@Article{HRS11,
author = { Michael J. Higgins and Ronald L. Rivest and Philip B. Stark },
title = { Sharper $p$-values for stratified election audits },
doi = { 10.2202/2151-7509.1031 },
url = { http://www.bepress.com/spp/vol2/iss1/7 },
OPTpages = { },
journal = { Statistics, Politics, and Policy },
date = { 2011 },
OPTyear = { 2011 },
volume = { 2 },
number = { 1, Article 7 },
abstract = { Vote-tabulation audits can be used to collect
evidence that the set of winners of an election (the
outcome) according to the machine count is correct---that
it agrees with the outcome that a full hand
count of the audit trail would show. The strength of
evidence is measured by the $p$-value of the
hypothesis that the machine outcome is
wrong. Smaller $p$-values are stronger evidence that
the outcome is correct.
\par
Most states that have
election audits of any kind require audit samples
stratified by county for contests that cross county
lines. Previous work on $p$-values for stratified
samples based on the largest weighted overstatement
of the margin used upper bounds that can be quite
weak. Sharper $p$-values can be found by solving a 0-1
knapsack problem. For example, the 2006 U.S. Senate
race in Minnesota was audited using a stratified
sample of 2-8 precincts from each of 87 counties,
202 precincts in all. Earlier work (Stark 2008b)
found that the $p$-value was no larger than 0.042. We
show that it is no larger than 0.016: much stronger
evidence that the machine outcome was correct.
\par
We also give algorithms for choosing how many batches
to draw from each stratum to reduce the counting
burden. In the 2006 Minnesota race, a stratified
sample about half as large---109 precincts versus
202---would have given just as small a $p$-value if
the observed maximum overstatement were the
same. This would require drawing 11 precincts
instead of 8 from the largest county, and 1 instead
of 2 from the smallest counties. We give analogous
results for the 2008 U.S. House of Representatives
contests in California.
},
}