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If the answer is "it depends", what does it depend on? Does convergence depend on the ratio of predictor variables to sample size, or the size of $R^2$, or something else?

I am mainly interested in CIs on Unadjusted $R^2$, but would also be interested to hear the answer to this question in relation to CIs on Adjusted $R^2$.

Primarily I'm interested in finding out the conditions under which convergence happens or doesn't happen, but I would also appreciated an explanation of why.

EDIT: I am at present using the calculator made here: http://www.danielsoper.com/statcalc3/calc.aspx?id=28

CI formula

Standard error

I'm interested to know whether this formula gets close to nominal coverage with larger sample size (or stays the same, or gets further away from nominal coverage). I'm also interested to know if this holds true for all $R^2$ CIs or just ones calculated using this formula.

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  • $\begingroup$ you have changed your original question and provided an additional link. Given the link, the answer to your question seems trivial (maybe I'm missing something): just replace the '30' there by increasingly larger numbers (3000,30000,....) until you see the pattern on the width of the CI's (hints: ceteris parebus they shrink). $\endgroup$
    – user603
    Apr 24, 2014 at 12:38

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I think for a general answer, we would need to know what sort of confidence interval you are using. However, let me try to cover some general aspects:

  • When using an exact confidence interval then the coverage is always 95%, regardless of sample size if all assumptions of the method are fulfilled. The confidence intervals just become more narrow as the sample size increases
  • If you are using a confidence interval based on asymptomatic theory, it will get closer to the nominal coverage as sample size increases. Again, this is if all the requirements are fulfilled
  • In practice, you usually have some bias caused by your sampling method, experimental design or using a method which has requirements that are not really fulfilled. Many of these sources cause bias that does is small but does not decrease with sample size. In this case, tthe coverage of the confidence interval will actually decrease when your sample size gets very large
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  • $\begingroup$ Edited my post to attempt to take into account the fact that it may depend on the method of calculating the CI. $\endgroup$ Apr 23, 2014 at 8:07

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