APPENDIX A: MEASURING CROSSOVER RATES THROUGH ECOLOGICAL INFERENCE
To estimate the percentage of Democrats and Republicans who cross over in legislative primaries and general elections, this chapter relies on a method of ecological inference (King 1997). While valuable, it is important
Knowing only the number of Republican and Democratic voters in each district and the vote totals for each party, a researcher needs to make inferences about the number of Republicans who vote for a Democratic candidate (and vice versa). The first step of King's procedure, the method of bounds, uses logic quite similar to the reasoning employed by Hall's (1923) analysis of Wisconsin primaries. Hall reports that in 1918, while 192,145 primary voters cast their ballots in favor of Republicans, general election results suggest that there were only 155,799 Republicans in the active electorate that year. Democrats must have provided some of the Republican primary votes. The fact that only 112,576 Democrats turned up in the general election sets an upper bound on how many Democrats voted in the Republican primary. These bounds imply that Democratic crossover in this race fell between 36,346 and 112,576.
King's method constructs a "tomography plot," with lines in this case representing each California district. The x-axis gives the proportion of Democrats voting for a Democratic candidate and the y-axis gives the proportion of Republicans voting for a Democratic candidate. King assumes that the crossover values are most likely to lie at the pinnacle of a "mountain" (taking the shape of a truncated bivariate normal distribution) that is located where the lines are most densely bunched. His ecological inference procedure yields point estimates of voting rates across all districts derived from the mountain's pinnacle and a confidence interval mapped by the mountain's contour lines. King's method then uses a simulation procedure to calculate district-level estimates of crossover rates.
Using EzI, a program designed to compute King's model, I computed crossover rates in the ninety contested California Assembly and Senate races in 1998.[12] I conducted separate runs of the program to estimate cross-over in the primaries and in the general election. Some of the assumptions made in both cases to provide input for EzI introduce substantive concerns in addition to the statistical criticisms of ecological inference. To calculate crossover rates, EzI needs to be fed the number of Republicans and Democrats who show up at the polls as well as the number of ballots cast for candidates from each party. While election returns give exact measures of the latter figures, researchers need to find a proxy for the partisan break-down of the electorate that actually turns out. The proxy that I employ here—Democratic and Republican registration as a percentage of major-party registration—is imperfect. In many elections, Republicans go to the polls at a rate higher than that of registered Democrats.
| Democratic Crossover | Republican Crossover | |||||
|---|---|---|---|---|---|---|
| Lower | Estimate | Upper | Lower | Estimate | Upper | |
| Primary election | 13.0% | 21.7% | 35.1% | 11.2% | 24.0% | 41.6% |
| General election | 6.0 | 11.5 | 19.9 | 8.8 | 16.8 | 27.9 |
The assumption that there is no turnout differential in this case, though, seems tenable. In part because of a large mobilization by organized labor to defeat a major anti-labor initiative, Proposition 226, Democratic participation in the primary did not lag behind Republican turnout, as it often has. General election turnout rates were also nearly equal.[13] Another flaw of this proxy is that it assumes that every voter is a member of one of the major parties. Since 9.5 percent of those who cast a ballot for a major-party candidate in the 1998 primary belonged to a minor party or had no declared party affiliation, the EzI procedure slightly overestimates primary crossover rates. In the general election, this figure was 8 percent (Los Angeles Times exit poll). The nonaffiliates in each district are functionally assigned a party identification based on the major-party breakdown, and some are counted as crossover voters if election returns diverge from party composition patterns.
Tam (1997a) warns of another potential problem with this application of ecological inference. King's ecological inference model assumes both that the parameters (crossover rates, here) are uncorrelated with the regressors (registration figures) and that the lines in the tomography plot are all "related to one common mode" (Tam 1997a). Since I found that crossover does vary with the partisan leanings of a district, and since the tomography plot for this application does appear to be bimodal, I recomputed the estimates using the model extensions described in chapter 9 of King (1997). Specifically, I provided information about the relative competitiveness of primary races expressed in two different ways to the EzI program.[14] The estimates were fairly robust to these changes. Estimates from each run of EzI correlated with analogous estimates at .95 or higher, and none of the means differed by more than 4 percent. Because the original crossover rates most closely match the survey data from five districts collected by Alvarez and Nagler (1998), I use these figures in my analysis.
Aware of the statistical and substantive challenges to the ecologically inferred crossover rates, one should not place great credence in the precise point estimates that EzI provides. Yet just as a weatherman will predict temperatures in a five-day forecast, this chapter reports and analyzes these figures because they represent a "best guess" about the direction and magnitude