680 research outputs found
A Sharper discrepancy measure for post-election audits
Post-election audits use the discrepancy between machine counts and a hand
tally of votes in a random sample of precincts to infer whether error affected
the electoral outcome. The maximum relative overstatement of pairwise margins
(MRO) quantifies that discrepancy. The electoral outcome a full hand tally
shows must agree with the apparent outcome if the MRO is less than 1. This
condition is sharper than previous ones when there are more than two candidates
or when voters may vote for more than one candidate. For the 2006 U.S. Senate
race in Minnesota, a test using MRO gives a -value of 4.05% for the
hypothesis that a full hand tally would find a different winner, less than half
the value Stark [Ann. Appl. Statist. 2 (2008) 550--581] finds.Comment: Published in at http://dx.doi.org/10.1214/08-AOAS171 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Testing earthquake predictions
Statistical tests of earthquake predictions require a null hypothesis to
model occasional chance successes. To define and quantify `chance success' is
knotty. Some null hypotheses ascribe chance to the Earth: Seismicity is modeled
as random. The null distribution of the number of successful predictions -- or
any other test statistic -- is taken to be its distribution when the fixed set
of predictions is applied to random seismicity. Such tests tacitly assume that
the predictions do not depend on the observed seismicity. Conditioning on the
predictions in this way sets a low hurdle for statistical significance.
Consider this scheme: When an earthquake of magnitude 5.5 or greater occurs
anywhere in the world, predict that an earthquake at least as large will occur
within 21 days and within an epicentral distance of 50 km. We apply this rule
to the Harvard centroid-moment-tensor (CMT) catalog for 2000--2004 to generate
a set of predictions. The null hypothesis is that earthquake times are
exchangeable conditional on their magnitudes and locations and on the
predictions--a common ``nonparametric'' assumption in the literature. We
generate random seismicity by permuting the times of events in the CMT catalog.
We consider an event successfully predicted only if (i) it is predicted and
(ii) there is no larger event within 50 km in the previous 21 days. The
-value for the observed success rate is : The method successfully
predicts about 5% of earthquakes, far better than `chance,' because the
predictor exploits the clustering of earthquakes -- occasional foreshocks --
which the null hypothesis lacks. Rather than condition on the predictions and
use a stochastic model for seismicity, it is preferable to treat the observed
seismicity as fixed, and to compare the success rate of the predictions to the
success rate of simple-minded predictions like those just described. If the
proffered predictions do no better than a simple scheme, they have little
value.Comment: Published in at http://dx.doi.org/10.1214/193940307000000509 the IMS
Collections (http://www.imstat.org/publications/imscollections.htm) by the
Institute of Mathematical Statistics (http://www.imstat.org
Limiting Risk by Turning Manifest Phantoms into Evil Zombies
Drawing a random sample of ballots to conduct a risk-limiting audit generally
requires knowing how the ballots cast in an election are organized into groups,
for instance, how many containers of ballots there are in all and how many
ballots are in each container. A list of the ballot group identifiers along
with number of ballots in each group is called a ballot manifest. What if the
ballot manifest is not accurate? Surprisingly, even if ballots are known to be
missing from the manifest, it is not necessary to make worst-case assumptions
about those ballots--for instance, to adjust the margin by the number of
missing ballots--to ensure that the audit remains conservative. Rather, it
suffices to make worst-case assumptions about the individual randomly selected
ballots that the audit cannot find. This observation provides a simple
modification to some risk-limiting audit procedures that makes them
automatically become more conservative if the ballot manifest has errors. The
modification--phantoms to evil zombies (~2EZ)--requires only an upper bound on
the total number of ballots cast. ~2EZ makes the audit P-value stochastically
larger than it would be had the manifest been accurate, automatically requiring
more than enough ballots to be audited to offset the manifest errors. This
ensures that the true risk limit remains smaller than the nominal risk limit.
On the other hand, if the manifest is in fact accurate and the upper bound on
the total number of ballots equals the total according to the manifest, ~2EZ
has no effect at all on the number of ballots audited nor on the true risk
limit
Conservative statistical post-election audits
There are many sources of error in counting votes: the apparent winner might
not be the rightful winner. Hand tallies of the votes in a random sample of
precincts can be used to test the hypothesis that a full manual recount would
find a different outcome. This paper develops a conservative sequential test
based on the vote-counting errors found in a hand tally of a simple or
stratified random sample of precincts. The procedure includes a natural
escalation: If the hypothesis that the apparent outcome is incorrect is not
rejected at stage , more precincts are audited. Eventually, either the
hypothesis is rejected--and the apparent outcome is confirmed--or all precincts
have been audited and the true outcome is known. The test uses a priori bounds
on the overstatement of the margin that could result from error in each
precinct. Such bounds can be derived from the reported counts in each precinct
and upper bounds on the number of votes cast in each precinct. The test allows
errors in different precincts to be treated differently to reflect voting
technology or precinct sizes. It is not optimal, but it is conservative: the
chance of erroneously confirming the outcome of a contest if a full manual
recount would show a different outcome is no larger than the nominal
significance level. The approach also gives a conservative -value for the
hypothesis that a full manual recount would find a different outcome, given the
errors found in a fixed size sample. This is illustrated with two contests from
November, 2006: the U.S. Senate race in Minnesota and a school board race for
the Sausalito Marin City School District in California, a small contest in
which voters could vote for up to three candidates.Comment: Published in at http://dx.doi.org/10.1214/08-AOAS161 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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