During oral argument in Rucho v. Common Cause, the North Carolina gerrymandering case, the Supreme Court and the attorneys for the parties considered a variety of criteria to identify what Justice Stephen Breyer called “real outliers” in terms of election results. In the first article of this series I considered his half-the-vote/one-third-the seats criterion. In the last posting I considered the notion of measuring deviations from some predicted baseline result. In that article I proposed a formula for estimating the baseline based on historical voting data:

Expected % Democratic Seats = 2 X (% Democratic Two-Party Votes) – 50

This formula provides a simple, yet historically accurate baseline for estimating the share of seats we should expect the Democrats to win given their share of the Congressional vote statewide. (I should note that this formula is entirely symmetric. We could use the Republican vote and seat shares and get the identical result.) Armed with a method for determining the baseline prediction, I turn now to a method for identifying deviant electoral results.

Measuring the Deviation

How big a deviation from that baseline should be considered “significant” depends on both statistical and legal/constitutional criteria. I will only be talking about “significance” in the statistical sense. As we’ll see, the size of the deviation you are willing to tolerate depends on the proportion of outcomes you consider to be possibly unconstitutional.  In that sense, Justice Potter Stewart’s famous comment about identifying pornography, “I know it when I see it,” applies to gerrymandering just as well.

In the discussion about proportionality, plaintiff’s attorney Paul Stewart suggested and dismissed a “one standard deviation” away from some baseline criterion for gerrymandering. I have dealt with his objection concerning estimating a baseline result, but just one standard deviation is much too low a bar. As this graph shows, about 32% of elections should fall outside the one-standard-deviation criterion, many too many to qualify for judicial review. Statisticians often use two standard deviations as a minimal criterion for “statistical significance.” That would subject about five percent of the elections to additional scrutiny. Justice Breyer’s criterion works out to about one election in a thousand, which corresponds to a standard deviation difference of about 2.5.

Now it turns out the regression method also generates an estimate of the “standard deviation” of the predicted values. This quantity is called the “standard error,” and for the regression using state-years as the unit of analysis, the estimated standard error for the percent of seats won is 10.2.  So, using two standard errors as a minimum criterion, we should look for results where the difference between the actual number of seats won, and the prediction from the formula above, is at least 20 percent.  Here are the elections held since 2010 where the actual outcome differs from the predicted value by a least 20 percent. The “standardized deviation” column measures the absolute value of the quantity (Actual – Predicted)/(Standard Error).  The larger the value the further the election deviated from the prediction. Using the absolute value treats both parties symmetrically.

All three elections identified by the “Breyer criterion” also appear in this list. However there are a number of elections that fail his criterion, but where the actual seat outcome differs from the predicted value by at least two standard errors.  Connecticut persistently sent five Democrats to Congress since 2010 when the vote suggests there should have been at least one Republican in the delegation. Connecticut would not be identified by Breyer’s criterion, but a reasonable observer would conclude that state’s Congressional district lines appear to have been gerrymandered in the Democrats’ favor. Democrats also got “too many” seats in Maryland in 2014, but that pattern did not recur in other elections since the 2010 Census. Similar “one-offs” like VA12, NJ18, and MI12 might also be attributed to chance rather than systematic discrimination via gerrymandering.

Most of the other elections in the list show an excess number of Republicans winning House seats given the statewide vote for Democrats. Both North Carolina and Pennsylvania appear twice as does Ohio, whose map was just thrown out by a Federal court.

By either Breyer’s criterion or by measuring deviations from a predicted baseline, the map created by the North Carolina legislature qualifies as gerrymandered. Ohio and Connecticut also deserve judicial scrutiny.

In my last post I considered what I called the “Breyer criterion” for identifying partisan gerrymandering — a party winning half the vote in a state receives only a third of the seats.  That criterion identified just seven races out of the nearly eight hundred I examined, or just 0.9 percent of the state-level elections to Congress where candidates of both major parties stood. Breyer proposed his criterion to identify “real outliers,” elections that are “really extraordinary.” A one-in-a-thousand criterion probably fits that definition.

However the Court also discussed the general concept of how to measure “proportionality” between seats and votes. The attorney for the plaintiffs, Paul Clement, brought up the notion of a “one standard deviation from proportional representation” criterion mostly as a straw horse. Leaving aside his use of “proportional representation,” which as the oral argument shows is fraught with constitutional issues, Clement then claimed that it is impossible to know what the correct baseline should be from which to measure seat outcomes.

So I think the fundamental problem is there is no one standard deviation from proportional representation clause in the Constitution. And, indeed, you can’t talk even generally about outliers or extremity unless you know what it is you’re deviating from.

Clement’s argument ignores decades of political science research into the relationship between votes won and seats awarded.  Studies dating back to at least 1948 have theorized about and examined empirically the relationship between seats and votes.

Measuring the Baseline

I’ve written a number of times about the relationship between votes won and seats awarded in “first-past-the-post” or “plurality” electoral systems like ours.  These types of electoral systems routinely award the majority winner of the vote a disproportionately greater share of seats. Here is a simple example, using national electoral results for Congress.

The dark blue line represents the “best-fit” relationship between the percent of votes won by the Democrats in each election year and the percent of House seats the party won using simple “ordinary least squares” regression. The historical relationship is substantially steeper than the thin line in the chart representing parity, or when a party’s share of seats equals its share of votes.¹

Using simple regression the equation that best describes this relationship is, in round numbers,²

% Seats Democrat = 2 X (% Votes Democrat) – 50

So, for instance, in a year when the Democrats win 55 percent of the vote, they should receive on average (2 X 55) – 50 = 60 percent of the seats.

Since gerrymanders take place at the state level, data from national elections do not provide the correct basis for determining whether a particular state’s election deviated “too far” from some predicted baseline. To develop such a baseline for Congressional elections I turn again to the MIT database of Congressional races I used in the preceding blog post.  Here is the relationship between votes and seats for state-year combinations. Each point represents a general election in a given state in a particular year, like Alabama in 1976.

A number of races resulted in one party or the other winning all the seats. These unanimous outcomes pose mathematical problems for our method, so I excluded those 84 races in the calculation of the slope and intercept for the regression line in the chart.

(The horizontal lines come from states with small numbers of districts where the number of outcomes is mathematically restricted. For instance, a state with four districts will often return a 3-1 result for one party. That leads to clustering at values of 25 or 75 percent.)

Using state-level election results gives us a model that is numerically quite similar to the simple method based on election years above:

% Seats Democrat = 2.3 X (% Votes Democrat) – 66

Here the slope of the line is slightly steeper than two and the intercept slightly more negative. In practice, though, the difference between these results and predictions using the simpler model from national-level data are negligible. The lines are so close that I could not represent them both on the chart.

Given the convergence between these two sets of estimates, I propose that

The best “baseline” estimate for the division of seats given the division of the vote in state-level Congressional elections is

% Seats Democrat = 2 X (% Votes Democrat) – 50

That formula uses simple numbers like two and fifty and produces results nearly identical to those using the estimated regression coefficients of 2.3 and -66.

The regression method also produces a measure of the “standard deviation” of actual outcomes around the predicted values. I use that quantity in the next post to identify potential gerrymanders using the deviation from proportionality method.

¹The results for the last two Democratic off-year House victories, retaking the chamber in 2006 and 2018, both fall on this parity line. Given the historical relationship, the Democrats did not receive the usual reward in the House for their victories in the popular vote. The elections in 2012 and 2018 also show significant negative effects for Democrats.

²Complete results for both models here.

On March 26th the Supreme Court heard oral argument in Rucho v. Common Cause. The case concerns whether North Carolina’s post-2010 electoral map so disadvantages Democratic candidates that it should be ruled unconstitutional. This case raises many Constitutional and legal issues that fall outside the purview of this blog; for instance, whether the Republican-controlled North Carolina legislature showed the intent to discriminate against Democrats in their choice of map. However some of the issues raised during oral argument lend themselves to empirical examination.

A persistent concern during oral argument was whether “proportionality” should be used as a Constitutional standard to determine if a particular electoral outcome might be ruled unconstitutional.  In one of these discussions Justice Stephen Breyer proposed that “when a party wins a majority of the votes in a state, … but the other party gets more than two-thirds of the seats” the result could be declared unconstitutional.

How frequently might Justice Breyer’s criterion apply to actual state-level results comparing votes cast for Congress and the proportion of seats awarded? The Court has an incentive to establish a highly-restrictive criterion to deter future filings by state parties hoping to overturn an unfortunate result. How restrictive is the Breyer criterion? How often might we see electoral results flagged as potentially unconstitutional by the workings of this rule?

What Elections to Analyze

To address these questions, I begin with an invaluable dataset compiled by the MIT Election Lab. It comprises election returns for all candidates who ran for Congress between 1976 and 2018. Using these candidate records as a basis, I created a new aggregated dataset containing results by party for each combination of state and election year.

In the process I eliminated a number of records from consideration. First, because it is impossible to gerrymander a state with just one Congressional district, I excluded any state-year combinations when the state was apportioned into a single district. Examples include Alaska and Wyoming throughout the 1976-2018 period, and states like Montana and Nevada in the years when they had but one district.

I further eliminated states with just two Congressional districts. In those cases an election would fit the criterion if one party won over half the vote and lost both seats. However that outcome would occur by random chance a quarter of the time if both seats had even odds of going to either party.  Courts would likely not be willing to rule a particular seat distribution was unconstitutional when the result could have happened by chance a quarter of the time. As a result I also removed state-years when the state was apportioned only two seats.

Even this set of races needs further refinement to use as a basis to examine Breyer’s criterion. The canonical notion of a two-party race between a Democrat and a Republican dissolves once we look at the data.  Most races include minor candidates and not every seat has both a Democratic and a Republican contender.  Many seats were left uncontested over this period by one or the other major party, especially in the South.  And with the introduction of “top-two” voting in California and Washington, general elections can pit two Democrats or two Republicans against one another.

So I further limited the sample by selecting only Congressional elections with both a Democratic and a Republican contender. That left a total of 7,701 eligible races which I then aggregated to the level of state-years, e.g., Alabama in 1976. Some state-year combinations then had fewer than three contested races; those observations were also excluded. That left me with a total of 799 state-years for the analysis to follow.

Justice Breyer’s Criterion

So in this sample of nearly eight hundred Congressional outcomes, how often do we find the particularly egregious combination where a party won at least half the Congressional vote in a state but was awarded fewer than a third of the seats.

In practice Breyer’s criterion turns out to be highly restrictive.  Of the 799 Congressional elections that qualified for my sample, only seven (0.9 percent) would have fit his rule.  Moreover, only four seats were contested in the three Alabama races and the one in South Carolina. Assuming even odds of each seat electing a Democrat, but a Democratic majority overall, the chance of getting an outcome with at least three Republican seats is 1/8.¹ Intuitively that seems too low a bar for declaring a particular result unconstitutional.

Of more interest is that three of the seven Alabama seats, and two seats in the South Carolina race, were uncontested. The totals for these states represent the votes cast and seats awarded in the contested districts. Leaving seats uncontested may itself be an indicator of gerrymandering, If maps are too distorted, it may make little sense for a party to invest resources in races where their opponents are certain to be victorious.

Pennsylvania and North Carolina are another story entirely though.

Breyer’s criterion flags three elections in those states. all of which took place after the 2010 Census. Since then both states have become poster children for gerrymandering. The Pennsylvania map that took effect in 2012 awarded Republicans fully thirteen of the state’s eighteen seats while the Democrats won the popular vote statewide by a small margin. The Pennsylvania State Supreme Court ruled in January, 2018, that the Congressional map was so unfair that it violated the state’s own Constitution. The Court threw out the map and later that month commissioned Stanford Law School professor Nate Persily to draw a new one.  The 2018 election using the redrawn district lines resulted in a 9-9 tie, compared to the 13-5 advantage Republicans had maintained since 2010.

North Carolina is, of course, the state at issue in Rucho v Common Cause, so it is appropriate that it should be flagged here as well. Twice since the 2010 Census have the Democrats won a small majority of the popular vote, but were awarded only three or four of the state’s thirteen Congressional seats. So if Justice Breyer wanted to establish a criterion that would pick out the most egregious partisan gerrymanders, his one-half the vote/one-third the seats rule seems to fit the requirement.

Justice Breyer’s rule was not the only criterion discussed in oral arguments that day. Both plaintiff’s attorney Paul Clement and Justice Neal Gorsuch discussed a measure based on the difference between a state’s actual seat distribution and some measure of what its “proportionate” share might be. I turn to that subject in my next posting.

¹Imagine a state with four districts. In three of them the Democrats and Republicans tie. In the fourth seat the Democrats win by one. That gives them a one-vote majority in the popular vote and one seat. If we flip a coin for each of the three tied districts, a result with three Republicans occurs one time in eight. I  thank my friend Jim Stodder for making me rethink the calculation of this probablity.