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Let's assume that we are investigating how tobacco smoking is associated with incident lung cancer in a population. In the full population, the true relative risk of lung cancer associated with tobacco smoking is 2.

Next, we collect 100 random samples from the population. For each sample, we calculate 95% confidence intervals for the relative risk of lung cancer associated with tobacco smoking.

What can we expect from the distribution of confidence intervals here:

A. 95/100 confidence intervals will include the true value RR=2
   5/100 confidence intervals will not include the true value RR=2
A2. As "A". Additionally, confidence intervals can freely include RR=1
B. 95/100 confidence intervals will not include RR=1
   5/100 confidence intervals will include RR=1

I also provided a simplified picture that presents 20 (100 would be too much) confidence intervals as vertical bars.

enter image description here

Edit: This post seemed to reach the consensus that "A" was true, but calculated means was used as an example instead of relative risk measures.

CarlAH
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    Any suggestions on how to make the question more interesting would be appreciated. I understand that there are many posts about confidence intervals, but it seems worthwhile to clarify how CIs of RR should be interpreted in particular – CarlAH Oct 03 '17 at 23:44

1 Answers1

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The answer is A2. Confidence intervals are agnostic to the "null value" of a parameter. They may freely include it or not. There may be some confusion stemming from the idea that a 95% CI will cause you to incorrectly reject H0 5% of the time; but this is only true if the true RR is the null RR (i.e., 1). This is not the case here because it was stated as a known premise that H0 is false and the true RR is not the null RR. If the true RR was 1.05, you might expect a 95% CI to include the null RR almost every time.

Noah
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