Election Audits Using a Trinomial Bound

In November 2008, we audited contests in Santa Cruz and Marin counties, California. The audits were risk-limiting: they had a prespecified minimum chance of requiring a full hand count if the outcomes were wrong. We developed a new technique for these audits, the trinomial bound. Batches of ballots are selected for audit using probabilities proportional to the amount of error each batch can conceal. Votes in the sample batches are counted by hand. Totals for each batch are compared to the semiofficial results. The ldquotaintrdquo in each sample batch is computed by dividing the largest relative overstatement of any margin by the largest possible relative overstatement of any margin. The observed taints are binned into three groups: less than or equal to zero, between zero and a threshold
d
, and larger than
d
. The number of batches in the three bins have a joint trinomial distribution. An upper confidence bound for the overstatement of the margin in the election as a whole is constructed by inverting tests for trinomial category probabilities and projecting the resulting set. If that confidence bound is sufficiently small, the hypothesis that the outcome is wrong is rejected, and the audit stops. If not, there is a full hand count. We conducted the audits with a risk limit of 25%, ensuring at least a 75% chance of a full manual count if the outcomes were wrong. The trinomial confidence bound confirmed the results without a full count, even though the Santa Cruz audit found some errors. The trinomial bound gave better results than the Stringer bound, which is commonly used to analyze financial audit samples drawn with probability proportional to error bounds.