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I am confused about how to determine statistical significance in election polls. I was taught that overlapping confidence intervals do not necessarily imply a statistically insignificant difference between two values. I was Googling around for an answer, and this article confirming it.

However, this article seems to give a different interpretation. In particular, I'm concerned about a simple yes/no poll question (or Candidate A / Candidate B), and not in the case where there is a 3rd option. The claim the second article makes is that, in this simplified case, one can simply add the confidence intervals and if they overlap, then the difference is not statistically significant i.e Candidate A at 53% and Candidate B at 47% with a +/- 3% MOE is not statistically significant.

The two articles seem to be at odds and I would like to understand why. Is it because, in the case of yes/no polling, the two values are perfectly negatively correlated?

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Note that the first article you mentioned is about two "independent" populations. In polling, only a "single" population is used, and by so the sample proportion values of candidate A and B are related. While in two independent population tests as suggested in the first article, the sample means and generally the two populations are assumed to be fully independent. Probably that is the reason that comparing the CIs for the yes/no question for a single population (as in presidential voting) works fine.

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For a yes/no question, the hypothesis of the two options having the same probability is the same as a fixed hypothesis $H_0: p=0.5$. If the a confidence interval for the "yes" option covers 0.5, then the formal test at the same level won't reject the null. By the virtue of the "no" answer being a perfect complement of the "yes" answer, you would have the confidence interval for "no" answer be a mirror reflection of the CI for the "yes" answer around 0.5 (or, to be more precise, anything that can be stated for the probability of the "yes" answer can be stated for 1-probability of the "no" answer). So if the CI for the "yes" answer covers 0.5, then the CI for the "no" answer covers 0.5, as well, and the two CIs overlap. The converse is that if they don't overlap, then neither covers 0.5, the probability of either answer is significantly different from 0.5, and the probabilities of the two answers significantly differ from one another. Of course, this argument hinges critically on you being an American and not being able to imagine elections with more than two candidates.

I could not really see where the second article says that you should "add the confidence intervals" -- I am not even sure I would understand how to perform such an operation. In general, I am opposed to shortcuts, as they get abused. It could make a cute methodological paper, but generally the shortcuts create more confusion than solve the computational problems they are supposed to solve (non-existent anymore, as nobody sits with the hand arithmometers cranking the poll results).

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  • $\begingroup$ The second article makes the argument that if % for B - 2 x MoE $\le$ % for A then the difference is not statistically significant at the bottom of page 2 and top of page 3. $\endgroup$ – Dan Oct 14 '12 at 13:56

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