A very brief comment on the recent presidential election in the United States. There has been some discussion of the differences between opinion poll estimates of the popular vote for the two main candidates, and the vote that they actually received. Anthony Wells has a short post on the topic and I'm sure many more can be found.

A peculiarity of US elections is the early publication of exit polls. There's a piece in Slate about this. Those who stayed up to watch the election coverage on television will have noticed how, early in the night, the Kerry campaign team were reported as being upbeat and the Bush team downbeat; this reversed later in proceedings. The reason, perhaps, was that exit poll data which became known early in the evening (US time) suggested that Kerry would win in critical states (by two points in Ohio and Florida, for instance) and, therefore, win the election; later, as real results came in, it seemed more likely that Bush would win, as he eventually did.

It is certainly true that the reported exit poll data look substantially different to the last reported pre-election polls: (significance: p-value < 0.0001)

(Pre-election polls from Andrew Tannenbaum's electoral-vote.com; exit polls as reported in Slate.)

I don't know enough about exit poll methodology to say anything very useful about this.

Next let us reach briefly into conspiracy theory territory.

A number of states in the United States use `Diebold' direct-recording electronic voting machines, which are touchscreen operated and lack paper receipts. These devices are manufactured by a company whose owners give money to the Republican party. Such machines are suitable only for people who do not care about keeping their elections honest, because it is impossible to audit their results. I have written before about how such devices are not appropriate for elections in this country (although unaccountably the government do not listen).

Suppose that pre-election opinion polls accurately (to within some margin of error) divine the voting intentions of electors. Suppose further that actual election results may be affected by fraud in certain states, but that opinion poll results are not.

In these circumstances, you would expect the distribution of differences between the lead of a candidate as reported in opinion polls to have approximately a zero mean for states which do not suffer from fraud, and to have a mean which differs significantly from zero where fraud has occured (assuming that all fraud is intended to give advantage to the same side).

Here is the plot of poll vs. election result discrepancies for states which use Diebold direct-recording electronic voting machines (per this list, but with the addition of Florida and Ohio, and the removal of California, to account for more recent developments):

States	Mean discrepancy (% points)	95% confidence interval (% points)
non-Diebold	-0.26	-2.22 < d < +1.70
Diebold	+2.71	+0.53 < d < +4.89

Note that positive numbers for the discrepancies mean that Kerry got fewer votes than predicted by the polls.

The two distributions -- for Diebold and non-Diebold states -- do differ significantly (p-value = 0.040, i.e. ``significant at the 95% level'', with the 95% confidence interval on the difference in means being 0.15 to 5.81). Therefore, on this very naive analysis, there is some evidence for voting-machine fraud in the recent US presidential election.

(Caveats: don't take this too seriously. In particular, it was done hastily for my own entertainment and I haven't checked the provenance of the polls from the Andrew Tannenbaum site or independently verified the types of voting machines in use in the various states. Plus, the implicit assumption above that polls should accurately predict electoral results even in states where the election is honest is... a bit of a leap. In particular methodology may differ from state to state or cultural differences between the states may bias the results of telephone polling. The states which use Diebold voting machines do not necessarily use them everywhere. Etc. etc. Wait for somebody to do the analysis carefully and properly before you go round to Diebold HQ with the pitchforks and the flaming torches. In my defence, this is probably more socially responsible than spreading rumours about electoral fraud -- I assume there are some -- and given that there is no other way to audit the results, I hope that there is further work on this issue. On the same subject, you may also want to read this piece about auditing the recent Venezuelan recall election, which suggests how such audits may be manipulated even where paper receipts are recorded.)