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I have two (classically) estimated coefficients from 2 different models, $\alpha_{m1}$ and $\alpha_{m2}$, say, with 95% confidence intervals. I maintain that model m2 is mis-specified and that the parameters are statistically distinct.

However a referee argues that since the central value of $\alpha_{m2}$ falls within the CIs of $\alpha_{m1}$ (albeit only marginally) that the parameters are indistinct. My understanding is that parameters can indeed have overlapping CIs but still be statistically distinct.

What's the best way to counter the referee's argument?

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    $\begingroup$ See answer&links at stats.stackexchange.com/questions/423609/…, stats.stackexchange.com/questions/99936/…, $\endgroup$ Dec 27, 2022 at 12:09
  • $\begingroup$ The referee sounds correct: their point is that these CIs overlap too much. After all, if a CI were to overlap a true value you would conclude there is no significant difference; instead of comparing a CI to a true value, you are comparing it to an uncertain estimate (I presume that's what you mean by "central value.") A fortiori, there is even less evidence of a significant difference. If there is something special about your situation leading you to believe this difference is significant, then apply an appropriate significance test to the difference: don't just eyeball it. $\endgroup$
    – whuber
    Dec 27, 2022 at 21:10

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