I keep pointing people to this article by Carl Morris so I thought I’d post it. The article is really hard to find because it has no title: it appeared in the Journal of the American Statistical Association as a discussion of a couple of other papers.
All 3 scenarios have the same p-value. And, given the way that statistics is typically taught (slogans about “statistical significance” vs. “practical significance”), one might think that scenario (a) is the best for the candidate, as the estimate is the largest in scenario A above.
Think about it: a higher estimate is good for Mr. Allen, and a smaller p-value is good for Mr. Allen. So it stands to reason that, with equal p-values, the larger estimate will be the better news.
And look at the 95% confidence intervals:
A: (.560, .940)
B: (.506, .644)
C: (.501, .545)
A looks best, no? The decision doesn’t even seem close.
But, no, as Carl explains, scenario C is the most encouraging for Mr. Allen.
Regular readers of this blog will recognize scenario A as the noisy-data situation where statistical significance tells you very little:
It might get you a publication in Psychological Science or the tabloids but that’s about it.