Type:	Package
Title:	Entropy-Based Segregation Indices
Version:	1.1.0
Description:	Computes segregation indices, including the Index of Dissimilarity, as well as the information-theoretic indices developed by Theil (1971) <isbn:978-0471858454>, namely the Mutual Information Index (M) and Theil's Information Index (H). The M, further described by Mora and Ruiz-Castillo (2011) <doi:10.1111/j.1467-9531.2011.01237.x> and Frankel and Volij (2011) <doi:10.1016/j.jet.2010.10.008>, is a measure of segregation that is highly decomposable. The package provides tools to decompose the index by units and groups (local segregation), and by within and between terms. The package also provides a method to decompose differences in segregation as described by Elbers (2021) <doi:10.1177/0049124121986204>. The package includes standard error estimation by bootstrapping, which also corrects for small sample bias. The package also contains functions for visualizing segregation patterns.
License:	MIT + file LICENSE
Depends:	R (≥ 3.5.0)
Imports:	data.table, checkmate, Rcpp, RcppProgress,
Encoding:	UTF-8
LazyData:	true
Suggests:	testthat, covr, knitr, rmarkdown, dplyr, ggplot2, scales, tidycensus, tigris, rrapply, dendextend, patchwork
URL:	https://elbersb.github.io/segregation/
BugReports:	https://github.com/elbersb/segregation/issues
RoxygenNote:	7.2.3
VignetteBuilder:	knitr
SystemRequirements:	C++17
LinkingTo:	Rcpp, RcppProgress
NeedsCompilation:	yes
Packaged:	2023-12-02 11:54:48 UTC; benjamin
Author:	Benjamin Elbers [aut, cre]
Maintainer:	Benjamin Elbers <be2239@columbia.edu>
Repository:	CRAN
Date/Publication:	2023-12-02 12:10:01 UTC

segregation: Entropy-based segregation indices

Description

Calculate and decompose entropy-based, multigroup segregation indices, with a focus on the Mutual Information Index (M) and Theil's Information Index (H). Provides tools to decompose the measures by groups and units, and by within and between terms. Includes standard error estimation by bootstrapping.

Author(s)

Maintainer: Benjamin Elbers be2239@columbia.edu (ORCID)

See Also

https://elbersb.com/segregation

Compresses a data matrix based on mutual information (segregation)

Description

Given a data set that identifies suitable neighbors for merging, this function will merge units iteratively, where in each iteration the neighbors with the smallest reduction in terms of total M will be merged.

Usage

compress(
  data,
  group,
  unit,
  weight = NULL,
  neighbors = "local",
  n_neighbors = 50,
  max_iter = Inf
)

Arguments

Value

Returns a data.table.

Calculates Index of Dissimilarity

Description

Returns the total segregation between group and unit using the Index of Dissimilarity.

Usage

dissimilarity(
  data,
  group,
  unit,
  weight = NULL,
  se = FALSE,
  CI = 0.95,
  n_bootstrap = 100
)

Arguments

Value

Returns a data.table with one row. The column est contains the Index of Dissimilarity. If se is set to TRUE, an additional column se contains the associated bootstrapped standard errors, an additional column CI contains the estimate confidence interval as a list column, an additional column bias contains the estimated bias, and the column est contains the bias-corrected estimates.

References

Otis Dudley Duncan and Beverly Duncan. 1955. "A Methodological Analysis of Segregation Indexes," American Sociological Review 20(2): 210-217.

Examples

# Example where D and H deviate
m1 <- matrix_to_long(matrix(c(100, 60, 40, 0, 0, 40, 60, 100), ncol = 2))
m2 <- matrix_to_long(matrix(c(80, 80, 20, 20, 20, 20, 80, 80), ncol = 2))
dissimilarity(m1, "group", "unit", weight = "n")
dissimilarity(m2, "group", "unit", weight = "n")

Calculates expected values when true segregation is zero

Description

When sample sizes are small, one group has a small proportion, or when there are many units, segregation indices are typically upwardly biased, even when true segregation is zero. This function simulates tables with zero segregation, given the marginals of the dataset, and calculates segregation. If the expected values are large, the interpretation of index scores might have to be adjusted.

Usage

dissimilarity_expected(
  data,
  group,
  unit,
  weight = NULL,
  fixed_margins = TRUE,
  n_bootstrap = 100
)

Arguments

Value

A data.table with one row, corresponding to the expected value of the D index when true segregation is zero.

Examples

# build a smaller table, with 100 students distributed across
# 10 schools, where one racial group has 10% of the students
small <- data.frame(
    school = c(1:10, 1:10),
    race = c(rep("r1", 10), rep("r2", 10)),
    n = c(rep(1, 10), rep(9, 10))
)
dissimilarity_expected(small, "race", "school", weight = "n")
# with an increase in sample size (n=1000), the values improve
small$n <- small$n * 10
dissimilarity_expected(small, "race", "school", weight = "n")

Calculates the entropy of a distribution

Description

Returns the entropy of the distribution defined by group.

Usage

entropy(data, group, weight = NULL, base = exp(1))

Arguments

Value

A single number, the entropy.

Examples

d <- data.frame(cat = c("A", "B"), n = c(25, 75))
entropy(d, "cat", weight = "n") # => .56
# this is equivalent to -.25*log(.25)-.75*log(.75)

d <- data.frame(cat = c("A", "B"), n = c(50, 50))
# use base 2 for the logarithm, then entropy is maximized at 1
entropy(d, "cat", weight = "n", base = 2) # => 1

Calculates pairwise exposure indices

Description

Returns the pairwise exposure indices between groups

Usage

exposure(data, group, unit, weight = NULL)

Arguments

Value

Returns a data.table with columns "of", "to", and "exposure". Read results as "exposure of group x to group y".

Create crosswalk after compression

Description

After running compress, this function creates a crosswalk table. Usually it is preferred to call merge_units directly.

Usage

get_crosswalk(compression, n_units = NULL, percent = NULL, parts = FALSE)

Arguments

Value

Returns a ggplot2 plot.

Returns a data.table.

Adjustment of marginal distributions using iterative proportional fitting

Description

Adjusts the marginal distributions for group and unit in source to the respective marginal distributions in target, using the iterative proportional fitting algorithm (IPF).

Usage

ipf(
  source,
  target,
  group,
  unit,
  weight = NULL,
  max_iterations = 100,
  precision = 1e-04
)

Arguments

Details

The algorithm works by scaling the marginal distribution of group in the source data frame towards the marginal distribution of target; then repeating this process for unit. The algorithm then keeps alternating between group and unit until the marginals of the adjusted data frame are within the allowed precision. This results in a dataset that retains the association structure of source while approximating the marginal distribution of target. If the number of unit and group categories is different in source and target, the data frame returns the combination of unit and group categories that occur in both datasets. Zero values are replaced by a small, non-zero number (1e-4). Note that the values returned sum to the observations of the source data frame, not the target data frame. This is different from other IPF implementations, but ensures that the IPF does not change the number of observations.

Value

Returns a data frame that retains the association structure of source while approximating the marginal distributions for group and unit of target. The dataset identifies each combination of group and unit, and categories that only occur in either source or target are dropped. The adjusted frequency of each combination is given by the column n, while n_target and n_source contain the zero-adjusted frequencies in the target and source dataset, respectively.

References

W. E. Deming and F. F. Stephan. 1940. "On a Least Squares Adjustment of a Sampled Frequency Table When the Expected Marginal Totals are Known". Annals of Mathematical Statistics. 11 (4): 427–444.

T. Karmel and M. Maclachlan. 1988. "Occupational Sex Segregation — Increasing or Decreasing?" Economic Record 64: 187-195.

Examples

## Not run: 
# adjusts the marginals of group and unit categories so that
# schools00 has similar marginals as schools05
adj <- ipf(schools00, schools05, "race", "school", weight = "n")

# check that the new "race" marginals are similar to the target marginals
# (the same could be done for schools)
aggregate(adj$n, list(adj$race), sum)
aggregate(adj$n_target, list(adj$race), sum)

# note that the adjusted dataset contains fewer
# schools than either the source or the target dataset,
# because the marginals are only defined for the overlap
# of schools
length(unique(schools00$school))
length(unique(schools05$school))
length(unique(adj$school))

## End(Not run)

Calculates isolation indices

Description

Returns isolation index of each group

Usage

isolation(data, group, unit, weight = NULL)

Arguments

Value

Returns a data.table with group column and isolation index.

Turns a contingency table into long format

Description

Returns a data.table in long form, such that it is suitable for use in mutual_total, etc. Colnames and rownames of the matrix will be respected.

Usage

matrix_to_long(
  matrix,
  group = "group",
  unit = "unit",
  weight = "n",
  drop_zero = TRUE
)

Arguments

Value

A data.table.

Examples

m <- matrix(c(10, 20, 30, 30, 20, 10), nrow = 3)
colnames(m) <- c("Black", "White")
long <- matrix_to_long(m, group = "race", unit = "school")
mutual_total(long, "race", "school", weight = "n")

Creates a compressed dataset

Description

After running compress, this function creates a dataset where units are merged.

Usage

merge_units(compression, n_units = NULL, percent = NULL, parts = FALSE)

Arguments

Value

Returns a data.table.

Decomposes the difference between two M indices

Description

Uses one of three methods to decompose the difference between two M indices: (1) "shapley" / "shapley_detailed": a method based on the Shapley decomposition with a few advantages over the Karmel-Maclachlan method (recommended and the default, Deutsch et al. 2006), (2) "km": the method based on Karmel-Maclachlan (1988), (3) "mrc": the method developed by Mora and Ruiz-Castillo (2009). All methods have been extended to account for missing units/groups in either data input.

Usage

mutual_difference(
  data1,
  data2,
  group,
  unit,
  weight = NULL,
  method = "shapley",
  se = FALSE,
  CI = 0.95,
  n_bootstrap = 100,
  base = exp(1),
  ...
)

Arguments

Details

The Shapley method is an improvement over the Karmel-Maclachlan method (Deutsch et al. 2006). It is based on several margins-adjusted data inputs and yields symmetrical results (i.e. data1 and data2 can be switched). When "shapley_detailed" is used, the structural component is further decomposed into the contributions of individuals units.

The Karmel-Maclachlan method (Karmel and Maclachlan 1988) adjusts the margins of data1 to be similar to the margins of data2. This process is not symmetrical.

The Shapley and Karmel-Maclachlan methods are based on iterative proportional fitting (IPF), first introduced by Deming and Stephan (1940). Depending on the size of the dataset, this may take a few seconds (see ipf for details).

The method developed by Mora and Ruiz-Castillo (2009) uses an algebraic approach to estimate the size of the components. This will often yield substantively different results from the Shapley and Karmel-Maclachlan methods. Note that this method is not symmetric in terms of what is defined as group and unit categories, which may yield contradictory results.

A problem arises when there are group and/or unit categories in data1 that are not present in data2 (or vice versa). All methods estimate the difference only for categories that are present in both datasets, and report additionally the change in M that is induced by these cases as additions (present in data2, but not in data1) and removals (present in data1, but not in data2).

Value

Returns a data.table with columns stat and est. The data frame contains the following rows defined by stat: M1 contains the M for data1. M2 contains the M for data2. diff is the difference between M2 and M1. The sum of the five rows following diff equal diff.

additions contains the change in M induces by unit and group categories present in data2 but not data1, and removals the reverse.

All methods return the following three terms: unit_marginal is the contribution of unit composition differences. group_marginal is the contribution of group composition differences. structural is the contribution unexplained by the marginal changes, i.e. the structural difference. Note that the interpretation of these terms depend on the exact method used.

When using "km", one additional row is returned: interaction is the contribution of differences in the joint marginal distribution of unit and group.

When "shapley_detailed" is used, an additional column "unit" is returned, along with six additional rows for each unit that is present in both data1 and data2. The five rows have the following meaning: p1 (p2) is the proportion of the unit in data1 (data2) once non-intersecting units/groups have been removed. The changes in local linkage are given by ls_diff1 and ls_diff2, and their average is given by ls_diff_mean. The row named total summarizes the contribution of the unit towards structural change using the formula .5 * p1 * ls_diff1 + .5 * p2 * ls_diff2. The sum of all "total" components equals structural change.

If se is set to TRUE, an additional column se contains the associated bootstrapped standard errors, an additional column CI contains the estimate confidence interval as a list column, an additional column bias contains the estimated bias, and the column est contains the bias-corrected estimates.

References

W. E. Deming, F. F. Stephan. 1940. "On a Least Squares Adjustment of a Sampled Frequency Table When the Expected Marginal Totals are Known." The Annals of Mathematical Statistics 11(4): 427-444.

T. Karmel and M. Maclachlan. 1988. "Occupational Sex Segregation — Increasing or Decreasing?" Economic Record 64: 187-195.

R. Mora and J. Ruiz-Castillo. 2009. "The Invariance Properties of the Mutual Information Index of Multigroup Segregation." Research on Economic Inequality 17: 33-53.

J. Deutsch, Y. Flückiger, and J. Silber. 2009. "Analyzing Changes in Occupational Segregation: The Case of Switzerland (1970–2000)." Research on Economic Inequality 17: 171–202.

Examples

## Not run: 
# decompose the difference in school segregation between 2000 and 2005,
# using the Shapley method
mutual_difference(schools00, schools05,
    group = "race", unit = "school",
    weight = "n", method = "shapley", precision = .1
)
# => the structural component is close to zero, thus most change is in the marginals.
# This method gives identical results when we switch the unit and group definitions,
# and when we switch the data inputs.

# the Karmel-Maclachlan method is similar, but only adjust the data in the forward direction...
mutual_difference(schools00, schools05,
    group = "school", unit = "race",
    weight = "n", method = "km", precision = .1
)

# ...this means that the results won't be identical when we switch the data inputs
mutual_difference(schools05, schools00,
    group = "school", unit = "race",
    weight = "n", method = "km", precision = .1
)

# the MRC method indicates a much higher structural change...
mutual_difference(schools00, schools05,
    group = "race", unit = "school",
    weight = "n", method = "mrc"
)

# ...and is not symmetric
mutual_difference(schools00, schools05,
    group = "school", unit = "race",
    weight = "n", method = "mrc"
)

## End(Not run)

Calculates expected values when true segregation is zero

Description

When sample sizes are small, one group has a small proportion, or when there are many units, segregation indices are typically upwardly biased, even when true segregation is zero. This function simulates tables with zero segregation, given the marginals of the dataset, and calculates segregation. If the expected values are large, the interpretation of index scores might have to be adjusted.

Usage

mutual_expected(
  data,
  group,
  unit,
  weight = NULL,
  within = NULL,
  fixed_margins = TRUE,
  n_bootstrap = 100,
  base = exp(1)
)

Arguments

Value

A data.table with two rows, corresponding to the expected values of segregation when true segregation is zero.

Examples

## Not run: 
# the schools00 dataset has a large sample size, so expected segregation is close to zero
mutual_expected(schools00, "race", "school", weight = "n")

# but we can build a smaller table, with 100 students distributed across
# 10 schools, where one racial group has 10% of the students
small <- data.frame(
    school = c(1:10, 1:10),
    race = c(rep("r1", 10), rep("r2", 10)),
    n = c(rep(1, 10), rep(9, 10))
)
mutual_expected(small, "race", "school", weight = "n")
# with an increase in sample size (n=1000), the values improve
small$n <- small$n * 10
mutual_expected(small, "race", "school", weight = "n")

## End(Not run)

Calculates local segregation scores based on M

Description

Returns local segregation indices for each category defined by unit.

Usage

mutual_local(
  data,
  group,
  unit,
  weight = NULL,
  se = FALSE,
  CI = 0.95,
  n_bootstrap = 100,
  base = exp(1),
  wide = FALSE
)

Arguments

Value

Returns a data.table with two rows for each category defined by unit, for a total of 2*(number of units) rows. The column est contains two statistics that are provided for each unit: ls, the local segregation score, and p, the proportion of the unit from the total number of cases. If se is set to TRUE, an additional column se contains the associated bootstrapped standard errors, an additional column CI contains the estimate confidence interval as a list column, an additional column bias contains the estimated bias, and the column est contains the bias-corrected estimates. If wide is set to TRUE, returns instead a wide dataframe, with one row for each unit, and the associated statistics in separate columns.

References

Henri Theil. 1971. Principles of Econometrics. New York: Wiley.

Ricardo Mora and Javier Ruiz-Castillo. 2011. "Entropy-based Segregation Indices". Sociological Methodology 41(1): 159–194.

Examples

# which schools are most segregated?
(localseg <- mutual_local(schools00, "race", "school",
    weight = "n", wide = TRUE
))

sum(localseg$p) # => 1

# the sum of the weighted local segregation scores equals
# total segregation
sum(localseg$ls * localseg$p) # => .425
mutual_total(schools00, "school", "race", weight = "n") # M => .425

Calculates the Mutual Information Index M and Theil's Entropy Index H

Description

Returns the total segregation between group and unit. If within is given, calculates segregation within each within category separately, and takes the weighted average. Also see mutual_within for detailed within calculations.

Usage

mutual_total(
  data,
  group,
  unit,
  within = NULL,
  weight = NULL,
  se = FALSE,
  CI = 0.95,
  n_bootstrap = 100,
  base = exp(1)
)

Arguments

Value

Returns a data.table with two rows. The column est contains the Mutual Information Index, M, and Theil's Entropy Index, H. The H is the the M divided by the group entropy. If within was given, M and H are weighted averages of the within-category segregation scores. If se is set to TRUE, an additional column se contains the associated bootstrapped standard errors, an additional column CI contains the estimate confidence interval as a list column, an additional column bias contains the estimated bias, and the column est contains the bias-corrected estimates.

References

Henri Theil. 1971. Principles of Econometrics. New York: Wiley.

Ricardo Mora and Javier Ruiz-Castillo. 2011. "Entropy-based Segregation Indices". Sociological Methodology 41(1): 159–194.

Examples

# calculate school racial segregation
mutual_total(schools00, "school", "race", weight = "n") # M => .425

# note that the definition of groups and units is arbitrary
mutual_total(schools00, "race", "school", weight = "n") # M => .425

# if groups or units are defined by a combination of variables,
# vectors of variable names can be provided -
# here there is no difference, because schools
# are nested within districts
mutual_total(schools00, "race", c("district", "school"),
    weight = "n"
) # M => .424

# estimate standard errors and 95% CI for M and H
## Not run: 
mutual_total(schools00, "race", "school",
    weight = "n",
    se = TRUE, n_bootstrap = 1000
)

# estimate segregation within school districts
mutual_total(schools00, "race", "school",
    within = "district", weight = "n"
) # M => .087

# estimate between-district racial segregation
mutual_total(schools00, "race", "district", weight = "n") # M => .338
# note that the sum of within-district and between-district
# segregation equals total school-race segregation;
# here, most segregation is between school districts

## End(Not run)

Calculates a nested decomposition of segregation for M and H

Description

Returns the between-within decomposition defined by the sequence of variables in unit.

Usage

mutual_total_nested(data, group, unit, weight = NULL, base = exp(1))

Arguments

Value

Returns a data.table similar to mutual_total, but with column between and within that define the levels of nesting.

Examples

mutual_total_nested(schools00, "race", c("state", "district", "school"),
    weight = "n"
)
# This is a simpler way to run the following manually:
# mutual_total(schools00, "race", "state", weight = "n")
# mutual_total(schools00, "race", "district", within = "state", weight = "n")
# mutual_total(schools00, "race", "school", within = c("state", "district"), weight = "n")

Calculates detailed within-category segregation scores for M and H

Description

Calculates the segregation between group and unit within each category defined by within.

Usage

mutual_within(
  data,
  group,
  unit,
  within,
  weight = NULL,
  se = FALSE,
  CI = 0.95,
  n_bootstrap = 100,
  base = exp(1),
  wide = FALSE
)

Arguments

Value

Returns a data.table with four rows for each category defined by within. The column est contains four statistics that are provided for each unit: M is the within-category M, and p is the proportion of the category. Multiplying M and p gives the contribution of each within-category towards the total M. H is the within-category H, and ent_ratio provides the entropy ratio, defined as EW/E, where EW is the within-category entropy, and E is the overall entropy. Multiplying H, p, and ent_ratio gives the contribution of each within-category towards the total H. If se is set to TRUE, an additional column se contains the associated bootstrapped standard errors, an additional column CI contains the estimate confidence interval as a list column, an additional column bias contains the estimated bias, and the column est contains the bias-corrected estimates. If wide is set to TRUE, returns instead a wide dataframe, with one row for each within category, and the associated statistics in separate columns.

References

Henri Theil. 1971. Principles of Econometrics. New York: Wiley.

Ricardo Mora and Javier Ruiz-Castillo. 2011. "Entropy-based Segregation Indices". Sociological Methodology 41(1): 159–194.

Examples

## Not run: 
(within <- mutual_within(schools00, "race", "school",
    within = "state",
    weight = "n", wide = TRUE
))
# the M for state "A" is .409
# manual calculation
schools_A <- schools00[schools00$state == "A", ]
mutual_total(schools_A, "race", "school", weight = "n") # M => .409

# to recover the within M and H from the output, multiply
# p * M and p * ent_ratio * H, respectively
sum(within$p * within$M) # => .326
sum(within$p * within$ent_ratio * within$H) # => .321
# compare with:
mutual_total(schools00, "race", "school", within = "state", weight = "n")

## End(Not run)

Student-level data including SES status

Description

Fake dataset used for examples. This is an individual-level dataset of students in schools.

Usage

school_ses

Format

A data frame with 5,153 rows and 3 variables:

school_id

school ID

ethnic_group

one of A, B, or C

ses_quintile

SES of the student (1 = lowest, 5 = highest)

Ethnic/racial composition of schools for 2000/2001

Description

Fake dataset used for examples. Loosely based on data provided by the National Center for Education Statistics, Common Core of Data, with information on U.S. primary schools in three U.S. states. The original data can be downloaded at https://nces.ed.gov/ccd/.

Usage

schools00

Format

A data frame with 8,142 rows and 5 variables:

state

either A, B, or C

district

school agency/district ID

school

school ID

race

either native, asian, hispanic, black, or white

n

n of students by school and race

Ethnic/racial composition of schools for 2005/2006

Description

Fake dataset used for examples. Loosely based on data provided by the National Center for Education Statistics, Common Core of Data, with information on U.S. primary schools in three U.S. states. The original data can be downloaded at https://nces.ed.gov/ccd/.

Usage

schools05

Format

A data frame with 8,013 rows and 5 variables:

state

either A, B, or C

district

school agency/district ID

school

school ID

race

either native, asian, hispanic, black, or white

n

n of students by school and race

Scree plot for segregation compression

Description

A plot that allows to visually see the effect of compression on mutual information.

Usage

scree_plot(compression, tail = Inf)

Arguments

Value

Returns a ggplot2 plot.

A visual representation of two-group segregation

Description

Produces one or several segregation curves, as defined in Duncan and Duncan (1955)

Usage

segcurve(data, group, unit, weight = NULL, segment = NULL)

Arguments

Value

Returns a ggplot2 object.

A visual representation of segregation

Description

Produces a segregation plot.

Usage

segplot(
  data,
  group,
  unit,
  weight,
  order = "segregation",
  secondary_plot = NULL,
  reference_distribution = NULL,
  bar_space = 0,
  hline = NULL
)

Arguments

Value

Returns a ggplot2 or patchwork object.