We show that in Palm Beach County in the 2000 election the discrepancy between the share of votes expected for Buchanan on the basis of previous election results and the share of votes Buchanan actually received was exceptionally large. The discrepancy in Palm Beach County was the second largest among the 3,015 counties in the U.S. for which discrepancies could be estimated.8 Most of the other counties for which we observe extremely large discrepancies either trace to documented balloting problems or occur in patterns that suggest that systematic and distinctive patterns of support for Buchanan developed in those places.
To measure discrepancy we use residuals from robust estimation of an overdispersed binomial regression model of the number of votes cast for Buchanan compared to the votes cast for all other presidential candidates.9 We estimate a separate model for each state. The binomial model respects the fact that the basic data are counts of votes. Following McCullagh and Nelder (1989), we allow for overdispersion because we believe that the county-level data (and, further below, precinct-level data) are subject to unobserved internal clustering effects.
We use robust estimators for several reasons. The primary reason is obvious: the voter complaints, legal cases and media reports strongly suggest that the processes that produced the electoral results in Palm Beach County were substantially different from the processes that produced the results elsewhere in Florida. The robust estimators we use have a high breakdown point (Donoho and Huber, 1983; Hampel, 1971) so that they are consistent and produce reliable measures of discrepancy even if unusual voting processes occurred in several counties in a state.10 A large anomaly in one county will not mask (Atkinson, 1986) comparable or perhaps somewhat smaller anomalies that occur in other counties (Hampel et al., 1986, 67). An estimator that lacks a high breakdown point will underestimate the frequency of highly anomalous election results. Another reason to use robust estimators is the fact that the regression models we use are at best rough approximations for the processes that produced the vote counts (Hampel et al., 1986, 82). The estimators we use produce reliable measures of discrepancy even under such conditions, as long as the model gives a pretty good approximation for Buchanan's expected vote share in most counties in each state. Data weakness is another reason for robust estimation. Because our regressors include functions of results from the previous election, there is a generic kind of problem. If anomalies occurred in the earlier election at roughly the same rate as in the current one, then the data include observations that have distorted regressor values. The robust estimators we use protect against the influence such distorted regressors might otherwise have on the results. An observation that has a substantially distorted regressor will not affect the results for the other observations and will itself appear as an observation that has a large discrepancy.
A county that has a discrepancy larger in magnitude than a certain threshold is an outlier. The threshold is defined in terms of quantiles of the standard normal distribution.11 If the threshold is large, it is highly likely that the relationship between current and previous election results in an outlier county differs from the relationship that approximates the data elsewhere in the state. The outlier county may have a different disturbance (e.g., a nonzero mean, a fat-tailed distribution), different regression coefficients or contaminated regressors. To explain why a county is an outlier requires investigation using additional information. There are many possible reasons for Palm Beach County to be an outlier. In later sections we provide further support for our claim that the cause of Palm Beach County's exceptional status is the butterfly ballot.