# 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:

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
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.