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Calculate poll odds

Most discussions of polls include a reference to the margin of error. For example, an Aug. 10, 2016 Bloomberg Politics poll found Hillary Clinton leading Donald Trump 50 percent to 44 percent with a margin of error plus-or-minus 3.6 percentage points.

Software developer Dan Vanderkam points out that most people don't quite know how to interpret margin of error. Does it mean Clinton's lead is statistically significant because it's greater than the 3.6-point margin of error? Or that Clinton's six-point lead is not statistically significant because it's less than twice the margin of error, or 7.4 percent?

In fact, the answer is that a lead is significant if it's greater than 1.6 times the margin of error, for gory mathematical reasons Vanderkam goes into. But Vanderkam's larger point is that there is a better way to think about error margins in polls: given the poll's results, what are the odds that a candidate is ACTUALLY ahead?

"It’s easier to interpret the statement 'There’s an 86.5% chance that Obama is ahead' than a statement about margins of error," Vanderkam writes.

For the Bloomberg poll, for example, there is a 95.5 percent that Hillary Clinton is actually ahead given the results and margin of error. There's a 4.5 percent chance that sampling error means Donald Trump is actually ahead. (This makes sense given the prior paragraph. Statistics traditionally concludes a result is significant if there's a 95 percent chance it didn't occur by random chance; the chance Clinton is really ahead is just above 95 percent — and her six-point lead is just above 1.6 times the 3.6 percent margin of error, or 5.76 percentage points.)

That can be calculated using Bayesian statistics and an integral, using a function called the "regularized incomplete beta function". It's a little tricky and every time I try to calculate it, it takes me about 10 minutes to re-figure out how to do it.

So I've whipped up this simple R function that does the math for me — or for you. Load winnerodds.R, run it with some basic data about the poll, and it will tell you the odds that each candidate is winning given the poll.

The variables you'll need:

  • samplesize: The number of respondents in the poll.
  • cand1: The name of the first candidate. Defaults to "Candidate 1".
  • cand2: The name of the second candidate. Defaults to "Candidate 2".
  • cand1.pct: The percent of the vote the first candidate got in the poll, in decimal form (e.g, .49).
  • cand2.pct: The percent of the vote the second candidate got in the poll.

Here's how the script works for the Bloomberg poll discussed here:

> winnerodds(749,"Hillary Clinton","Donald Trump",.5,.44)
The poll showed Hillary Clinton with 50% of the vote and Donald Trump with 44% of the vote.
There is a 95.5% chance Hillary Clinton is winning based on this poll.
There is a 4.5% chance Donald Trump is winning based on this poll.

This code, and all code published on the Pioneer Press Github, is made available under the MIT License.