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I'm looking to check my logic here.

Say you measure a quantity in group A, and find the mean is 2 and your 95% confidence interval ranges from 1 to 3. Then you measure the same quantity in group B and find a mean of 4 with a 95% confidence interval that ranges from 3.5 to 4.5. Assuming that A & B are independent, what is the 95% confidence interval for the difference between the groups? Presumably you can compute this using standard t-statistics, but I'd like to know if it's also possible to compute an estimate based on the CI's alone.

I reason that the lower bound of the CI of the difference should be the minimum credible difference between A & B; that is, the lower bound of the interval for B (3.5) minus upper bound of the interval for A (3), which yields a lower bound for the difference of 0.5. Similarly, the upper bound of the CI of the difference should be the maximum credible difference between A & B; that is, the upper bound of the interval for B (4.5) minus lower bound of the interval for A (1), which yields a lower bound for the difference of 3.5. This reasoning thus yields a confidence interval for the difference that ranges from 0.5 to 3.5.

Does that make sense, or is this a case where logic and statistics diverge?

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1 Answer

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No, you can't compute a CI for the difference that way I'm afraid, for the same reason you can't use whether the CIs overlap to judge the statistical significance of the difference. See, for example,

"On Judging the Significance of Differences by Examining the Overlap Between Confidence Intervals" Nathaniel Schenker, Jane F Gentleman. The American Statistician. August 1, 2001, 55(3): 182-186. doi:10.1198/000313001317097960. http://pubs.amstat.org/doi/abs/10.1198/000313001317097960

or:

Overlapping confidence intervals or standard error intervals: What do they mean in terms of statistical significance? Mark E. Payton, Matthew H. Greenstone, and Nathaniel Schenker. Journal of Insect Science 2003; 3: 34. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC524673/

The correct procedure requires you also know the sample sizes of both groups. You can then back-compute the two standard deviations from the CIs and use those to conduct a standard two-sample t-test, or to calculate a standard error of the difference and hence a CI for the difference.

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Ah, yes, I knew that overlapping CI's didn't necessitate zero difference, but hadn't connected that to this example, where clearly the above procedure would always compute difference "CI" that includes zero whenever the individual CIs overlap. Thanks! – Mike Lawrence Aug 3 '10 at 12:39
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A useful fact given in these references is that non-overlapping 84% confidence intervals do approximate a 95% level test. – Aniko Aug 3 '10 at 14:08
May I ask what your intention is in offering a bounty on someone else's question when you have already provided a good answer? – whuber♦ Oct 30 '10 at 21:02
I was taking an experimental approach to a question i've just discovered is answered at meta.stackoverflow.com/questions/16065/…, namely 'Can I award a bounty to my own answer?' – onestop Oct 31 '10 at 9:06
.. but i forgot i'd have to wait 24 hours before i could award the bounty, during which time the question would be 'featured'. Sorry about that! Now i've learnt of the existence of the Community FAQ for the original trinity (or triumvirate, or whatever it's called) of SE sites i won't need to experiment again. meta.stackoverflow.com/questions/7931 – onestop Oct 31 '10 at 16:02

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