Skip to contents

Foster-Greer-Thorbecke (FGT) poverty measures

Source: R/fgt_poverty.R
fgt_poverty.Rd

Calculates the FGT metrics, a family of poverty measures originally proposed by Foster et al. (1984) that capture the extent and severity of poverty within an accessibility distribution. The FGT family is composed of three measures that differ based on the \(\alpha\) parameter used to calculate them (either 0, 1 or 2) and which also changes their interpretation. Please see the details section for more information on the interpretation of the measures.

Usage

fgt_poverty(
  accessibility_data,
  sociodemographic_data,
  opportunity,
  population,
  poverty_line,
  poor_below_threshold = TRUE,
  group_by = character(0)
)

Arguments

accessibility_data

A data frame. The accessibility levels whose poverty levels should be calculated. Must contain the columns id and any others specified in opportunity.

sociodemographic_data

A data frame. The distribution of sociodemographic characteristics of the population in the study area cells. Must contain the columns id and any others specified in population.

opportunity

A string. The name of the column in accessibility_data with the accessibility levels to be considered when calculating accessibility poverty.

population

A string. The name of the column in sociodemographic_data with the number of people in each cell. Used to weigh accessibility levels when calculating poverty.

poverty_line

A numeric. The poverty line below which individuals are considered to be in accessibility poverty.

poor_below_threshold

A logical. If TRUE (default) the observations below the poverty line are considered to be poor. This is the correct approach for primal accessibility measures (e.g. cumulative accessibility). If FALSE, then observations above the poverty line are considered to be poor. This is the correct approach for dual accessibility measures (e.g. travel time to the closest facility). When set to FALSE, FGT 1 and 2 do not have an upper bound.

group_by

A character vector. When not character(0) (the default), indicates the accessibility_data columns that should be used to group the poverty estimates by. For example, if accessibility_data includes a race column that specifies the racial category of the population (e.g. "black" and "white") that each entry refers to, passing "race" to this parameter results in poverty estimates grouped by race.

Value

A data frame containing the three poverty estimates (FGT0, FGT1 and FGT2) for the study area.

Interpretation of FGT measures

With a poverty line \(z\), the FGT measures are calculated as the population-weighted average of \(g_i^\alpha\), in which \(g_i\) is the normalised shortfall of each observation \(i\) with accessibility \(y_i\): \(g_i = (z - y_i) / z\) if \(y_i < z\) and 0 otherwise. When poor_below_threshold is FALSE, \(g_i\) is instead the normalised excess \((y_i - z) / z\) if \(y_i > z\), and 0 otherwise. The interpretation of each measure depends on the \(\alpha\) parameter:

  • with \(\alpha = 0\) (FGT0) the measure captures the incidence (or extent) of poverty as a simple headcount - i.e. the proportion of people in poverty;

  • with \(\alpha = 1\) (FGT1) the measure, also known as the "poverty gap index", captures the depth (or intensity) of poverty as the average normalised shortfall from the poverty line, calculated over the whole population (people not in poverty have a shortfall of 0);

  • with \(\alpha = 2\) (FGT2) the measure, also known as the "squared poverty gap index", captures the severity of poverty. Squaring the shortfalls gives more weight to the people furthest from the poverty line, which makes the measure sensitive to inequality among the poor.

A value of 0 indicates that no one is in poverty. When poor_below_threshold is TRUE, FGT values range from 0 to 1: FGT0 is 1 when everyone is below the poverty line, and FGT1 and FGT2 are 1 only when everyone has an accessibility of 0. When poor_below_threshold is FALSE, FGT0 still ranges from 0 to 1, but FGT1 and FGT2 have no upper bound: they can be greater than 1 (e.g. when travel costs are more than twice the poverty line) and are Inf if any accessibility value is Inf. This happens, for example, with the travel costs of origins that can't reach the required opportunities, as returned by cost_to_closest(). The function raises a warning in this case, and these values should be replaced by a maximum travel cost (e.g. the highest cost in the travel matrix) before calculating the FGT measures.

References

Foster J, Greer J, Thorbecke E (1984). “A Class of Decomposable Poverty Measures.” Econometrica, 52(3), 761–766. ISSN 0012-9682. doi:10.2307/1913475 . 1913475.

See also

Other equity: concentration_index(), gini_index(), palma_ratio(), theil_t()

Examples

data_dir <- system.file("extdata", package = "accessibility")
travel_matrix <- readRDS(file.path(data_dir, "travel_matrix.rds"))
land_use_data <- readRDS(file.path(data_dir, "land_use_data.rds"))

access <- cumulative_cutoff(
  travel_matrix,
  land_use_data,
  cutoff = 30,
  opportunity = "jobs",
  travel_cost = "travel_time"
)

poverty <- fgt_poverty(
  accessibility_data = access,
  opportunity = "jobs",
  sociodemographic_data = land_use_data,
  population = "population",
  poverty_line = 50000
)
poverty
#>         FGT0      FGT1      FGT2
#>        <num>     <num>     <num>
#> 1: 0.3923817 0.1776241 0.1010123