accessibility 1.2.0

Daniel Herszenhut 2023-09-21 4 min read

{accessibility} v1.2.0 has been released. This version includes two new functions to calculate inequality measures, concentration_index() and theil_t(). This post presents an overview of both functions. But first, let’s create an accessibility dataset to use in this demonstration.

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
#> 1: 89a881a5a2bffff 14561
#> 2: 89a881a5a2fffff 29452
#> 3: 89a881a5a67ffff 16647
#> 4: 89a881a5a6bffff 10700
#> 5: 89a881a5a6fffff  6669
#> 6: 89a881a5b03ffff 37029

Concentration Index

concentration_index() calculates the Concentration Index (CI) of a given accessibility distribution. This measures 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.

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. 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
#> 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 of 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
#> 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.
# ps: 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
#> 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
#>  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
#>  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

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.