Technical Appendix: Seats and Votes in the 2018 Election

I am extending the simple model I presented in 2012 relating the proportion of House seats won by Democrats against that party’s share of the (two-party) national popular vote for Congressional candidates.  It uses dummy variables to represent each redistricting period (e.g., the 2000 Census was used to redistrict elections from 2002-2010), and a slope change that starts with the Republican House victory of 1994.

To review, the earlier model showed this pattern of partisan advantage for elections conducted since 1940:

The results for the 2010 redistricting were based solely on the 2012 election.  As we’ll see in a moment, adding in 2014 and 2016 only made that result more robust.

As I argued earlier, not all of this trend results from partisan gerrymandering.  Americans have sorted themselves geographically over the past half-century with Democrats representing seats from urban areas and Republicans holding seats from suburban and rural areas.  As partisans self-segregate, the number of “safe” seats rises, and electoral competitiveness declines.

Partisan self-segregation also makes gerrymandering easier.  Opponents can be “packed” into districts where they make up a super-majority.  House Minority Leader Nancy Pelosi routinely wins 80 percent or more of the voters in her tiny, but densely populated San Francisco district.  Many of these seats are held by minority Members of Congress because of our national policy of encouraging “majority-minority” districts.   These efforts were well-motivated as a response to racist gerrymandering that would “crack” minority areas and distribute pieces of them in a number of majority-white districts.  Unfortunately for the Democrats these policies have meant that too many of the party’s voters live in heavily-Democratic districts.

Here is the result of an ordinary least squares regression for the share of House seats won by the Democrats in elections since 1940:

If I plot the predicted and actual values for Democratic seats won, the model unsurprisingly follows the historical pattern quite closely:

The Democrats routinely won around sixty percent of House seats between 1940 and 1992.  Since then they have only held a majority in the House twice, in 2006 and 2008.  Notice, too, that both the actual and predicted values for 1994 to the present show much less variance than the earlier decades.  The results above show that the “swing ratio” relating seats and votes has become much smaller falling from 1.92 before 1994 to 1.33 (=-0.59+1.92) since.  A smaller swing ratio indicates that House elections have become less competitive since Bill Clinton was elected President in 1992.  Changes in vote shares are still amplified in seat outcomes, as they are in all first-past-the-post electoral systems like ours, but the effect has been diminished because of the increase in the number of safe seats on both sides of the aisle.

We can use this model to estimate the share of votes required in order for the Democrats to win a majority in the House.  This chart shows the predicted relationships between seats and votes for two historical periods, one through the election of Bill Clinton in 1992, and the other beginning with the Republican victory in the House election of 1994 under Newt Gingrich and his “Contract with America.”

The slope in the latter period is substantially flatter than in the earlier period, meaning that Congressional elections have become somewhat less competitive since 1992.  Changes in vote shares have a smaller effect on changes in seat shares than they did before 1994.

Finally, the third line represents an estimate for the relationship in 2018, using the 1994-2016 slope and only the post-2010 intercept shift.  The chart shows that for the Democrats to win half the seats in 2018 they will need to garner a bit over 53 percent of the two-party popular vote for the House.



*The intercepts in these charts represent weighted averages of the adjustments for the various Census years. For instance, the 1994-2016 line includes the coefficients for the 1990, 2000, and 2010 Census weighted by the number of elections in each decade. So in this case the 1990 and 2000 adjustments would have weights of five, and the 2010 adjustment a weight of three. The 2018 line applies only the 2010 redistricting adjustment.

Most Vulnerable Republicans Voting against the Shutdown Resolution


This table presents those Republicans who voted against H.R. 2775, the resolution to end the government shutdown and extend the debt ceiling until February 7, 2014.  The “incumbent” column indicates whether the Member already held the seat during the 2012 election.  Eight of the twenty are newcomers, though all but one come from districts carried by Mitt Romney in last year’s Presidential election.

According to Roll Call, Michelle Bachmann has already announced her intention to retire. The remainder have made no announcements about whether they will run for re-election in 2014.

Gerrymandering: The Final Reckoning

Over the past few posts I have narrowed down the list of states where we might claim gerrymandering affected the outcome of the 2012 Congressional election.  There is still one more task remaining — identifying those states where the partisan composition of the legislature and governorship, and the laws governing redistricting, enabled one party or the other to draw lines in a favorable manner.  This table combines information on each state’s constellation of partisanship and the method by which the state allocates Congressional seats.

There are four types of apportionment methods identified by Ballotpedia.  Most states place control over redistricting in the hands of the state legislature with the governor having a veto in all such states except North Carolina.  Nine states use nonpartisan commissions, and another five states can appoint a commission if the legislature fails to agree on a plan.  I treat those states as equivalent to states where the legislature is entirely in control.  Iowa reverses the backup system, with the legislature brought in only if the commission fails to come up with a plan.  I consider Iowa a commission state.

In the table above, I have divided the states into four groups.  At the top we have the eight states with unified Democratic control of state government and laws that grant the legislature and governor control over the apportionment process.  Some of the states identified in the last post as showing a pro-Democratic bias in their seat allocations like Massachusetts and Connecticut appear on this list of states with conditions favorable to Democratic gerrymandering.

Then follows a much larger group of states, nineteen, which had unified Republican state houses with control over redistricting.  Again we see some familiar faces from earlier tables like Pennsylvania and Ohio.  The third block of fourteen states have more uncertain gerrymandering conditions because of split partisan control either within the legislature or between the legislature and the governor.  The last group of nine states rely on nonpartisan commissions to draw their lines.  Both California and Arizona appear on this list despite showing a Republican and a Democratic excess of seats respectively.

So the last step is to combine the earlier list of states where gerrymandering might have taken place with the lists of states in the top two groups of the table above where the arrangement of political forces in the state might have encouraged gerrymandering.

Democratic Gerrymanders

Four of the eight states with pro-Democratic seat outcomes seem likely candidates for gerrymanderers.  All told, the Democrats probably won between four and six additional seats in Massachusetts, Connecticut, Illinois, and Maryland than they would have under a fair allocation of seats.  Gerrymandering seems much less likely to explain the additional Democratic victories in Georgia and Maine, where the Republicans were in control of the apportionment process, or in Arizona where district lines are drawn by a commission.

In Georgia, the entire process was controlled by the Republicans and was expected to produce a result favorable to that party.  However one of the targeted Democratic incumbents, John Barrow, moved after his district was redrawn, contested the 12th CD, and won with 54% of the vote.  Barrow’s dogged pursuit of his seat probably accounts for the “extra” Democrat in the Georgia delegation.

In New Hampshire, a Republican legislature faced off against a Democratic governor, though most of political struggles took place within Republican ranks.  Both seats had been captured from the Democrats in 2010, and the two new incumbents squabbled over the small changes that needed to be made to balance the two districts population.  In the event both seats were retaken by the Democrats in 2012.  This Democratic surge in New Hampshire probably has much to do with the regional trends to the Democrats across New England,  and the efforts by the Obama campaign to mobilize Democratic voters in a swing state.

Republican Gerrymanders

Two states that showed a pro-Republican bias had lines drawn by a commission rather than the legislature, while in Virginia legislative control was split between a Democratic Senate and a Republican House.  That leaves ten states with unified Republican control that show evidence of gerrymandering.  At the top of the list we have Pennsylvania, Ohio, Michigan and North Carolina, all states that have been repeatedly cited by observers as being heavily gerrymandered in the Republicans’ favor.  Unfortunately these observers often tend to claim the Republicans maintained control of the House entirely by gerrymandering and neglect the effects of incumbency.

At one extreme gerrymandering might have given the Republicans seventeen more seats in the House.  That figure combines the minimal estimate for Democratic gerrymanders, four, with the maximal estimate for the Republicans of twenty-one. If those seventeen seats had been won by Democrats, they would have eked out a one-seat majority in the House.  To achieve that value, though, we have to assume that urbanism only exerted effects in the Democratic states and had no effects in the Republican ones.  The opposite set of assumptions leads to an estimate of gerrymandering effects of just eight seats for the Republicans, far too few to have changed the outcome in the House.

These results provide a lower- and upper-bound on the effects of gerrymandering.  The actual effects probably lie somewhere in between.  Perhaps about a dozen seats remained with the Republicans because of gerrymanders, hardly an insignificant number to be sure, but not sufficient to explain why the Democrats could not win control of the House despite winning a slight majority of the popular vote for Congress.


Accounting for Geography

Updated, November 24, 2012, with complete results for all seats except the NC 7th.

If seats in Congress were allocated in an unbiased fashion, the Democrats might have won as many as twenty additional seats than they did on November 6th and would have taken control of the House of Representatives with 221 seats.

Where did this large Democratic deficit come from?  Democratic politicians and left-leaning pundits point their fingers at partisan gerrymandering by Republican state governments elected in the off-year landslide of 2010.  Students of the redistricting process itself point to a more fundamental problem for the Democrats, the geographic distribution of their supporters.

In an intriguing paper based on a careful simulation model of the redistricting process, political scientists Jowei Chen and Jonathan Rodden show that the tendency for Democratic voters to be tightly clustered in urban areas naturally advantages the Republicans when lines are drawn:

We show that in many urbanized states, Democrats are highly clustered in dense central city areas, while Republicans are scattered more evenly through the suburban, exurban, and rural periphery. Precincts in which Democrats typically form majorities tend to be more homogeneous and extreme than Republican-leaning precincts. When these Democratic precincts are combined with neighboring precincts to form legislative districts, the nearest neighbors of extremely Democratic precincts are more likely to be similarly extreme than is true for Republican precincts. As a result, when districting plans are completed, Democrats tend to be inefficiently packed in homogeneous districts.

In another study of the 2012 redistricting Nicholas Goedert observes that measures of urbanization correlate with the degree to which the Democrats gain a smaller or larger share of seats than what their votes share would predict.  So before we join the critics in claiming Republican gerrymandering as the source of the Democratic seat deficit, we need to first consider the role of urbanism.

The Census Bureau defines two types of urban areas — “urbanized areas” which contain a minimum of 50,000 people, and “urban clusters” which contain between 2,500 and 50,000 inhabitants.  The Bureau provides detailed information by state for both these types of urban areas.  I have tested a variety of these measures of urbanism by adding them to the baseline logit model for Democratic seats and votes.  Typically the measures for urban clusters have no significant effect on either vote or seat shares, but the data for urbanized areas, places with at least 50,000 people, matter considerably.

To get a sense of how urbanized areas and urban clusters are distributed across the country I recommend looking at two maps on this page at the Census Bureau website. The map on the left displays the density of the urbanized areas and urban clusters.  We can easily identify the large urban conglomerates like the Northeast Corridor, Atlanta, Chicago, Houston, Los Angeles, and Seattle. The second map codes entire counties and shows how California’s geography differs from most of the rest of the nation. Whole counties stretching back to the Nevada border are counted as urbanized even though most of the population living in the urbanized areas are along the coast.  California also dominates the list of urbanized areas when they are sorted by population density.  Of the top-thirty urbanized areas ranked by population density only six are outside California.

It is certainly the case that Democrats do better in states with a larger percentage of their populations living in urbanized areas.  About fourteen percent of the variation in Democratic Congressional vote across states can be accounted for by the proportion living in urbanized areas.  When it comes to the relationship betweens seats and votes, however, simply measuring how urbanized a state is does not affect the share of seats a party receives.  What turns out to matter much more is the population density of urbanized areas.  Adding that variable to our simple seats and votes model significantly improves our ability to predict the share of Democratic seats in a state given their share of its votes.  It also makes theoretical sense that urban density should play an important role given the relationship between clustering and apportionment bias Chen and Rodden explore.

To see how urban density influences affects the distribution of Congressional seats, look at this table which  shows the expected Democratic share of the seats given different values of the predictors.

Look first at the 50% column.  Even if the Democrats win half the vote in a state, they can only be assured of winning half the seats in the most heavily urbanized states.  Even in states like Maryland or Texas, with levels of urbanism higher than three-quarters of the states, winning half the vote does not guarantee a commensurate share of seats.  The effects of urban population density give the Democrats a boost in the most urbanized states, but they are few in number.  There are many more states where the Democrats need to poll well above 50% to claim half the seats in those states.

Given this powerful effect of urban density, I have rerun my seat estimates adjusting for the effect of urban density.  Not surprisingly, the Democratic deficit compared to the unbiased allocation shrinks when political geography is taken into account, but the amount of shrinkage is striking.

Let us start with the totals at the bottom of the table.  Using the method of “unbiased allocations” I estimate an 17 seat deficit for the Democrats in these states based solely on the share of the vote they won.  Adjusting for urban density accounts for fully 12 of those seats leaving a total deficit of just five.

Two of those five seats are in California, where a nonpartisan commission draws district boundaries.  As the maps above attest, the definition of “urbanism” applies rather differently to California than to the other states with densely populated urbanized areas.  So we might be a bit hesitant to claim that those two seats reflect gerrymandering.

Was Gerrymandering the Culprit? — Part I

Results updated on November 23, 2012, with final Congressional results for 434 races; NC 7th is still undecided.

It is now time to put some of the findings from earlier postings together and try to determine the extent of gerrymandering in the 2012 Congressional Elections.

Three factors should influence the number of House seats a party wins in a state Congressional election:

I have taken two separate measurements of the first item, the relationship between seats and votes.  I have calculated both a longitudinal measurement using elections from 1942 on, and a cross-sectional measurement using state results for 2012.  In both approaches I estimate the coefficients α and β of this “logit” model:

log(Democratic Seats/Republican Seats) = α + β log(Democratic Votes/Republican Votes)

The two models produce very different estimates for α, the seat “bias,” because it varies historically.  However the two estimates for β are nearly identical. The longitudinal estimate was 1.92; the cross-sectional estimate is 2.08.  For simplicity, I will just use two for the value of β.  (Mathematically, that implies that the ratio of Democratic to Republican seats varies in proportion to the square of the ratio of their votes.)

In this Technical Appendix, I explain why, if the Democrats win exactly half the vote, the only way they can win exactly half the seats is if the “bias” term α is zero. We can use this fact to create an “unbiased” distribution of seats.  I simply substitute two for β and apply it to the logit of the state-wide Democratic vote for Congress.  I will call this the “unbiased allocation.”  For each state I compare this estimate to the number of seats the Democrats actually won. Here are the results:

I have included all states where the difference between the predicted and actual number of Democratic seats was at least 0.7.  The state that gave us the word “gerrymander,” Massachusetts, shows the largest pro-Democratic deviation.  While the unbiased allocation model would award the Democrats only seven or eight of the nine seats in that state, not one Republican represents the Commonwealth of Massachusetts in Congress. The other state where Democrats did better than expected is Arizona, where they won a majority of the state’s Congressional seats with a minority of the popular vote.  Arizona had two of the closest races in the country, and they both fell to the Democrats by slim margins. All told, eight states including four New England states, have new Congressional delegations with an “extra” Democratic member in their numbers.

Many more states deviate from the unbiased allocation on the Republican side, with half-a-dozen states showing a pro-Republican bias of two, three, or, in the case of Pennsylvania, four seats. All told, sixteen states met our 0.7 criterion.  Compared to an unbiased allocation, the results in these sixteen states probably cost the Democrats 28 seats.  When we subtract out the eight extra seats the Democrats won in the pro-Democratic states, we get a net Democratic deficit in 2012 of some twenty seats compared to an “unbiased” allocation based solely on the popular vote for Congress in each state.

Before we start attributing all those seats to Republican gerrymandering, we first need to consider what other factors might influence the translation of Democratic votes to Democratic seats.  There is good reason to believe that the geographic distribution of Democratic voters by itself creates a pro-Republican bias when district lines are drawn.

 Accounting for Geography


Seats and Votes 2012: Evidence from the States

Last week I published a chart showing the historical relationship between House seats and votes since 1940.  Using the district-level level 2012 results compiled by David Wasserman, I plotted the percent of votes and seats for the Democrats in the 33 states where both parties had at least one member of the delegation.  That graph bears a striking similarity to the earlier chart depicting the historical relationship.


If I apply the same method to measure bias as I described here using the “logit” of the seats and votes shares, I get a slope that is nearly identical to what I estimated for the historical relationship.

Slope (β) 1.92 2.08
Intercept (α) 0.034 -0.189
Democratic Seat Bias 3.7 -20.5

However the intercept term, α, shows a much larger bias in the Republican direction than the four-seat pro-Democratic bias we found historically.  In these 32 states where both parties won at least one seat in 2012, the two parties garnered nearly identical numbers of votes, but the Democrats were rewarded with just 176 or 177 of the 388 seats (45.5%) in these states.

In the eighteen states with homogeneous delegations, the Democrats netted only one seat.  They swept all fourteen seats in Massachusetts and Connecticut and picked up ten more in six other states. The Republicans scored best in Oklahoma and Arkansas, but Massachusetts alone has as many seats (nine) as those two states combined.  All told the Republicans won twenty-three seats in ten of these homogeneous states.  The Democrats’ one-seat gain the homogenous states left them facing an estimated 19-20 seat deficit after adjusting for the votes they won.

The Effects of Reapportionment

The reapportionment process following the 2010 Census transferred twelve Congressional districts from ten largely Northern and Midwestern states to eight largely Southern and Western ones.  The largest beneficiary was Texas which gained four seats and increased its representation in Congress to 36 Members.  Florida gained two seats, and Arizona, Georgia, Nevada, South Carolina, Utah and Washington added one seat each.  New York and Ohio both lost two seats, while Illinois, Iowa, Louisiana, Massachusetts, Michigan, Missouri, New Jersey, and Pennsylvania lost one apiece.

While all states must redistrict every ten years, the effects are greatest in states gaining or losing representation in Congress.  Popular candidates from the same party in nearby districts can be pitted against each other in the primaries after their two seats are reduced to one.  New seats provide opportunities for creative cartographers to draw districts that not only command a majority for their parties today but over the next ten years as well.

Over the next few postings I will be exploring the question of why the Democrats fared so poorly in the Congressional elections.  I begin today not with the issue of gerrymandering, but the more fundamental question of whether reapportionment itself might have changed the partisan balance of the House.  Adding four seats to a state like Texas whose Congressional delegation was already 23-9 in favor of the Republicans should on average have given that party three of those four seats. Removing a district from Massachusetts where all the Members were Democrats means the sure loss of a Democratic seat.

How many seats did reapportionment affect?
On the face of it, this question should be easy to answer.  Since twelve seats were moved, the largest effect it could have is twelve, and only then if every seat lost belonged to one party and every new seat was won by the other party.  Since both Democratic New York and Republican Louisiana lost seats in 2012, and Republican Texas and Democratic Washington gained seats, calculating the actual effects of reapportionment requires a bit more effort.

Start again with Texas with its four new seats and a Congressional delegation that had a 23-9 margin in favor of the Republicans.  If we draw the new districts proportionately to maintain the current partisan balance, we would expect the 2012 Texas delegation to include, on average, 2.9 (=4*23/(23+9)) new Republicans and 1.1 (=4*9/(23+9)) new Democrats. Using just the 2010 results overstates the extent of Republican sentiments because of the size of the 2010 Republican landslide.  I used instead the average partisan split in both the 2008 and 2010 Congressional delegations to estimate the division in 2012.  For Texas, this increases the Democrats’ estimate a bit from 1.1 to 1.3 when we base that figure on both the 2008 and the 2010 results. The top half of the table includes the data for states that gained a seat in 2012; those that lost a seat appear in the bottom half.

In 2008 Republicans held 55 of the 99 Congressional seats in states that added a district in 2012.  In the 2010 Republican landslide they pushed their advantage up to 68 seats. Depending on the outcome in the Arizona 2nd, Republicans will hold 72 or 73 of the seats in these eight states when Congress convenes in January.  Is this a strong showing for the Republicans across these states?  How might we tell?

The last three columns provide an answer to that question.  If we allocate each state’s new seat(s) in proportion to the average division in that state’s Congressional delegation in 2008 and 2010, the Democrats should have gained 4.6 of the 12 new seats and the Republicans 7.4. The Democrats actually won either seven or eight seats in these states depending on Arizona. While the Democrats added not one of the four new seats in Texas, they did pick up four seats in Florida, one each in Nevada and Washington, and either one or two seats in Arizona.

Democrats also fared slightly better than expected in the states that lost a Congressional seat in 2012.  Applying the same proportional model predicts that the Democrats should have lost 6.7 seats.  Their strong showing in Illinois held their losses to just five.

Overall, then, it seems like the Democrats not only survived the process of reapportionment in the states most immediately affected, they actually carried four or five more seats than we might have expected.  Strong performances in Florida and Illinois were largely responsible for this result.

Reapportionment and the 2012 House Results
In the opening posting on the 2012 Congressional elections, I estimated that the Democrats now face a ten-seat disadvantage in the House based on that party’s share of the popular vote.  The method above suggests that we can attribute the loss of perhaps two of these seats to the effects of reapportionment itself. Using the method of extrapolating the pro-Republican trend from 1960-2010 results in an estimate of four seats.  That leaves some six to eight seats that constitute the Republican “bulwark” still to be explained.




Technical Appendix – Estimating the Effects of Reapportionment

Estimating the Effects of Redistricting
I took a rather simple-minded approach to the task of measuring redistricting effects.  I began by measuring the relationship between the national vote for Democratic Congressional candidates and the number of seats the Democrats won for all Congressional elections beginning in 1942.  (The 2012 data do not include four undecided seats.) I picked 1942 because it was the first election fought on seats whose boundaries were based on a New Deal Census.

I began by estimating the relationship between the proportion of seats won by the Democrats and the proportion of their vote.  I used a slightly more sophisticated statistical method here than I did when looking at the Electoral College.  I again used linear regression to estimate the relationship between seats and votes but only after first transforming each of the proportions using the “logistic” function.  In brief, I am estimating the model:

ln(Democratic Seats/Republican Seats) = α + β ln(Democratic Votes/Republican Votes)

Instead of measuring the proportion or percent of seats and votes, I transformed each variable into the (natural) logarithm of its “odds ratio.”  Suppose the Democrats hold 60% of the House of Representatives.  If I choose a seat from the House at random, the odds of my drawing a Democratic seat are 60:40.  That is the odds ratio; in this case we could also call it 3:2 after simplifying the fraction.  If I take the logarithm of this odds ratio, I get a very “well-behaved” variable.  It is no longer constrained to the range between zero and one like proportions are, and the logarithmic transformation turns the non-negative odds ratio into a continuous variable that encompasses the whole number line.  At even odds, or a odds ratio of one, the logarithm is zero.  For values below 50:50 we get negative values; for values above 50:50 the values are positive ones.  This is commonly called a “logit” transformation.

We can use this fact that the logit is zero at 50% to determine whether an electoral system is “biased.”  If  we define an unbiased system as one that awards half the seats to a party winning half the vote, that definition of unbiasedness requires that the intercept term α be zero.  (This is true no matter what value we use for β.)  If α is not zero, a vote share of 50%, whose logit is zero, will predict a share either greater or less than 50% depending on the sign of α.  A positive value means the party wins more seats than it “deserves” based on its share of the popular vote; a negative value means the party was “shortchanged.”

Using this equation as the basis, we can measure the bias associated with different apportionments by testing the “null hypothesis” that the value of α in an apportionment period is equal to zero.  I define an “apportionment period” as the five elections that begin two years after a Census and end in the next Census year.  All apportionment periods begin with an election in a year ending in two and end in the next election when the year ends in zero. As an example, the apportionment period associated with the 1960 Census begins in 1962 and ends in 1970.

This is a very crude measure of differences in apportionment to be sure.  I could create a more fine-grained measure that includes important factors like the use of nonpartisan commissions, the partisan divisions of the state legislatures in the apportionment year, and whether a party controlled all three branches of state government.  These are worthwhile tasks best engaged in after we see whether we can detect any partisan effects from apportionment simply by examining variations across the apportionment periods.

As a crude first step, then, I have created “dummy” variables for each apportionment period.  These variables have the value one for elections held in that period and zero otherwise.  For instance, the Census_40 variable in the results below has the value one for each of the five elections from 1942 to 1950 and zero after that.  Here are the results:

OLS, 36 Congressional Elections, 1942-2012
Dependent variable: log(% Dem Seats/(100 - % Dem Seats))

              coefficient   std. error   t-ratio   p-value 
  Apportionment Biases (α)  
  Census_40    0.0673105    0.0171312     3.929    0.0005   ***
  Census_50    0.0220976    0.0185816     1.189    0.2447  
  Census_60    0.0741743    0.0191011     3.883    0.0006   ***
  Census_70    0.0645399    0.0210222     3.070    0.0048   ***
  Census_80    0.0431043    0.0204332     2.110    0.0443   **
  Census_90    0.0134422    0.0171097     0.7856   0.4389  
  Census_00   −0.0124388    0.0171423    −0.7256   0.4743  
  Census_10   −0.0881460    0.0382346    −2.305    0.0291   **

  Responsiveness (β)  
  lgt_D_Vote   1.77527      0.146841     12.09     2.09e-12 *** 

Mean dependent var   0.100578   S.D. dependent var   0.121026
Sum squared resid    0.039463   S.E. of regression   0.038231
R-squared            0.923023   Adjusted R-squared   0.900215
F(8, 27)             40.46924   P-value(F)           4.57e-13

*p<0.10; **p<0.05; ***p<0.01

This model does not have a constant term, for I have included all the dummies for the apportionment periods instead.  In this formulation each coefficient is measured as a deviation from zero, our standard for unbiasedness.  According to the statistical tests the period from 1992 until 2010 had no measurable bias.  The two tiny values we measure for those decades are not even as large as their standard errors.  The elections fought in the seats drawn after the 1950 Census also stand out as much less Democratic than in any other decade before the 1990s.

Looking more closely at the estimated coefficients, or at the graph they generate, makes it clear there are three distinct periods in these results.  From 1942 to 1990 the Democrats were the beneficiaries of a seven-seat advantage in the House of Representatives, excluding the one apportionment period beginning in 1952.  This period of Democratic dominance was followed by two decades of parity where the electoral system advantaged neither party.

One other way to view these historical patterns trends more clearly is to simplify the model above.  By removing the Census_40 variable and replacing it with a constant term, we can envision how the period of Democratic dominance gave way to parity.  Remember that statistically this model is identical to the one above, but the interpretation of the coefficients is different.

  Constant     0.0673105    0.0171312     3.929    0.0005   ***
  lgt_V_Dem    1.77527      0.146841     12.09     2.09e-12 ***
  Census_50   −0.0452129    0.0255820    −1.767    0.0885   *
  Census_60    0.00686379   0.0260132     0.2639   0.7939  
  Census_70   −0.00277057   0.0276003    −0.1004   0.9208  
  Census_80   −0.0242061    0.0271129    −0.8928   0.3799  
  Census_90   −0.0538683    0.0241829    −2.228    0.0344   **
  Census_00   −0.0797493    0.0242902    −3.283    0.0028   ***
  Census_10   −0.155457     0.0419114    −3.709    0.0010   ***

What we see now is a period of Democratic dominance that stretched from the 1940s through the 1980s with the exception of the decade following the 1952 reapportionment.  Republicans had taken control of 26 state legislatures  in 1950 compared to just 16 for the Democrats and thus controlled the redistricting process in many states.  Our data suggest they were able to reduce the Democrats’ advantage substantially in that decade, though we will see in a moment that the situation is even more complicated.

Elections over the two decades from 1962 to 1980 show the same pro-Democrat advantage that elections held in the 1940s do.  Reapportionment actions in those three decades show no significant deviation from the 1940s baseline.  After 1992, though, the situation changes.  After the 1992 elections the Democrats’ advantage fades quickly.  The effects for 1992-2010 essentially eliminate that advantage and usher in two decades where neither party was advantaged or disadvantaged by the workings of the electoral system.  However the trends in the Republicans’ favor reach historic proportions in the 2012 election.  The Republicans have now turned things in their favor beginning with the election of 2012.

I have not spoken at all about β, the slope coefficient that measures how changes in the popular vote odds translate into changes in the number of seats won. Larger values of this coefficient increase the steepness of the relationship between seats and votes. British statisticians as early as 1950 talked about  “cube law” relationship between seats and votes.  In terms of our model that translates into a value for β of three.  For Congressional elections since 1940 I estimate a value of 1.78, considerably below the “cube-law” value, but still substantially higher than one, which would indicate pure proportionality or “parity” as I call it in the graph to the right.   A party whose share of the Congressional vote rises from 40% to 60% should expect to see their share of seats increase from 33% to 67%.  A cube law system is much more ruthless giving the party at 40% a mere 23% of the seats, while one that wins 60% of the vote gets an enormous bonus winning 77 % of the seats in the legislature.


The New Republican Bulwark in the House

For only the third time since 1940, the winner of the national popular vote for Congress failed to take control of the House of Representatives.  The Democrats won a slim majority of the (two-party) vote, 50.2%, but failed to gain back the House, winning just 197 of the 431 seats currently decided,* or 45.7%. This is the largest adverse gap between the Democratic Party’s share of seats and its share of votes since the New Deal.

For most of the years between 1942 and 1994, the Democrats enjoyed a “bonus” in terms of the seats they were awarded in the House of Representatives.  A good portion of that bonus came from the workings of our electoral system.  Political scientists and statisticians have long known that the method of voting used in Congressional elections, called “plurality voting” or “first-past-the-post,” exaggerates the size of majorities in the elected assembly.  A party that wins 51% of the vote gains more than 51% of the seats, and the bonus increases as the party’s share of the popular vote grows.** This is not a uniquely American phenomenon; we see the same exaggeration at work in countries like the United Kingdom which also employ plurality voting.  The bonus is quite evident if we plot the share of seats won by the Democrats against their share of the (two-party) popular vote.

The actual relationship between seats and votes is much steeper than the “parity” line which awards a party a share of the seats equal to its share of the vote.  Much of the Democratic advantage we saw in the first graph reflects this feature of our electoral system.  In the time-series plot we see the Democrats winning dramatically oversized House majorities in 1974 and 1976, but in the context of this historical relationship between seats and votes, those elections are not out of line.

I will use this relationship between seats and votes to see how Congressional reapportionment shifted the balance of power in the House.  The long-term relationship gives us a method for estimating how many seats the Democrats should have won given their share of the popular vote.  We can then look at the differences between these estimates and the actual share of seats won to measure any partisan bias.

Reapportionment by itself can change the partisan balance of the House.  As populations shifted from the Northeast to the Southwest, the fixed size of the House at 435 Members meant that seats in Democratic bastions like Massachusetts and New York were transferred to Republican strongholds like Texas.

What draws more attention from citizens and pundits alike is the possibility of partisan gerrymandering.   American political institutions grant parties enormous power to mold the structure of electoral competition.   Every ten years Constitutionally-mandated reapportionment requires state governments to redraw the boundaries of Congressional districts.  Nine states use a nonpartisan commission to draw district lines.  Of the remaining 41, the state legislature and governor have full control of the redistricting process in 28 of them.  The other thirteen states use a “hybrid” approach with a commission that is usually subservient, and sometimes purely advisory, to the elected state legislature.

These institutions invest an enormous amount of power in one specific set of state legislators and governors, those elected in a year ending in zero.  These partisan elected officials have the power to structure competition for the House of Representatives for the decade to come.  The development of sophisticated computer software combined with geographic marketing databases places enormous power in the hands of determined modern gerrymanderers.

One way we might begin the measure the extent of partisan gerrymandering is to look at whether individual “apportionment periods,” the five elections conducted in the same seat boundaries, showed a bias toward one party or another.  We can use the seats/votes relationship to provide a baseline expectation of what seat outcomes ought to be, then calculate the average deviation for each five-election apportionment period like this.

I date apportionment periods by the date of the Census, so the 1940 apportionment covers elections held from 1942 through 1950. The vertical bars measure the difference between the average number of seats the Democrats won in the five Congressional elections following each Census and the number of seats they “should” have won in that period based on their share of the popular vote.  Remember that this technique already accounts for the bonus resulting from our plurality-voting electoral system.  These bars measure any additional partisan bias associated with a particular apportionment period.

The electoral system showed a distinct pro-Democratic bias of five to eight seats for most of the period between the New Deal and the 1990s.  The only exception came in the 1950s when the bonus fell to an estimated two seats, a value not statistically different from zero.  The Democrats’ bonus began to slip after 1960 and has moved in the Republicans’ direction ever since.  Extrapolating the pro-Republican trend from 1960 through 2010 would predict an advantage for the GOP of about three or four seats in 2012. The actual Republican bias in 2012 looks closer to ten seats.  Not an auspicious start to the decade ahead if you are a Democrat.

While many commentators have pointed to Republican gerrymandering as the primary explanation for this result, I want to take things a bit more slowly and consider first how the process of reapportionment itself may have altered the balance of power in the House by shifting seats from Democratic states to Republican ones.  I begin that discussion in the next post.

The Effects of Reapportionment


*I awarded two of the six undecided House races from the November 12th list at CNN. I gave both AZ 9 and FL 18 to the Democrats, leaving two races in California, and one each in Arizona and North Carolina as undecided. That puts the current tally at 234 to 197 in favor of the Republicans. All four of these uresolved contests show slim majorities for the Democrats.  If they took all four it would raise their total to 201.  Even with those seats added, that’ hypothetical mark of 46.2% of the seats would remain the worst result by a majority winner since the New Deal.  Both the 1942 Republicans and the 1996 Democrats failed to win the House, but they both won a larger share of the seats than the 2012 Democrats.  (Return)

**The claim that the bonus increases as the popular vote share increases holds true for values in the range observed historically.  Eventually the bonus must shrink as the popular vote approaches 100%.  In the historical period I am examining here, the Democrats never won less than 45.3% of the popular vote (1946) or more than 58.3% (1974). (Return)