Abstract

This is a study of gerrymandering in Alabama. We will test three methods of shape-based compactness scores, assess representativeness of districts based on prior presidential elections and race. We will then extend prior studies by calculating representativeness of the convex hull of district polygons.

Study metadata

Study design

This is an original study based on literature on gerrymandering metrics.

It is an exploratory study to evaluate usefulness of a new gerrymandering metric based on the convex hull of a congressional district and the representativeness inside the convex hull compared to the congressional district.

Materials and procedure

Computational environment

I plan on using: - the tidyverse package for general dataset processing - the here package for file locations - the sf package for spatial processing - the tmap package for displaying data - the tidycensus package for gathering census data - the lwgeom package for minimum bounding circles - the knitr package for better table displays - the patchwork package allows for multiple graphs in one display - the cowplot package allows for extraction of plot legends

Data and variables

We plan on using data scources: precincts20 districts23, blockgroups2020

Precincts 2020 (Secondary)

  • Title: Voting Precincts 2020
  • Abstract: Alabama voting data for 2020 elections by precinct.
  • Spatial Coverage: Alabama
  • Spatial Resolution: Voting precincts
  • Spatial Reference System: EPSG 4269 NAD 1983 Geographic Coordinate System
  • Temporal Coverage: voting precincts used for tabulating the 2020 election
  • Temporal Resolution: annual election
  • Lineage: Saved as geopackage format. Processing prior to download is explained in al_vest_20_validation_report.pdf
  • Distribution: Data available at Redistricting Data Hub
  • Constraints: Permitted for noncommercial and nonpartisan use only. Copyright and use constraints explained in redistrictingdatahub_legal.txt
  • Data Quality: State any planned quality assessment
  • Variables: For each variable, enter the following information. If you have two or more variables per data source, you may want to present this information in table form (shown below)
    • Label: variable name as used in the data or code
    • Alias: intuitive natural language name
    • Definition: Short description or definition of the variable. Include measurement units in description.
    • Type: data type, e.g. character string, integer, real
    • Accuracy: e.g. uncertainty of measurements
    • Domain: Expected range of Maximum and Minimum of numerical data, or codes or categories of nominal data, or reference to a standard codebook
    • Missing Data Value(s): Values used to represent missing data and frequency of missing data observations
    • Missing Data Frequency: Frequency of missing data observations: not yet known for data to be collected
Label Alias Definition Type Accuracy Domain Missing Data Value(s) Missing Data Frequency
VTDST20 … Voting district ID … … … … …
GEOID20 … Unique Geographic ID … … … … …
G20PRERTRU … total votes for Trump in 2020 … … … … …
G20PREDBID … total votes for Biden in 2020 … … … … … 
precincts20 <- st_read(here("data","raw","public","districts.gpkg"), layer = "precincts20")
## Reading layer `precincts20' from data source 
##   `/Users/jorredahl/Documents/GitHub/OR-Gerrymander-Alabama/data/raw/public/districts.gpkg' 
##   using driver `GPKG'
## Simple feature collection with 1972 features and 8 fields
## Geometry type: MULTIPOLYGON
## Dimension:     XY
## Bounding box:  xmin: -88.47323 ymin: 30.14442 xmax: -84.88825 ymax: 35.00803
## Geodetic CRS:  NAD83

Districts 2023 (Primary)

  • Title: Voting Districts 2024
  • Abstract: Alabama voitng districts approved in 2023 for use in 2024
  • Spatial Coverage: Alabama
  • Spatial Resolution: US Congressional Districts
  • Spatial Reference System: EPSG 3857 WGS 1984 Web Mercator projection
  • Temporal Coverage: approved in 2023 for 2024 use
  • Temporal Resolution: districting based on 2020 census data (updated every 10 years)
  • Lineage: Loaded into QGIS as ArcGIS feature service layer and saved in geopackage format. Extraneous data fields were removed and the FIX GEOMETRIES tool was used to correct geometry errors.
  • Distribution: Alabama State GIS via ESRI feature service at https://services7.arcgis.com/jF2q3LPxL7PETdYk/arcgis/rest/services/2023_Court_Ordered_Congressional_Plan/FeatureServer
  • Constraints: Public Domain data free for use and redistribution.
  • Data Quality: State any planned quality assessment
  • Variables: For each variable, enter the following information. If you have two or more variables per data source, you may want to present this information in table form (shown below)
    • Label: variable name as used in the data or code
    • Alias: intuitive natural language name
    • Definition: Short description or definition of the variable. Include measurement units in description.
    • Type: data type, e.g. character string, integer, real
    • Accuracy: e.g. uncertainty of measurements
    • Domain: Expected range of Maximum and Minimum of numerical data, or codes or categories of nominal data, or reference to a standard codebook
    • Missing Data Value(s): Values used to represent missing data and frequency of missing data observations
    • Missing Data Frequency: Frequency of missing data observations: not yet known for data to be collected
Label Alias Definition Type Accuracy Domain Missing Data Value(s) Missing Data Frequency
DISTRICT … US Congressional District Number … … … … …
POPULATION … total population (2020 census) … … … … …
WHITE … total white population (2020 census) … … … … …
BLACK … total black population (2020 census) … … … … … 
districts23 <- st_read(here("data","raw","public","districts.gpkg"), layer = "districts23")
## Reading layer `districts23' from data source 
##   `/Users/jorredahl/Documents/GitHub/OR-Gerrymander-Alabama/data/raw/public/districts.gpkg' 
##   using driver `GPKG'
## Simple feature collection with 7 features and 4 fields
## Geometry type: MULTIPOLYGON
## Dimension:     XY
## Bounding box:  xmin: -88.47323 ymin: 30.14443 xmax: -84.88825 ymax: 35.00803
## Geodetic CRS:  NAD83

Block Groups 2020 (Primary)

  • Title: 2020 Census block groups
  • Abstract: Vector polygon layer of census block groups and their demographic data
  • Spatial Coverage: Alabama
  • Spatial Resolution: Census block groups
  • Spatial Reference System: EPSG 4269 NAD 1983 Geographic Coordinate System
  • Temporal Coverage: 2020 census
  • Temporal Resolution: 10-year census
  • Lineage: Downloaded data from US Census API “p1” public law summary file using tidycensus in R.
  • Distribution: US Census API
  • Constraints: Public Domain data free for use and redistribution.
  • Data Quality: State any planned quality assessment
  • Variables: For each variable, enter the following information. If you have two or more variables per data source, you may want to present this information in table form (shown below)
    • Label: variable name as used in the data or code
    • Alias: intuitive natural language name
    • Definition: Short description or definition of the variable. Include measurement units in description.
    • Type: data type, e.g. character string, integer, real
    • Accuracy: e.g. uncertainty of measurements
    • Domain: Expected range of Maximum and Minimum of numerical data, or codes or categories of nominal data, or reference to a standard codebook
    • Missing Data Value(s): Values used to represent missing data and frequency of missing data observations
    • Missing Data Frequency: Frequency of missing data observations: not yet known for data to be collected
Label Alias Definition Type Accuracy Domain Missing Data Value(s) Missing Data Frequency
GEOID … Code to uniquely identify tracts … … … … …
P4_001N … Total Population, 18 years or older … … … … …
P4006N … Total: Not Hispanic or Latino, Population of one race, Black or African American alone, 18 years or older … … … … …
G20PREDBID … Total institutionalized population in correctional facilities for adults 18 years or older … … … … … 
census_metadata_file <- here("data", "metadata", "census2020pl_vars.csv")
if(file.exists(census_metadata_file)){
  census2020pl_vars <- read.csv(census_metadata_file)
} else {
  census2020pl_vars <- load_variables(2020, "pl")
  write.csv(census2020pl_vars, here("data", "metadata", "census2020pl_vars.csv"))
}
blockgroup_file <- here("data", "raw", "public", "block_groups.gpkg")

# if the data is already downloaded, just load it
# otherwise, query from the census and save
if(file.exists(blockgroup_file)){
  blockgroups <- st_read(blockgroup_file)
} else {
  blockgroups <- get_decennial(geography = "block group",
                               sumfile = "pl",
                               table = "P3",
                               year = 2020,
                               state = "Alabama",
                               output = "wide",
                               geometry = TRUE,
                               keep_geo_vars = TRUE)
  st_write(blockgroups, blockgroup_file)
}
## Reading layer `block_groups' from data source 
##   `/Users/jorredahl/Documents/GitHub/OR-Gerrymander-Alabama/data/raw/public/block_groups.gpkg' 
##   using driver `GPKG'
## Simple feature collection with 3925 features and 83 fields (with 1 geometry empty)
## Geometry type: MULTIPOLYGON
## Dimension:     XY
## Bounding box:  xmin: -88.47323 ymin: 30.22333 xmax: -84.88908 ymax: 35.00803
## Geodetic CRS:  NAD83

Prior observations

I have not looked at this data before.

Bias and threats to validity

Modifiable Areal Unit Problem: demographic data is collected at both the district and the block level - This experiment tries to combat many of the problems with edge effects associated with studies of gerrymandering.

Data transformations

Calculate Percent Black

Find the sum of Black or African American people by gathering all variables that include the term “Black”.

black_vars <- census2020pl_vars |> 
  dplyr::filter(str_detect(name, "P3"),
                str_detect(label, "Black")) |> 
  select(-concept)

black_vars |> kable()
X name label
151 P3_004N !!Total:!!Population of one race:!!Black or African American alone
158 P3_011N !!Total:!!Population of two or more races:!!Population of two races:!!White; Black or African American
163 P3_016N !!Total:!!Population of two or more races:!!Population of two races:!!Black or African American; American Indian and Alaska Native
164 P3_017N !!Total:!!Population of two or more races:!!Population of two races:!!Black or African American; Asian
165 P3_018N !!Total:!!Population of two or more races:!!Population of two races:!!Black or African American; Native Hawaiian and Other Pacific Islander
166 P3_019N !!Total:!!Population of two or more races:!!Population of two races:!!Black or African American; Some Other Race
174 P3_027N !!Total:!!Population of two or more races:!!Population of three races:!!White; Black or African American; American Indian and Alaska Native
175 P3_028N !!Total:!!Population of two or more races:!!Population of three races:!!White; Black or African American; Asian
176 P3_029N !!Total:!!Population of two or more races:!!Population of three races:!!White; Black or African American; Native Hawaiian and Other Pacific Islander
177 P3_030N !!Total:!!Population of two or more races:!!Population of three races:!!White; Black or African American; Some Other Race
184 P3_037N !!Total:!!Population of two or more races:!!Population of three races:!!Black or African American; American Indian and Alaska Native; Asian
185 P3_038N !!Total:!!Population of two or more races:!!Population of three races:!!Black or African American; American Indian and Alaska Native; Native Hawaiian and Other Pacific Islander
186 P3_039N !!Total:!!Population of two or more races:!!Population of three races:!!Black or African American; American Indian and Alaska Native; Some Other Race
187 P3_040N !!Total:!!Population of two or more races:!!Population of three races:!!Black or African American; Asian; Native Hawaiian and Other Pacific Islander
188 P3_041N !!Total:!!Population of two or more races:!!Population of three races:!!Black or African American; Asian; Some Other Race
189 P3_042N !!Total:!!Population of two or more races:!!Population of three races:!!Black or African American; Native Hawaiian and Other Pacific Islander; Some Other Race
195 P3_048N !!Total:!!Population of two or more races:!!Population of four races:!!White; Black or African American; American Indian and Alaska Native; Asian
196 P3_049N !!Total:!!Population of two or more races:!!Population of four races:!!White; Black or African American; American Indian and Alaska Native; Native Hawaiian and Other Pacific Islander
197 P3_050N !!Total:!!Population of two or more races:!!Population of four races:!!White; Black or African American; American Indian and Alaska Native; Some Other Race
198 P3_051N !!Total:!!Population of two or more races:!!Population of four races:!!White; Black or African American; Asian; Native Hawaiian and Other Pacific Islander
199 P3_052N !!Total:!!Population of two or more races:!!Population of four races:!!White; Black or African American; Asian; Some Other Race
200 P3_053N !!Total:!!Population of two or more races:!!Population of four races:!!White; Black or African American; Native Hawaiian and Other Pacific Islander; Some Other Race
205 P3_058N !!Total:!!Population of two or more races:!!Population of four races:!!Black or African American; American Indian and Alaska Native; Asian; Native Hawaiian and Other Pacific Islander
206 P3_059N !!Total:!!Population of two or more races:!!Population of four races:!!Black or African American; American Indian and Alaska Native; Asian; Some Other Race
207 P3_060N !!Total:!!Population of two or more races:!!Population of four races:!!Black or African American; American Indian and Alaska Native; Native Hawaiian and Other Pacific Islander; Some Other Race
208 P3_061N !!Total:!!Population of two or more races:!!Population of four races:!!Black or African American; Asian; Native Hawaiian and Other Pacific Islander; Some Other Race
211 P3_064N !!Total:!!Population of two or more races:!!Population of five races:!!White; Black or African American; American Indian and Alaska Native; Asian; Native Hawaiian and Other Pacific Islander
212 P3_065N !!Total:!!Population of two or more races:!!Population of five races:!!White; Black or African American; American Indian and Alaska Native; Asian; Some Other Race
213 P3_066N !!Total:!!Population of two or more races:!!Population of five races:!!White; Black or African American; American Indian and Alaska Native; Native Hawaiian and Other Pacific Islander; Some Other Race
214 P3_067N !!Total:!!Population of two or more races:!!Population of five races:!!White; Black or African American; Asian; Native Hawaiian and Other Pacific Islander; Some Other Race
216 P3_069N !!Total:!!Population of two or more races:!!Population of five races:!!Black or African American; American Indian and Alaska Native; Asian; Native Hawaiian and Other Pacific Islander; Some Other Race
218 P3_071N !!Total:!!Population of two or more races:!!Population of six races:!!White; Black or African American; American Indian and Alaska Native; Asian; Native Hawaiian and Other Pacific Islander; Some Other Race

Next, calculate new columns. Black : sum of all columns as any combination of groups including black Total : equal to P3_001N, total population 18 or over PctBlack : Percentage of people listed in Black CheckPct : sum of P3_003N and Black percentages, this value should not exceed 100%.

blockgroups_calc <- blockgroups |> 
  rowwise() |> 
  mutate(Black = sum(c_across(all_of(black_vars$name)))) |> 
  ungroup() |> 
  mutate(Total = P3_001N,
         PctBlack = Black / Total * 100,
         CheckPct = (Black + P3_003N) / Total * 100) |> 
  select(GEOID, Black, Total, PctBlack, CheckPct)

Save the results as blockgroups_calc.gpkg

st_write(blockgroups_calc, 
         here("data", "derived", "public", "blockgroups_calc.gpkg"),
         append=FALSE)
## Deleting layer `blockgroups_calc' using driver `GPKG'
## Writing layer `blockgroups_calc' to data source 
##   `/Users/jorredahl/Documents/GitHub/OR-Gerrymander-Alabama/data/derived/public/blockgroups_calc.gpkg' using driver `GPKG'
## Writing 3925 features with 5 fields and geometry type Multi Polygon.

Calculate District Variables

Calculate area and perimeter of districts

sf_use_s2(FALSE)
## Spherical geometry (s2) switched off
districts23 <- districts23 |>
  mutate(
    area = st_area(geom),
    perim = st_length(st_cast(st_cast(geom, "MULTIPOLYGON"), "MULTILINESTRING"))
  )

Area weighted re-aggregation to gather Percent Democrat in each district. Using precincts20 layer and assuming population stays constant across each voting precinct.

The intersection of voting precincts and districts was gathered, calculating the area of each precinct beforehand. Then an area weight was assigned to each fragment, where the weighted democrat vote count and weighted total vote count is counted. Using these values to re-aggregate by district, I summarize by district to get vote counts for each, getting percent democrat in each district.

For area weighted re-aggregation, there is a level of bias stemming from the assumption that pieces of the geometry becoming fragmented are spatially homogenous. While this experiment acknowleges that this is not possible, an area weighted re-aggregation at a small scale of voting precinct or census block group minimizes this bias.

sf_use_s2(FALSE)

precincts20 <- precincts20 |>
  st_transform(crs = 4269) |>
  mutate(
    precinctarea = st_area(geom),
    total_vote = G20PRERTRU + G20PREDBID + G20PRELJOR + G20PREOWRI
  )

precincts_int_districts <- st_intersection(precincts20, districts23) |>
  mutate(f_area = st_area(geom),
         aw = as.numeric(f_area / precinctarea),
         aw_dem = aw * G20PREDBID,
         aw_total = aw * total_vote)
## although coordinates are longitude/latitude, st_intersection assumes that they
## are planar
## Warning: attribute variables are assumed to be spatially constant throughout
## all geometries
districts_dem <- precincts_int_districts |>
  group_by(DISTRICT) |> 
  summarize(
    sumvote = sum(aw_total),
    sumdem = sum(aw_dem)
  ) |>
  mutate(
    pct_dem = as.numeric(sumdem / sumvote)
  )
## although coordinates are longitude/latitude, st_union assumes that they are
## planar
districts23 <- districts23 |>
  mutate(
    pct_dem = districts_dem$pct_dem[match(DISTRICT, districts_dem$DISTRICT)]
  )

Area weighted re-aggregation to gather Percent Black in each district. Using blockgroups_calc layer and assuming population stays constant across each voting precinct.

The intersection of census block groups and districts was gathered, calculating the area of each block group beforehand. Then an area weight was assigned to each fragment, where the black population and weighted total population is counted. Using these values to re-aggregate by district, I summarize by district to get demographics for each, getting percent black in each district.

sf_use_s2(FALSE)

blockgroups_calc <- blockgroups_calc |>
  st_transform(crs = 4269) |>
  mutate(
    blockarea = st_area(geom),
  )

blocks_int_districts <- st_intersection(blockgroups_calc, districts23) |>
  mutate(f_area = st_area(geom),
         aw = as.numeric(f_area / blockarea),
         aw_black = aw * Black,
         aw_total = aw * Total)
## although coordinates are longitude/latitude, st_intersection assumes that they
## are planar
## Warning: attribute variables are assumed to be spatially constant throughout
## all geometries
districts_black <- blocks_int_districts |>
  group_by(DISTRICT) |> 
  summarize(
    sumblack = sum(aw_black),
    sumpop = sum(aw_total)
  ) |>
  mutate(
    pct_black = as.numeric(sumblack / sumpop)
  )
## although coordinates are longitude/latitude, st_union assumes that they are
## planar
districts23 <- districts23 |>
  mutate(
    pct_black = districts_black$pct_black[match(DISTRICT, districts_black$DISTRICT)]
  )

Analysis

Calculate Compactness

Find compactness using the Polsby-Popper isoperimetric ratio

districts23 <- districts23 |>
  mutate(
    ppir_compact = round(as.numeric((4 * pi * area) / perim^2),2)
  )

Calculate the compactness of each district using the ratio of convex-hull area to actual area

sf_use_s2(FALSE)

districts23_hull <- districts23 |> st_convex_hull()

districts23_hull <- districts23_hull |>
  mutate(
    hullarea = st_area(geom)
  )

districts23 <- districts23 |>
  mutate(
    hullarea = districts23_hull$hullarea[match(DISTRICT, districts23_hull$DISTRICT)],
    ch_compact = round(as.numeric(area / hullarea),2)
  )

Calculate the compactness of each district using the ratio of minimum bounding circle area to actual area

sf_use_s2(FALSE)

districts23_mbcircle <- districts23 |> st_minimum_bounding_circle()

districts23_mbcircle <- districts23_mbcircle |>
  mutate(
    circlearea = st_area(geom)
  )

districts23 <- districts23 |>
  mutate(
    circlearea = districts23_mbcircle$circlearea[match(DISTRICT, districts23_mbcircle$DISTRICT)],
    mbc_compact = round(as.numeric(area / circlearea),2)
  )

Find if districts are abnormally black

Using the convex hulls of the districts, we can perform area weighted re-aggregation on each for the block groups to find the percent black of areas surrounding the hull.

sf_use_s2(FALSE)

blocks_int_districtshull <- st_intersection(blockgroups_calc, districts23_hull) |>
  mutate(f_area = st_area(geom),
         aw = as.numeric(f_area / hullarea),
         aw_black = aw * Black,
         aw_total = aw * Total)
## although coordinates are longitude/latitude, st_intersection assumes that they
## are planar
## Warning: attribute variables are assumed to be spatially constant throughout
## all geometries
districtshull_black <- blocks_int_districtshull |>
  group_by(DISTRICT) |> 
  summarize(
    sumblack = sum(aw_black),
    sumpop = sum(aw_total)
  ) |>
  mutate(
    pct_black = as.numeric(sumblack / sumpop)
  )
## although coordinates are longitude/latitude, st_union assumes that they are
## planar
districts23 <- districts23 |>
  mutate(
    hull_pct_black = districtshull_black$pct_black[match(DISTRICT, districtshull_black$DISTRICT)]
  )

Now, I can calculate a metric for how unusually concentrated a district is with black or non-black voters through abs(pct_black - hull_pct_black).

districts23 <- districts23 |>
  mutate(
    concentration_val = abs(pct_black - hull_pct_black)
  )

Results

Where are districts placed

Here are values for districts and their percentages of black voting-age people.

districts_labels <- tm_shape(districts23) +
  tm_polygons(fill_alpha = 0,
              col = "red") +
  tm_labels(text = "DISTRICT",
          col="red",
          bgcol = "white",
          bgcol_alpha = 0.5,
          on_surface = TRUE,
          just = c("center", "center")
          )
## 
## ── tmap v3 code detected ───────────────────────────────────────────────────────
## [v3->v4] `tm_text()`: migrate the layer options 'just' to 'options =
## opt_tm_text(<HERE>)'
## [tm_text()] Argument `on_surface` unknown.
black_blocks <- tm_shape(blockgroups_calc) + 
  tm_polygons(
    fill = "PctBlack",
    col_alpha = 0.2,
    lwd = 0.1,
    col = "grey90"
  )

black_blocks +
  districts_labels

Here are values for districts and their percentages of democrat votes on main-ticket candidates in the 2020 presidential election.

precincts20 <- precincts20 |>
  mutate(
    pct_dem = G20PREDBID / total_vote
  )

dem_precincts <- tm_shape(precincts20) + 
  tm_polygons(
    fill = "pct_dem",
    col_alpha = 0.2,
    lwd = 0.1,
    col = "grey90"
  )

dem_precincts +
  districts_labels

Here are values for districts on their score for black concentration. A higher number means black people or are more unusually concetrated in this district.

districts_representation <- tm_shape(districts23) + 
  tm_polygons(
    fill = "concentration_val",
    col_alpha = 0.2,
    lwd = 0.1,
    col = "grey90"
  )

districts_representation +
  districts_labels

Lastly here’s a table with each District, and each of the calculated metrics.

districts23 |> 
  st_drop_geometry() |> 
  select(DISTRICT, ppir_compact, ch_compact, mbc_compact, pct_dem,
         pct_black, concentration_val) |>
  kable()
DISTRICT ppir_compact ch_compact mbc_compact pct_dem pct_black concentration_val
1 0.15 0.59 0.19 0.2483071 0.1622268 0.0220864
2 0.14 0.60 0.20 0.5535744 0.4861677 0.1284007
3 0.35 0.81 0.47 0.2891686 0.2069872 0.0609139
4 0.20 0.61 0.32 0.1890682 0.0761442 0.0004853
5 0.40 0.91 0.32 0.3547979 0.1832956 0.0919157
6 0.20 0.71 0.46 0.3021344 0.1753046 0.0186869
7 0.21 0.71 0.47 0.6394241 0.5186634 0.1297934

How well does a compactness score show representation?

Each of the three compactness scores ranks the scores of each district differently. As a result, it’s unclear which score most aptly fits to the cacluated concentration_val statistic for representation. Here, each compactness metric is mapped against each other to find correlation with both each other and representation.

districts23_results_plot1 <- districts23 |> ggplot() +
  aes(x = ch_compact, y = ppir_compact) +
  geom_smooth(method="lm", col = "grey30") +
  geom_label(aes(label = DISTRICT, fill = pct_black)) +
  scale_fill_distiller(type = "div", palette = "PRGn") +
  theme(legend.position = "none")

districts23_results_plot2 <- districts23 |> ggplot() +
  aes(x = mbc_compact, y = ch_compact) +
  geom_smooth(method="lm", col = "grey30") +
  geom_label(aes(label = DISTRICT, fill = pct_black)) +
  scale_fill_distiller(type = "div", palette = "PRGn") +
  theme(legend.position = "none")

districts23_results_plot3 <- districts23 |> ggplot() +
  aes(x = ppir_compact, y = mbc_compact) +
  geom_smooth(method="lm", col = "grey30") +
  geom_label(aes(label = DISTRICT, fill = pct_black)) +
  scale_fill_distiller(type = "div", palette = "PRGn") +
  theme(legend.position = "none")

legend_plot <- districts23 |> ggplot(aes(x = ppir_compact, y = ch_compact, fill = concentration_val)) +
  geom_point() +
  scale_fill_distiller(type = "div", palette = "PRGn") +
  theme_minimal() +
  theme(legend.position = "right") 

legend <- patchwork::wrap_elements(get_legend(legend_plot)) 
## Warning in get_plot_component(plot, "guide-box"): Multiple components found;
## returning the first one. To return all, use `return_all = TRUE`.
(districts23_results_plot1 + districts23_results_plot2) / (districts23_results_plot3 + legend)
## `geom_smooth()` using formula = 'y ~ x'
## `geom_smooth()` using formula = 'y ~ x'
## `geom_smooth()` using formula = 'y ~ x'

Discussion

Here, you can see that districts 2 and 7 with the highest concentrations of black people over the age of 18 had the highest value of unusual concentration of black people, with an over 12% difference in black population percentage inside the district as compared to a district with the same shape as its convex hull.

Overall, compactness metrics did not show this representation value well. While compactness might not be a great indicator of whether a district was unfairly created, statistics like these can illuminate the advantages for some and disadvantages for others maps like these create.

Integrity Statement

I completed this preregistration to the best of my knowledge and that no other preregistration exists pertaining to the same hypotheses and research.

Acknowledgements

This report is based upon the template for Reproducible and Replicable Research in Human-Environment and Geographical Sciences, DOI:[10.17605/OSF.IO/W29MQ](https://doi.org/10.17605/OSF.IO/W29MQ)

References