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I'm a little confused regarding the intraclass correlation coefficient I am now reading "Face to Face interviews " by loosveldt 2008 . He pointed out on page 218 that the intraclass correlation coefficient could be negative . Screenshot of the page

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Could someone explain how the intraclass correlation coefficient could be negative ?

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Please note that it is the estimate that can be negative - not the intraclass correlation itself. A common way to obtain the estimate of the numerator is via the method of moments -- in simple one-way anova cases this involves $MS(between)-MS(within)$. This difference will turn out to be negative whenever the $F$ ratio for the "between" effect is less than $1$.

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  • $\begingroup$ Although I agree that the excerpt in the question is talking about the estimate being negative, I hope you're not implying here that the estimand must also be non-negative? $\endgroup$
    – Jake Westfall
    Commented Nov 4, 2015 at 5:08
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    $\begingroup$ @JakeWestfall - Exactly. The estimand -- intraclass correlation -- is always non-negative. $\endgroup$
    – Russ Lenth
    Commented Nov 4, 2015 at 15:00
  • $\begingroup$ But that's just not true. The lower bound of the ICC is -1/(m-1) where m is the number of times each unit is measured. It approaches being non-negative as m increases, but in real data it can certainly be negative. For small m it can even be substantially negative. Classic example is asking both members of married couples to estimate % of housework they typically complete: generally, the higher one's guess, the lower the other's guess, leading to strong negative ICC. Why do you say that the ICC must be non-negative? $\endgroup$
    – Jake Westfall
    Commented Nov 4, 2015 at 15:08
  • $\begingroup$ @JakeWestfall - I misread your earlier comment. Yes, I definitely did mean to imply that the true ICC must be nonnegative. It is a variance divided by the sum of two variances. Variances can't be negative. Your latest comment is about the estimate -- not the estimand. $\endgroup$
    – Russ Lenth
    Commented Nov 4, 2015 at 15:54
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    $\begingroup$ @JakeWestfall I'm appreciating your comment more and think it adds valuable insights. I had been thinking too narrowly in terms of the classical "model II" and the words "in real data" and distracted me from what you were really saying. $\endgroup$
    – Russ Lenth
    Commented Nov 6, 2015 at 1:32

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