s ′ id ρ ′ i ( k ) ≥ 0 , whic h is strictly p ositiv e if the p erson is (or b ecomes) p o or so that ρ ′ i ( k ) = 1, and 0 if i remains non-p o or ( ρ ′ i ( k ) = 0). Therefore aggregate dimensional monotonicit y holds under anchored CDFs. In all cases, a w orsening cannot reduce P i ; and under anchored CDFs the aggregate index P ( X ; z , k ) w eakly increases (strictly when a p o or p erson w orsens in a deprived indicator or b ecomes newly depriv ed and p o or). V erification Subgroup consistency and decomp osabilit y . Let the p opulation b e partitioned in to disjoin t subgroups S 1 , . . . , S m with sizes n ℓ (so P ℓ n ℓ = n ), and let X ℓ denote the restriction of X to S ℓ . Because the CDFs are anc hored and ( z , k , w ) are common, eac h cell term g α ij ( k ) takes the same v alue whether computed in the full sample or within an y subgroup. (i) Sub gr oup c onsistency. Let X ′ differ from X only within subgroup ℓ ′ , and assume subgroup sizes are unchanged. Then, grouping the a verage by subgroups, P ( X ′ ; z , k ) − P ( X ; z , k ) = n ℓ ′ n  P ( X ′ ℓ ′ ; z , k ) − P ( X ℓ ′ ; z , k )  . 36 Hence the ov erall index mo ves in the same dir e ction as the index of the c hanged subgroup: if P ( X ′ ℓ ′ ; z , k ) < P ( X ℓ ′ ; z , k ), then P ( X ′ ; z , k ) < P ( X ; z , k ) (and con versely if it worsens). (ii) Population sub gr oup de c omp osability. Grouping the population sum b y subgroups, P ( X ; z , k ) = 1 n m X ℓ =1 X i ∈ S ℓ d X j =1 w j g α ij ( k ) = m X ℓ =1 n ℓ n 1 n ℓ X i ∈ S ℓ d X j =1 w j g α ij ( k ) ! = m X ℓ =1 n ℓ n P ( X ℓ ; z , k ) , where the last equality uses that, under the common (anc hored) CDFs, the paren thesised term equals the subgroup index P ( X ℓ ; z , k ). Weighte d analo gue. With sampling w eights, replace n ℓ /n by W ℓ /W , where W ℓ = P i ∈ S ℓ ω i and W = P n i =1 ω i . Pro of Theorem 1 . F or the axioms depriv ation fo cus, p o vert y fo cus, monotonicit y , subgroup consistency and decomp osabilit y pro ofs follo w from the verifications ab ov e. F or the remaining properties, the verification is di- rect. 37