Abstract
Using the 2006 round of the British Household Panel Study dataset, I explore the statistical behavior of three fuzzy measures of poverty through a simulation (Monte Carlo) method. The measures [totally fuzzy (TF), totally fuzzy and relative (TFR), and integrated fuzzy and relative (IFR)] acknowledge that (1) poverty is a multidimensional concept, and (2) the ‘poor’ and ‘non-poor’ are not two mutually exclusive sets and the distinction can be ‘fuzzy’. I find that the sampling distributions of the fuzzy measures are quite normally distributed, and they are robust to arbitrary choice in the estimation as well as reliable with relatively small sample size, though there is some differences between the methods. Also, I show that they are robust to measurement errors: allowing random measurement errors in all indicators, the measures still yield strongly reliable results. Finally, I investigate the identification performance of each measure and show that IFR measure has strong consistency, while both TF and TFR measures significantly underestimate the number of people whose fuzzy index values are very high.
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Notes
The title of Sen (1979a)’s Tanner lecture at Stanford University—“Equality of what?” is very suggestive in this regard.
On the contrary, Ravallion (2011) argues that the aggregation is unnecessary since for policy purposes, disaggregation will be required for poverty measurements. In addition, he asserts that due to the “implicit rate of substitution” in the process, the aggregate poverty measures are not consistent with consumer welfare theory.
The first suggestion is based on econometric analysis of the data. The main idea is to compute ‘appropriate’ weights to create an index as the weighted average of multiple dimensions, which can be obtained by applying a statistical technique that reveals the weight structure in the data (Dewilde 2004; Kuklys 2005; Lelli 2001). The second one is grounded on a set of desirable axioms. For instance, Bourguignon and Chakravarty (2003) show that one simple functional form for multidimensional poverty measurement that satisfies focus, transfer, and monotonicity axioms as well as subgroup decomposability axiom. The third approach considers poverty to be a ‘fuzzy’ concept and I follow this approach.
However, he emphasizes that they should be distinguished from both (1) having commodities, and (2) happiness from the use of commodities.
Thus, the function can reflect individual difference in well-being due to personal characteristics. For example, the possession of a car can be a source of functioning “free transportation” for a normal person. However, it is not for a blind person.
For instance, two people with same income can be entirely different in terms of well-being for many reasons: one may be in situation where freedom to use the income is prohibited, or one simply does not have a market to use it.
Tsui (2002) says that “the capability of a person is an opportunity set of bundles of functionings and not the functionings achieved”.
For concrete indicators for the dimensions, accepting Sen (1997)’s advice on the problem that “Openness to critical scrutiny, combined with public consent, is a central requirement of non-arbitrariness of valuation in a democratic society”, I choose each indicator based on previous empirical research without assuming that this is a ‘universal’ list (For detailed list, see Appendix “2”.).
The concept of ‘fuzziness’ originates from the inherent vagueness in any representational system, such as languages, though the external world seems to be continuous (Dubois and Prade 2000). Based on the graduality principle which extends the two-valued classical logic to a more general case, the fuzzy proposition is related to the degree of truth of a statement (Fustier 2006). Rather than saying one statement is either true or false, this proposition mentions a proposition can be “true”, “untrue”, or “more or less true”. By Zadeh (1965)’s fuzzy logic, the degree of truth is assumed to be between zero and one.
Although there have been several proposals that try to embody this “gradation” concept in pre-existing measurements (Atkinson 1987), still no one alternative stands out.
According to the fuzzy-set theory, most of the concepts we use in social science (Keefe and Smith 1996) or in language (Dubois and Prade 2000; Zadeh 1965) actually do not have clear and sharp borderline of application. For example, we can always say one person is either tall or small, but it is very hard to show the exact height of the person to be qualified as either tall or small—so-called Sorites paradox (Klir and Yuan 1995).
In ordinal variable case, the maximum and minimum values can be determined more easily by assuming the value of the lowest category as minimum, the highest as maximum. For example, if ‘health status’ variable would take five values from “very good” (five) to “very bad” (one), then one can be the minimum value, and five the maximum. Here, the order by the membership function is reversed because poor health indicates people to be closer to ‘poor’ group, which means bigger membership function value.
Simple algebraic expression for ‘color TV’ is as follows: \(\mu_{j} \left( i \right) = 1\), if a person does not own a color TV; or \(\mu_{j} \left( i \right) = 0\), if a person owns a color TV. Still, it is entirely possible to include these variables in constructing a poverty measurement because traditional set can be considered as just a special case of fuzzy set (Dubois and Prade 2000; Fustier 2006; Klir and Yuan 1995).
This notation is a bit confusing because important variable such as income or expenditure usually moves in the opposite direction to the membership function.
The choice of appropriate formula depends on the specific property of the dimension. For income which decreases poverty conceptually, the former part of Eq. (6) is appropriate.
For ordinal variables, fundamentally the same idea is applied. However, this specification can be sometimes problematic if one of the frequencies of extreme modalities is very high. For example, Appendix “3” shows that by Eq. (6) the membership function for the ordinal variable \(x\) with five scales cannot be zero, though the membership function should be between zero and one. In order to overcome this problem, Cheli and Lemmi (1995) suggest following formula (7) for ordinal variables which has k categories in them (j (k) indicates k-th category of indicator j):
$$\mu_{j} \left( i \right) = \mu_{j(k)} \left( i \right) = \left\{ {\begin{array}{*{20}c} o & {if\, k = 1} \\ {\mu_{j(k - 1)} \left( i \right) + \frac{{F\left( {J_{i}^{(k)} } \right) - F\left( {j_{i}^{(k - 1)} } \right)}}{{1 - F\left( {j_{i}^{(1)} } \right)}}} & {otherwise} \\ \end{array} } \right.$$(7)For binary variables, Cheli and Lemmi (1995) adopted the same functions as in the TF method (see note 14).
The Lorenz function is firstly suggested as a measure of wealth concentration—a mapping from cumulated percents of a population according to wealth to the total wealth held by the percents (Lorenz 1905). More generally, it can be considered the cumulative distribution function of the empirical probability distribution function of an indicator (Kakwani 1977).
Betti et al. (2005a) claim that this measure is more sensitive to the actual disparities in a dimension (e.g., income) compared to the simple cumulative distribution function which is just the proportion of individuals less poor than the person concerned. Furthermore, it has one important advantage over previous methods: it has a close relationship with the Gini coefficient since the mean of the membership function is (2 + G)/6. So, it can be concluded that IFR measure as an aggregate index is sensitive to the distribution of each indicator. In addition, they argue that separate measures should be estimated for monetary and non-monetary dimensions because monetary dimension still has a ‘fundamental role’ in poverty research (Betti et al 2002; Betti and Verma 2008). Still, integrating the two dimensions into one index is a more attractive strategy for policy makers, they introduce the concept of “manifest” and “latent” poverty. While the former indicates a subgroup of population who is poor for both of the dimensions, the latter indicates who is poor for either one of the dimensions (Betti and Verma 1998, 2008). For non-monetary dimensions that mainly consist of ordinal and dichotomous variables, they first calculate a deprivation indicator for each indicator, d ji (based on the TF method) where j indicates each dimension and i denotes each individual, and then integrate each indicator into one index using a weight function that is discussed below.
This shows ‘relative’ characteristic of the measure clearly. If everyone is poor in a dimension, its contribution to individual’s poverty - its weight - should be zero in a relative point of view.
Betti et al (2005a) suggest that most of the time, it turns out that ρH is a correlation coefficient with an indicator itself, in other words, unity. For more general approach, Betti and Verma (2008) propose a ‘largest gap’ criterion, which assumes ρH exists between the biggest gap among correlation coefficients.
High correlation between indicators implies that they basically measure the same under- lying phenomenon. In other words, some variables can be ‘redundant’ in terms of information. The authors also reported that in most of the application the second factor in this weight goes to one.
This is why Table 1 seems incorrect, compared to the actual situations of the country. For example, a data from Eurostat (http://appsso.eurostat.ec.europa.eu/nui/show.do?dataset=une_rt_aamp;lang=en) shows that the unemployment rate in 2006 was 5.4 %, though it is almost 33 % in the dataset.
By definition, mean shortfall in income = poverty gap ratio × poverty line. From Table 1, the poverty line is £13,919.
Technically, the bootstrap and Monte Carlo methods are separate techniques with different traditions. But a clear distinction between the two concepts does not provide much information. In fact, Hall (1992) argues that Efron’s biggest contribution is to recognize the benefits of combining the two techniques.
From the generated data, weight functions for each index are calculated and finally, fuzzy indices for each individual are computed a number of times. Since there is no guideline for ‘right’ number of iterations (Efron and Tibshirani 1993), I adopt several numbers of simulations, i.e., 100, 1,000, or 5,000 times to see whether the number of iterations can make a difference in results.
Each of the focus can be considered as the first step of examining two desirable statistical properties of a ‘good’ estimator: unbiasedness and consistency. According to Sage and Melsa (1971), there are four desirable properties of a ‘good’ estimator: unbiasedness, minimum-variance and unbiased, consistency, and efficiency.
Originally, the fuzzy measures range from zero to one by definition. However, for readability, all fuzzy measures in this paper are multiplied by 100.
In the graphs, \(\widehat{\theta }\) indicates a calculated value from BHPS data because strictly speaking, it also is an estimate of an unknown population parameter, θ. \(E(\widehat{\theta })\) is the mean of simulated values.
p values from Kolmogorov–Smirnov test for normality are 0.6068 (TF measure), 0.3625 (TFR measure), and 0.1058 (IFR measure), respectively.
I calculate the size of bias using mean square error, and the size of biases are 0.168 (TF), 0.033 (TFR), and 0.208 (IFR), all of which are <1 % of each fuzzy measure.
All the results from nonparametric bootstrapping also fail to reject the null hypothesis of normality test.
In standard method, a confidence interval of \(\widehat{\theta }\) with confidence level α is defined in \(\left[ {\widehat{\theta } - \widehat{\sigma }z^{(\alpha )} ,\quad \widehat{\theta } + \widehat{\sigma }z^{(\alpha )} } \right]\), where z (α) denotes a value x which satisfies the condition z(p < x) = α in the standard normal distribution, and in percentile method, confidence interval is \(\left[ {\widehat{G}^{ - 1} \left( \alpha \right),\quad \widehat{G}^{ - 1} \left( {1 - \alpha } \right)} \right]\), where G denotes the cumulative bootstrap sample distribution for \(\widehat{\theta }\). Finally, “Bias-corrected” confidence intervals can be computed by fundamentally same principle as percentile method. But the confidence level is adjusted considering possible bias in the bootstrap distribution. The bias-corrected confidence interval can be expressed as \(\left[ {\widehat{G}^{ - 1} \left( {\Phi \left\{ {2z_{0} + z^{\left( \alpha \right)} } \right\}} \right),\quad \widehat{G}^{ - 1} \left( {\Phi \left\{ {2z_{0} + z^{{\left( {1 - \alpha } \right)}} } \right\}} \right)} \right]\), where \(\Phi^{ - 1}\) is the inverse of the cumulative Gaussian distribution, \(z_{0} = \Phi^{ - 1} \left\{ {\widehat{G}\left( {\widehat{\theta }} \right)} \right\}\), and Z (α) satisfies \(\Phi \left( {z^{\left( \alpha \right)} } \right) = \alpha\).
Due to the possible sensitivity of the test, I perform three additional normality tests, and the results show consistent failure of rejection. See Appendix “7”.
Although there are different opinions on the definition of “precision”, it is in general considered as a measure of the variations of one measurement around its nominal value, regardless of the accuracy of the nominal value (Dodge and Marriott 2003; Pearson 2011; Rice 2007; Taylor and Cihon 2004). van Belle (2002) argues that the concept corresponds to ‘reliability’ in social sciences, while accuracy to validity.
Since the fuzzy measures are “nonpivotal” in the sense of Hall (1992), in other words, we cannot be sure of the exact distribution of parameters, this bootstrap confidence interval may differ from the true confidence interval. However, as the difference decreases at the rate of the square root of the sample size, it is reasonable to assume the bootstrap error in this study is negligible, considering the big sample size.
Assuming we only have four variables, with 10 % of the observations missing independently for each indicator, then it is possible that we end up with only 65.6 % of total observations that can be used for any statistical analysis.
In a sense, small-sample behavior can be connected to the property of consistency of a statistic, which indicates an estimator becomes more accurate as the number of observations increases. Formally, it can be expressed as follows(Lehmann and Casella 1998; Sage and Melsa 1971; Voinov and Nikulin 1993): \(E\left[ {\left( {\widehat{\theta }_{n} - \theta } \right)^{2} } \right] \to 0\) as n → ∞ where \(\widehat{\theta }_{n}\) indicates an estimator when the number of observations is n.
Altman et al (2004) argue that a good estimator should yield ‘stable’ results, such as, monotonic and smooth reduction of variance, with increasing sample size to make a correct inference.
For each sample size, 5,000 simulations are run.
The mean square error (MSE) can be decomposed into the sum of variance of an estimator and squared bias as following equation: MSE \(\left( {\widehat{\theta }} \right) = E\left\{ {\left( {\widehat{\theta } - \theta } \right)^{2} } \right\} = Var\left( {\widehat{\theta }} \right) + \left( {Bias\left( {\widehat{\theta },\theta } \right)} \right)^{2}\) (Deutsch 1965; Kennedy 2008; Lehmann and Casella 1998; Sage and Melsa 1971).
This can be interpreted as one evidence of consistency, which indicates the property of asymptotic unbiasedness.
In traditional sense, this is a problem of setting a “poverty line”.
This is equivalent to saying that we can agree unanimously that a person is ‘poor’ when the person’s income is lower than 60 % of median income.
It is certain that the decision itself is not free from the criticism for arbitrariness. However, the arbitrariness is defensible in that the arbitrariness fundamentally originates from the fact that poverty is a complex and ‘fuzzy’ concept.
By this criterion, 996 out of 6,339 in the BHPS data (15.7 %) are definitely nonpoor.
It turns out that 1,846 observations, or 29.1 % of the sample in the BHPS data have saving greater than £600. In fuzzy perspective, it is not unreasonable to assume that this level of saving makes a person definitely nonpoor at least with regard to saving.
Since quite few people have inheritance (only 125 cases, 2.0 % of sample, receive money from their ancestors), the inheritance variable is not considered here.
Each line is drawn from a uniform distribution.
In a normally distributed random variable, say \(Z\sim N\left( {k,k^{2} } \right),\Pr \left( {Z > - k} \right) = 0.841\).
It is certainly possible to make the errors more relevant by modeling correlated part in error, instead of assuming a simple random error. For example, let a variable \(X_{simulated} = X_{BHPS} + \epsilon^{*}\), in place of a seemingly random error ϵ*, I can put error \(\rho X_{BHPS} + \epsilon\), where ρ represents a relationship between error and observation, and ϵ truly random error. By the formulation, a tendency in error can be modeled explicitly, i.e., people with higher income are more inclined to under-report their income.
If non-fuzzy perspective is adopted, then this sentence can be replaced by “the number of people who are in the phenomenon of poverty”.
For graphical presentation, see Appendix 15.
Robeyns (2000) reviews twelve researchers adopting the capability approach, and all of them regard health as an important functioning.
Tomer (2002) puts it in this way, “It is not about how much food one consumes; it is about eating tasty food and being well-nourished.”
These phrases indicate that there is still a room for inevitable arbitrariness in terms of choosing specific indicators, because the concept of “modern American society” or “every-day life activities” implies cultural or relative aspects of poverty.
Within the general concept of social participation, political participation is especially emphasized by many theorists, including Sen (1999) and Nussbaum (2003) [see also Robeyns (2005a); Wagle (2008); Clark and Hulme (2010); Anand et al. (2010)]. Since many problems associated with poverty can be attributed to the lack of political voice of the poor (Sen 1983), including this dimension seems appropriate. However, considering both the context of U.K., a developed country with long history of democracy, and the lack of proper indicators (only a variable of political party preference exists in the dataset), this study does not specify political dimension. Still it is important dimension especially when multidimensional measures are to be applied to developing countries’ contexts, where often democratic system is weak or nonexistent.
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Appendices
Appendix 1: Seven Dimensions of Poverty
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Economic resources: Strictly speaking, this is not a functioning per se (Brandolini and D’Alessio 1998). However, since economic resources can be directly linked to diverse functionings (e.g., buying healthy food), this dimension is usually included (Whelan 1993a, b; Kangas and Ritakallio 1998; Lelli 2001). Certainly the term does not indicate income or consumption exclusively. On the contrary, as the concept of functioning includes appropriate control over the resources, various forms of economic resources can be included as indicators.
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Health: This is one of the most basic functionings of human beings because without it proper ‘function’ of an individual in any society is impossible (Anand and Sen 1997; Doyal and Gough 1991; Duclos et al. 2006a; Federman and Garner 1996). Therefore, this functioning is included in almost every research adopting the capability approach.Footnote 54
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Employment: It can be considered as an important functioning because it does not just imply having a job, but also having an opportunity to participate in social interactions [“the life of the community”, according to Anand and Sen (1997)]. Also, the importance of employment in obtaining proper economic resources cannot be ignored.
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Housing: The dimension is regarded as an inevitable factor even in consumption-based traditional approach. From Orshansky (1965) to Citro and Michael (1995), the cost of housing constitutes an important part of the minimum cost-of-living. In the capability approach, not only the cost but also the conditions of housing matter because housing indicates a crucial functioning of “security” or “protection” (Blank 2008; Doyal and Gough 1991).
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Durable goods: Though it is certain that material possession itself is not a functioning,Footnote 55 some part of it—for example, having a telephone or a refrigerator—can be included as a functioning. Bauman (2003) understands those specific possessions as “minimum standards of functioning in modern American society”, and Boarini and d’Ercole (2006) also consider the possession of durable goods as “essential to perform every-day life activities.”Footnote 56 According to Townsend (1979), the lack of possession for certain goods can even be understood as a manifestation of poverty. Therefore, for certain types of goods, material possession can be understood as a functioning.
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Social capital: It is broadly understood to be the extent of participation in social networks (Narayan et al. 2000). This functioning emphasizes that human well-being can increase through relationships that make individuals more capable (Tomer 2002).
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Social participation: Though this overlaps with social capital, here the functioning represents something more general.Footnote 57 In a sense, the underlying motivation for this functioning comes from the social exclusion perspective which emphasizes the importance of participation in major social opportunities of the society (Dagum 2002).
Appendix 2
See Table 9.
Appendix 3
See Table 10.
Appendix 4
See Fig. 3.
The membership function in IFR method
Appendix 5
See Table 11.
Appendix 6
See Fig. 4.
Sampling distributions by nonparametric bootstrapping
Appendix 7
See Table 12.
Appendix 8
See Fig. 5.
Q–Q plots for simulated fuzzy measures
Appendix 9
See Table 13.
Appendix 10
See Fig. 6.
Replicated 95 % confidence intervals
Appendix 11
See Fig. 7.
Distribution of the simulated TF index, change in income criteria
Appendix 12
See Fig. 8.
Distribution of the simulated TF index, change in saving criteria
Appendix 13
See Fig. 9.
Distribution of the simulated TF index, change in both variables
Appendix 14
See Fig. 10.
Distribution of the simulated fuzzy indices, according to measurement errors
Appendix 15
See Fig. 11.
Simulation results for identification performance
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Kim, SG. Fuzzy Multidimensional Poverty Measurement: An Analysis of Statistical Behaviors. Soc Indic Res 120, 635–667 (2015). https://doi.org/10.1007/s11205-014-0616-8
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DOI: https://doi.org/10.1007/s11205-014-0616-8