Advanced Measures of Bias & Responsiveness
When you click on the Advanced tab in DRA 2020, two sections show many advanced measures of bias & responsiveness.
Methodology
To estimate the partisan characteristics of a map, we use:
- A composite of statewide elections, and
- Fractional seat probabilities instead of all-or-nothing accounting, except as noted.
Bias Measures
This section reports four clusters of metrics.
The first cluster includes straightforward linear measures of bias calculated at the statewide vote share:
- Proportional representation (PR) — This is the simple deviation from proportionality — the difference between the likely share of fractional seats won by Democrats and their statewide vote share.⁹
- Efficiency gap (EG) — This is calculated by taking one party’s total wasted votes in an election, subtracting the other party’s total wasted votes, and dividing by the total number of votes cast. It measures the extent to which district lines crack and pack one party’s voters more than the other party’s voters.⁷ Note: When the winner’s bonus (R) is between one (proportionality) and two (EG=0) inclusive, you may wish to interpret the plan as having an acceptable level of bias according to EG.
- Gamma (γ) — This is a new measure of bias that combines seats and responsiveness.⁵
The Analyze view also shows disproportionality. In contrast to PR here, that shows the disproportionality with respect to the Democratic seats closest to proportional not the statewide Democratic vote share like PR. See Two Notions of “Fair” for some background on the difference between that and these other measures.
The second cluster includes measures of bias that depend on the seats–votes curve away from that statewide vote share:
- Seats bias (αₛ) — The seats bias at 50% Democratic vote share. This is the fraction of seats less than (or greater than) half that Democrats win with half the votes. Alternatively, you can think of this as the difference in seats won by the two parties when the vote is evenly split.¹
- Votes bias (αᵥ) — The votes bias at 50% Democratic seat share. This is the fraction of votes more than (or less than) half that Democrats need to win half the seats.²
- Partisan bias (β) — The seats bias at the statewide Democratic vote share, not 50%. IOW, this estimates the difference in seats won by the two parties at the statewide Democratic vote share.⁸
- Global symmetry (GS) — This measures a combination of seats and votes bias.⁴
- Partisan bias rating — A combined rating of seats bias & votes bias.¹⁷
The third section shows measures of partisan gerrymandering via packing & cracking:
- Declination (δ) — This is the value of the declination angle (in degrees) calculated using fractional seats and votes.³ Declination measures the packing and cracking in a plan. When shown graphically, it visually illustrates “walls” of safe seats that are characteristic of unfair maps. The declination lines are overlaid on the rank-vote graph at the top. Declination is not defined for states with fewer than five districts or when one party might “sweep” all the seats (like Massachusetts).
- Mean–median (mM) — This is the mean Democratic vote share by district minus the median Democratic vote share. When the mean and the median diverge significantly, the district distribution is skewed in favor of one party and against its opponent.¹⁰
- Turnout bias (TO) — This measures bias in voter turnout between the parties as the difference between the statewide Democratic vote share and the average their average district vote share.¹¹
- Lopsided outcomes (LO) — This measures discriminatory packing. The ideal is that the average excess vote share for districts won by the two parties is the same. You can gauge this using the rank-vote graph at the top as the difference between the average vote shares for the Democratic and Republican wins.¹² LO is not defined for states when one party might “sweep” all the seats (like Massachusetts).
The last group of metrics helps you understand the contribution of political geography to overall bias:
- Proportional seats — This is the fractional number of Democratic seats that corresponds to the statewide Democratic vote share.
- Geographic seats — This is the estimated number of fractional Democratic seats implied by political geography, i.e., the likely number of Democratic seats if every county were contested like a district & the results weighted by the county’s total population.¹³ Note: In some states that have counties with lots of people — like Maricopa County in Arizona — these ‘big’ counties are disaggregated into constituent jurisdictions. When that happens, ‘county’ will say ‘jurisdiction’ instead.
- Geographic bias — This measures the implicit bias due to political geography, by subtracting geographic seats from the proportional seats and dividing by the number of districts.
- Map seats — This is the estimated number of fractional Democratic seats, given the district lines for the map.
- Boundary bias — This measures the explicit bias due to where the district lines are drawn in the map, by subtracting map seats from geographic seats and dividing by the number of districts.
Sometimes the fractional Democratic seats implied by precinct political geography is also reported in the Notes section:
- This can occur when whole counties have been used to calculate geographic seats or when ‘big’ counties have been disaggregated into sub-county jurisdictions.
- In competitive states where the most populous counties-or-jurisdictions still contain more than twice the population of an ideal district, the counties-or-jurisdictions political geography may hide a different partisan majority in subunits (cities, towns, or groups of precincts) worth more than one seat. If so, the measure of geographic seats would take a very different value if it were computed it with respect to (further) sub-county units. If these most populous counties-or-jurisdictions contain a large fraction of the total population of the state, the boundary bias measure would also be very different if the boundaries considered were those of smaller geographic units.
- So, in competitive states in which the most populous counties-or-jurisdictions contains more than twice the population of an ideal district, if the difference between the value of geographic seats computed with respect to counties-or-jurisdictions geography and the value computed with respect to precinct geography is large, we add this note.
For more background on political geography and the notions of implicit & explicit gerrymandering, see Compact Districts Aren’t Fair.
Responsiveness Metrics
The section section, shown below, reports several measures of responsiveness:
- Responsiveness (ρ) — This is the slope of the seats-votes curve at the statewide Democratic vote share.¹⁴
- Responsive districts — This the likely number of responsive districts, using fractional seat probabilities.¹⁵
- Overall responsiveness (R) — This is an overall measure of responsiveness which you can think of as a winner’s bonus.¹⁶ R is not defined when the statewide Democratic vote share is approximately 50%.
Note: We use a composite of multiple elections to compute all these metrics, rather than any single election. You can choose a different election dataset, using the Data Selector.
Update: We reorganized this post and left the many footnotes as is, so some appear out of order.
Footnotes
You can find more information about each of the advanced metrics in these resources.
- See What criteria should be used for redistricting reform? (Nagle, 2019)
- Ditto
- See Introduction to the declination function for gerrymanders (Warrington, 2018)
- See On measuring two-party partisan bias in unbalanced states (Nagle & Ramsay, 2021)
- Ditto
- Ditto
- See Efficiency Gap (PlanScore.org)
- See Theoretical Foundations and Empirical Evaluations of Partisan Fairness in District-Based Democracies (Katz, King, and Rosenblatt, 2020)
- See Nagle 2019
- See Wondering how the tests work? (Princeton Gerrymandering Project)
- The Arithmetic of Electoral Bias, with Applications to U.S. House Elections (McDonald, 2009)
- See Princeton Gerrymandering Project
- See A Measure of Partisan Advantage in Redistricting (Eguia, 2021)
- See Nagle 2019
- The details are described in this white paper
- See Nagle & Ramsay
- See How We Rate Partisan Bias
Acknowledgements
I would like to thank John Nagle at Carnegie Mellon University and Sam Wang with the Princeton Gerrymandering Project for their sustained and in-depth conversations about partisan analytics.
