Simplified Declination
An simplified formulation of the declination is sometimes used to avoid using normalizing constants and arctangents. This instead looks at the difference in means of wins over the seat proportions:
\[\textrm{Simplified Declination} = \frac{\bar{D}_{\textrm{voteshare}_D} - 0.5}{\textrm{S}_D(\textrm{voteshare}_D)} - \frac{0.5 - \bar{R}_{\textrm{voteshare}_D}}{1 - S_D(\textrm{voteshare}_D)}\]
with \(\bar{D}_{\textrm{voteshare}_D}\) as the average Democratic voteshare in Democratic-won districts and \(\bar{R}_{\textrm{voteshare}_D}\) as the average Democratic voteshare in Republican-won districts. \(S_D\) is the Democratic seat share.
As in the declination, a negative score indicates a pro-Democratic bias and positive score indicates a pro-Republican bias.
part_decl_simple(plans = nh$r_2020, shp = nh, dvote = ndv, rvote = nrv)
#> [1] -0.04681234 -0.04681234Mean-Median Score
The Mean Median Score is the difference between the mean Democratic vote share and the median Democratic district. This means that a negative score is in favor of Democrats and a positive score is in favor of Republicans. The score can range from -1 to 1.
Formally, this can be written as:
\[\textrm{Mean-Median} = \textrm{mean}(\textrm{voteshare}_D) - \textrm{median}(\textrm{voteshare}_D)\]
The Mean-Median score can be computed with:
Lopsided Wins
The Lopsided Wins score indicates if a party tends to win by a larger margin than the other party. A negative score indicates a pro-Democratic bias, whereas a positive score indicates a pro-Republican bias. Scores can range from -1 to 1.
Formally, this can be written as:
\[\textrm{Lopsided Wins} = \bar{D}_{\textrm{voteshare}_D} - (1 - \bar{R}_{\textrm{voteshare}_D})\]
with \(\bar{D}_{\textrm{voteshare}_D}\) as the average Democratic voteshare in Democratic-won districts and \(\bar{R}_{\textrm{voteshare}_D}\) as the average Democratic voteshare in Republican-won districts.
The Lopsided Wins score can be computed with:
The output of this can be used for a t-test.
Efficiency Gap
The Efficiency Gap describes the difference in wasted votes by a party. Wasted votes are those which don’t lead to a win (votes from the party lost that district) and those which weren’t necessary for a win (the excess votes above 50% + 1 vote by the party that won the district). These are normalized by the votes for the party, leading to a score between -1 and 1. A negative score indicates a pro-Democratic bias, whereas a positive score inidcates a pro-Republican bias.
Formally, this can be written as:
\[\textrm{Efficiency Gap} = \frac{W_D - W_R}{n_{votes}}\]
with \(W_R\) as the wasted votes by Republicans and \(W_D\) as the wasted votes by Democrats and \(n_{votes}\) are the total votes.
Wasted votes can be formalized as:
$$ W_D =
{ \[\begin{array}{ll} n_{\textrm{votes}_D} -( \textrm{floor}(\frac{n_\textrm{votes}}{2}) + 1) & \textrm{if voteshare}_D > 0.5 \\ n_{\textrm{votes}_D} & \textrm{if voteshare}_D < 0.5\\ \end{array}\]. $$
Republican wasted votes are defined analogously.
The Efficiency Gap can be computed with:
Efficiency Gap (Equal Population)
This is the efficiency gap, with the assumption that districts have equal voter turnout and population. The scale is the same as for the regular efficiency gap, where all scores are between -1 and 1, with negative implying pro-Democratic bias and positive implying pro-Republican bias.
Formally, this can be written as:
\[ \textrm{Efficiency Gap (Eq. Pop.)} = 2 * \textrm{voteshare}_D - S_D - 0.5\] with \(S_D\) as the seat share for Democrats.
The Efficiency Gap (Equal Population) can be computed with:
Tau Gap
The \(\tau\) gap is a generalization of the efficiency gap. When \(\tau = 1\), it is exactly twice the efficiency gap. The \(\tau\) parameter is a weighting parameter which describes how wasted votes should be weighted.
Formally, this can be written as:
\[ \tau\textrm{-Gap} = \frac{w_{D,\tau} - w_{R,\tau} }{w_{D,\tau} + w_{R,\tau}}\] where
\[w_{D, \tau} = \sum_{d = 1}^{n_\textrm{dists}} \frac{1}{\tau + 1} (2 * voteshare_D - 1)^{\tau + 1}) \]
and \(w_R\) is defined analogously.
The \(\tau\) Gap can be computed with:
part_tau_gap(plans = nh$r_2020, shp = nh, dvote = ndv, rvote = nrv)
#> [1] -0.009275831 -0.009275831The default is tau = 1.
Smoothed Seat Count Deviation
The Smoothed Seat Count Deviation offers an interpolation between seats when describing the number of seats won by a party. This can be added to the Democratic seats to describe the smoothed seat count.
As such, values are between 0 and 1, where 0 favors Republicans and 1 favors Democrats.
Formally, this is written as:
\[\textrm{SSCD} = \frac{0.5 - \min(D_{\textrm{voteshare}_D} )}{(1 - \max(R_{\textrm{voteshare}_D} )) - (1 - \min(D_{\textrm{voteshare}_D}))}\] where \(D_\textrm{voteshare}_D\) is the Democratic vote share in Democratic-won districts and \(R_\textrm{voteshare}_D\) is the Democratic vote share in Republican-won districts.
The Smoothed Seat Count Deviation can be computed with:
Ranked Marginal Deviation
The Ranked Marginal Deviation is also known as the Gerrymandering Index. Given a set of plans, we can first order them by Democratic percent in each seat. The Ranked Marginal Deviation is the square root of the sum of the squared distances between the average Democratic percent in each seat and the Democratic percent in a given plan.
Formally, this can be written as:
\[\textrm{Ranked Marginal Deviation} = \sum_{d = 1}^{n_\textrm{dists}} \textrm{voteshare}^s_{D}(d) - \overline{\textrm{voteshare}^s_{D}(d)}\]
where \(\textrm{voteshare}^s_{D}(d)\) is the \(s\)orted Democratic voteshare in district \(d\) and is the average of those across plans.
As such, it is sensitive to the plans inputted. Here we use the
nh_m data described in the “Using
redistmetrics” vignette, which contains 2 reference plans
and 50 simulated plans.
data(nh_m)
part_rmd(plans = nh_m, shp = nh, dvote = ndv, rvote = nrv)
#> [1] 0.428306762 0.428306762 24.219447110 24.219447110 2.552987343
#> [6] 2.552987343 1.807712702 1.807712702 1.430841789 1.430841789
#> [11] 0.014897340 0.014897340 0.126702397 0.126702397 0.319358751
#> [16] 0.319358751 0.243484554 0.243484554 0.372941780 0.372941780
#> [21] 0.361636666 0.361636666 0.148703808 0.148703808 1.430841789
#> [26] 1.430841789 0.243484554 0.243484554 1.327059351 1.327059351
#> [31] 0.014897340 0.014897340 0.484383034 0.484383034 1.807712702
#> [36] 1.807712702 0.509814359 0.509814359 0.837801241 0.837801241
#> [41] 0.816016501 0.816016501 0.484383034 0.484383034 4.297252277
#> [46] 4.297252277 0.350898444 0.350898444 1.094269004 1.094269004
#> [51] 0.509814359 0.509814359 0.006160546 0.006160546 0.816016501
#> [56] 0.816016501 0.509814359 0.509814359 0.412033985 0.412033985
#> [61] 0.210313258 0.210313258 0.407710508 0.407710508 0.982443775
#> [66] 0.982443775 0.126702397 0.126702397 0.432856285 0.432856285
#> [71] 2.715581106 2.715581106 0.014897340 0.014897340 1.281534954
#> [76] 1.281534954 0.574665802 0.574665802 0.484383034 0.484383034
#> [81] 0.406929519 0.406929519 0.008614078 0.008614078 0.008614078
#> [86] 0.008614078 0.361636666 0.361636666 0.350898444 0.350898444
#> [91] 0.465927075 0.465927075 1.316510943 1.316510943 1.316510943
#> [96] 1.316510943 0.465927075 0.465927075 0.803319255 0.803319255
#> [101] 2.610432184 2.610432184 1.281534954 1.281534954This score is intended for comparison, which we can display as a plot such as:
library(ggplot2)
dplyr::tibble(rmd = part_rmd(plans = nh_m, shp = nh, dvote = ndv, rvote = nrv),
plan = rep(colnames(nh_m), each = 2)) %>%
ggplot() +
geom_histogram(aes(x = rmd, fill = plan), bins = 50) +
theme_bw()
In this (small) set of simulations, the Democratic 2020 proposal is more similar to the simulated plans than the Republican 2020 proposal.