# Calculating accessibility inequality and poverty

accessibility not only includes functions to calculate accessibility levels from travel time matrices and land use datasets, but also to calculate accessibility inequality and poverty from an accessibility distribution. This vignette briefly presents these functions.

# Demonstration on sample data

We first need to calculate the accessibility levels that we are going to use in this demonstration. To do so, we use the `cumulative_cutoff()` function and the sample data included in the package.

    library(accessibility)

    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,
      opportunity = "jobs",
      travel_cost = "travel_time",
      cutoff = 30
    )
    head(access)
    #>                 id  jobs
    #>             <char> <int>
    #> 1: 89a881a5a2bffff 14561
    #> 2: 89a881a5a2fffff 29452
    #> 3: 89a881a5a67ffff 16647
    #> 4: 89a881a5a6bffff 10700
    #> 5: 89a881a5a6fffff  6669
    #> 6: 89a881a5b03ffff 37029

The functions we’ll be demonstrating in this section take much of the same information as input, including:

- the accessibility distribution;
- a sociodemographic dataset containing information such as the distribution of people in the study area and their income;
- the name of the column in the accessibility dataset with the opportunities whose accessibility should be considered in the calculation; and
- the name of the column in the sociodemographic dataset with the number of people in each cell, used to weigh accessibility levels when calculating inequality/poverty.

## Palma Ratio

`palma_ratio()` calculates the Palma Ratio of a given accessibility distribution. Originally defined as the income share of the wealthiest 10% of a population divided by the income share of the poorest 40%, this measure has been adapted in transport planning as the average accessibility of the wealthiest 10% divided by the average accessibility of the poorest 40%. Palma Ratio values higher than 1 indicate a scenario in which the wealthiest population has higher accessibility levels than the poorest, whereas values lower than 1 indicate the opposite situation.

This function includes an additional `income` parameter, used to list the column in the sociodemographic dataset with the income variable that should be used to classify the population in socioeconomic groups. Please note that this variable should describe income per capita (e.g. mean income per capita, household income per capita, etc), instead of the total amount of income in each cell.

    palma <- palma_ratio(
      access,
      sociodemographic_data = land_use_data,
      opportunity = "jobs",
      population = "population",
      income = "income_per_capita"
    )
    palma
    #>    palma_ratio
    #>          <num>
    #> 1:    3.800465

## Gini Index

`gini_index()` calculates the Gini Index of a given accessibility distribution. Probably the most frequently used inequality measure in transport planning, this index estimates how much a distribution deviates from a hypothetical situation in which everyone has the exact same accessibility conditions. Gini Index values range from 0 to 1. A value of 0 indicates a scenario of perfect equality, in which everyone has the same accessibility levels, whereas a value of 1 indicates a scenario of perfect inequality, in which the accessibility levels in a study area are all concentrated into a single cell.

    gini <- gini_index(
      access,
      sociodemographic_data = land_use_data,
      opportunity = "jobs",
      population = "population"
    )
    gini
    #>    gini_index
    #>         <num>
    #> 1:  0.4715251

## Concentration Index

`concentration_index()` calculates the Concentration Index (CI) of a given accessibility distribution. This indicator estimates the extent to which accessibility inequalities are systematically associated with individuals’ socioeconomic levels. CI values can theoretically vary between -1 and 1 (when all accessibility is concentrated in the most or in the least disadvantaged cell, respectively). Negative values indicate that inequalities favor the poor, while positive values indicate a pro-rich bias.

Just like `palma_ratio()`, this function includes an `income` parameter to indicate which variable from the sociodemographic dataset should be used to rank the population from the least to the most privileged groups. Unlike the Palma Ratio function, however, any variable that can be used to describe one’s socioeconomic status, such as education level, for example, can be passed to this argument, as long as it can be numerically ordered (in which higher values denote higher socioeconomic status).

`concentration_index()` also includes a `type` parameter, used to indicate which Concentration Index to calculate. This parameter currently supports two values, `"standard"` and `"corrected"`, which respectively identify the standard relative CI and the corrected CI, proposed by Erreygers (2009).

    ci <- concentration_index(
      access,
      sociodemographic_data = land_use_data,
      opportunity = "jobs",
      population = "population",
      income = "income_per_capita",
      type = "corrected"
    )
    ci
    #>    concentration_index
    #>                  <num>
    #> 1:           0.3346494

## Theil T

`theil_t()` calculates the Theil T Index of a given accessibility distribution. Values range from 0 (when all individuals have exactly the same accessibility levels) to the natural log of n, in which n is the number of individuals in the accessibility dataset.

If the individuals can be classified into mutually exclusive and completely exhaustive groups (i.e. into groups that do not overlap and cover the entire population), the index can be decomposed into a between- and a within-groups inequality component. The function includes a `socioeconomic_groups` parameter to indicate which variable from the sociodemographic dataset should be used identify the socioeconomic groups used to calculate these components.

Please note that the output `theil_t()` varies based on the value of `socioeconomic_groups`. If `NULL` (the default), the between- and within-groups components are not calculated, and the function returns a data frame containing only the total aggregate inequality for the returned area. If `socioeconomic_groups` is not `NULL`, however, the function returns a list containing three dataframes: one summarizing the total inequality and the between- and within-groups components, one listing the contribution of each group to the between-groups component and another listing the contribution of each group to the within-groups component. Both behaviors are shown below.

    theil_without_groups <- theil_t(
      access,
      sociodemographic_data = land_use_data,
      opportunity = "jobs",
      population = "population"
    )
    theil_without_groups
    #>      theil_t
    #>        <num>
    #> 1: 0.3616631

    # some cells are classified as in the decile NA because their income per capita
    # is NaN, as they don't have any population. we filter these cells from our
    # accessibility data, otherwise the output would include NA values (note that
    # subsetting the data like this doesn't affect the assumption that groups are
    # completely exhaustive, because cells with NA income decile don't have any
    # population)

    na_decile_ids <- land_use_data[is.na(land_use_data$income_decile), ]$id
    no_na_access <- access[! access$id %in% na_decile_ids, ]
    sociodem_data <- land_use_data[! land_use_data$id %in% na_decile_ids, ]

    theil_with_groups <- theil_t(
      no_na_access,
      sociodemographic_data = sociodem_data,
      opportunity = "jobs",
      population = "population",
      socioeconomic_groups = "income_decile"
    )
    theil_with_groups
    #> $summary
    #>        component     value share_of_total
    #>           <char>     <num>          <num>
    #> 1: between_group 0.1280753      0.3541287
    #> 2:  within_group 0.2335878      0.6458713
    #> 3:         total 0.3616631      1.0000000
    #>
    #> $within_group_component
    #>     income_decile       value share_of_component
    #>            <fctr>       <num>              <num>
    #>  1:             1 0.009181454         0.03930622
    #>  2:             2 0.011413697         0.04886255
    #>  3:             3 0.019320622         0.08271246
    #>  4:             4 0.023606928         0.10106232
    #>  5:             5 0.031470429         0.13472633
    #>  6:             6 0.023539337         0.10077296
    #>  7:             7 0.033329635         0.14268567
    #>  8:             8 0.032585905         0.13950173
    #>  9:             9 0.020299031         0.08690107
    #> 10:            10 0.028840780         0.12346868
    #>
    #> $between_group_component
    #>     income_decile        value
    #>            <fctr>        <num>
    #>  1:             1 -0.037573783
    #>  2:             2 -0.036276865
    #>  3:             3 -0.031829123
    #>  4:             4 -0.021600054
    #>  5:             5 -0.009938574
    #>  6:             6 -0.004401762
    #>  7:             7  0.025936879
    #>  8:             8  0.042240708
    #>  9:             9  0.075742415
    #> 10:            10  0.125775443

## Foster-Greer-Thorbecke (FGT) poverty measures

`fgt_poverty()` calculates the FGT metrics, a family of poverty measures originally proposed by Foster, Greer, and Thorbecke (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:

- with \(\alpha = 0\) (FGT0) the measure captures the extent of poverty as a simple headcount - i.e. the proportion of people below the poverty line;
- with \(\alpha = 1\) (FGT1) the measure, also know as the “poverty gap index”, captures the severity of poverty as the average percentage distance between the poverty line and the accessibility of individuals below the poverty line;
- with \(\alpha = 2\) (FGT2) the measure simultaneously captures the extent and the severity of poverty by calculating the number of people below the poverty line weighted by the size of the accessibility shortfall relative to the poverty line.

This function includes an additional `poverty_line` parameter, used to define the poverty line below which individuals are considered to be in accessibility poverty.

    poverty <- fgt_poverty(
      access,
      sociodemographic_data = land_use_data,
      opportunity = "jobs",
      population = "population",
      poverty_line = 50000
    )

    poverty
    #>         FGT0      FGT1      FGT2
    #>        <num>     <num>     <num>
    #> 1: 0.3923817 0.1776241 0.1010123

# References

Erreygers, Guido. 2009. “Correcting the Concentration Index.” Journal of Health Economics 28 (2): 504–15. https://doi.org/10.1016/j.jhealeco.2008.02.003.
Foster, James, Joel Greer, and Erik Thorbecke. 1984. “A Class of Decomposable Poverty Measures.” Econometrica 52 (3): 761–66. https://doi.org/10.2307/1913475.