Partisan metrics: some notes

Some notes on mean-median & partisan bias scores, and building seats-votes curves using lower house state legislative election results in the USA from 1971-2018. Also a place to organize some different non-geographical approaches to identifying partisan gerrymandering – eg – Warrington (2019); Warrington (2018); Gelman and King (1994); Katz, King, and Rosenblatt (2020).

State legislative election results

State House election results are made available here by Princeton University folks – which is apart of a larger ecosystem of tools/resources for investigating gerrymandering. Results are limited to lower houses for states with bicameral state legislatures.

library(tidyverse)
election_results <- read.csv(url(git_url)) %>% 
  janitor::clean_names() %>%
  mutate(d_voteshare = round(d_voteshare, 3))

A sample of the data set is detailed below. Results are presented year, state, and state house district; the d_voteshare column specifies the vote share received by the Democratic candidate.

election_results %>%
  filter(state == 'CA', year == 1980) %>%
  slice(1:5) %>%
  select(-incumbent) %>%
  knitr::kable()
state year district dem_votes gop_votes d_voteshare party
CA 1980 1 0 114547 0.000 R
CA 1980 2 81884 36504 0.692 D
CA 1980 3 55339 75547 0.423 R
CA 1980 4 76013 35894 0.679 D
CA 1980 5 52121 75998 0.407 R

A super simple imputation method: Per approach described in Gelman and King (1994), winning parties of uncontested elections are re-assigned a vote share of 0.75, and losing parties 0.25.

election_results1 <- election_results %>%
  mutate(d_voteshare = ifelse(d_voteshare == 1, 0.75, d_voteshare),
         d_voteshare = ifelse(d_voteshare == 0, 0.25, d_voteshare))

Summarizing election results

Next, we summarize election results per legislature. Summary stats include:

  • the number of seats in legislature,
  • the number/proportion of seats won by Democrats,
  • the average vote share received by Democratic candidates; and
  • the median Democratic vote share.

Important to emphasize here is that the v_mean value specifies the average Democratic vote share across individual state house races in a given year, and not the aggregate statewide house results.

full_summary <- election_results1 %>%
  group_by(state, year) %>%
  mutate(dseat = ifelse(dem_votes > gop_votes, 1, 0),
         rseat = ifelse(dem_votes < gop_votes, 1, 0),
         d_above = ifelse(d_voteshare > mean(d_voteshare),
                          1, 0)) %>%
  
  summarize(district_n = n(),
            d_seats = sum(dseat),
            r_seats = sum(rseat),
            v_mean = mean(d_voteshare),
            v_median = median(d_voteshare),
            d_above = sum(d_above)) %>%
  mutate(seat_bar = d_seats/district_n) %>%
  ungroup()

Election results for Colorado during the 2010s are presented below. So, Dems took the majority - and then some - in the Colorado State House during the previous decade.

full_summary %>%
  filter(state == 'CO',
         year > 2008) %>%
  select(-r_seats) %>% 
  mutate(across(c(v_mean, v_median, seat_bar), ~round(., 2))) %>%
  knitr::kable()
state year district_n d_seats v_mean v_median d_above seat_bar
CO 2010 65 32 0.48 0.50 35 0.49
CO 2012 65 37 0.51 0.54 37 0.57
CO 2014 65 34 0.48 0.51 37 0.52
CO 2016 65 37 0.50 0.54 37 0.57
CO 2018 65 41 0.55 0.59 36 0.63

The plots below illustrate the shifting partisan balance for a selection of state houses since 1972.

south <- c('CO', 'FL', 'AL', 'TX', 
           'AR', 'TN', 'OK', 'KY')

full_summary %>%
  filter(state %in% south) %>%
  ggplot() +
  geom_line(aes(x = year, 
                y = d_seats/district_n * 100), 
            color = '#437193', size = 1) +
  geom_line(aes(x = year, 
                y = r_seats/district_n * 100), 
            color = '#ae4952', size = 1) +
  facet_wrap(~state, ncol = 4) +
  theme_minimal() +
  theme(axis.text.x = element_text(angle = 90, hjust = 1)) +
  scale_x_continuous(breaks=seq(1972, 2018, 8)) +
  ggtitle('State house partisan trends')

Historical vote distributions

details1 <- election_results1 %>%
  #filter(party %in% c('D', 'R')) %>% 
  
  left_join(full_summary) %>%
  group_by(state, year) %>%
  
  mutate(swing = 0.5 + v_mean - d_voteshare,
         rank = rank(d_voteshare, ties.method = 'first'),
         seat_share = rank/n(),
         seat_share = 1 - seat_share) %>%
  
  ## still not correct exactly --
  mutate(swing = ifelse(seat_share == seat_bar, v_mean, swing)) %>%
  arrange(seat_share) %>%
  ungroup()

Vote distributions for election results in Wisconsin since 1972 are illustrated below. Districts have been sorted in increasing order of Democratic vote share.

details1 %>% 
  filter(state == 'WI') %>%
  
  ggplot() +
  geom_point(aes(x = factor(rank), 
                 y = d_voteshare),
             color = 'steelblue',
             size = .5) + 
  
  geom_hline(yintercept = 0.5, lty = 3) +
  facet_wrap(~year, ncol = 6) +
  theme_minimal() +
  theme(axis.text.x=element_blank()) +
  xlab('Districts ordered from least to most Democratic') +
  ylab('Percentage of votes for a Democrat') +
  labs(title = 'Wisconsin State House election results',
       subtitle = 'Democratic vote share by district, ranked')

Seats-votes curves

There are a host of metrics that aim to capture partisan asymmetries in vote distributions (see Warrington 2019 for a comparison). Here, we focus on mean-median scores and partisan bias scores, mainly because they are closely tied to the seats-votes curve.

The mean-median score is the difference between a party’s median vote share and its mean vote share – divergence between these two values suggests a vote distribution that is skewed in favor of a particular party. In contrast, the partisan bias score is the difference between (1) a party’s actual seat share and (2) that party’s hypothetical seat share if it garnered 50% of the statewide vote share. Both metrics are calculated below:

full_summary1 <- full_summary %>%
  mutate(mm = 0.5 + v_mean - v_median,
         pb = (d_above - 1) /district_n) ## 

As an example, we consider results from the Wisconsin State House in 2018. Again, results are presented from the perspective of Democrats.

xmm <- full_summary1 %>% filter(state == 'WI', year == 2018) 
district_n d_seats v_mean v_median d_above seat_bar mm pb
99 36 0.51 0.44 35 0.36 0.57 0.34

Per plot below, the green bar specifies the mean-median value; the red bar specifies the partisan bias score. So, if a seats-votes curve populates quadrant I, Democrats are over-represented in the legislature based on their statewide vote share; quadrant III, under-represented. The star specifies actual election results. Extreme values in either quadrant are symptomatic of gerrymandering.

see <- details1 %>%
  filter(state == 'WI', year == 2018)  

see %>%
  ggplot() +
  geom_hline(yintercept = .50) +
  geom_vline(xintercept = .50) +
  geom_step(aes(x = swing, 
                y = seat_share,
                color = factor(year)),
            size = 1) + 
  geom_point(aes(x = v_mean, 
                 y = seat_bar),
             pch="\u2605",
             size = 4) +
  
  annotate('segment', 
           x = 0.5, 
           y = xmm$pb, 
           xend = 0.5, 
           yend = 0.5,
           color = '#913a40', size = 3, alpha = .5) +
  annotate('segment', 
           x = xmm$mm, 
           y = 0.5, 
           xend = 0.5, 
           yend = 0.5,
           color = '#3c811a', size = 3, alpha = .5) +
  
  theme_minimal() +
  theme(legend.position = 'right')+
  ggthemes::scale_color_stata() +
  coord_equal(xlim = c(0.3, 0.7),
              ylim = c(0.3, 0.7)) + 
  ggtitle('Seate-Votes Curve: Wisconsin 2018') +
  ylab('Democratic share of lower house seats') +
  xlab('Democratic vote share')

A historical example from the state of Colorado –

Resources

Gelman, Andrew, and Gary King. 1994. “A Unified Method of Evaluating Electoral Systems and Redistricting Plans.” American Journal of Political Science, 514–54.

Katz, Jonathan N, Gary King, and Elizabeth Rosenblatt. 2020. “Theoretical Foundations and Empirical Evaluations of Partisan Fairness in District-Based Democracies.” American Political Science Review 114 (1): 164–78.

Warrington, Gregory S. 2018. “Quantifying Gerrymandering Using the Vote Distribution.” Election Law Journal 17 (1): 39–57.

———. 2019. “A Comparison of Partisan-Gerrymandering Measures.” Election Law Journal: Rules, Politics, and Policy 18 (3): 262–81.

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