ICC (Intra-class correlation coefficient) vs "9% of this variation in mortality was attributable solely to the surgeon."

[Christoph Schnelle]

I am trying to find out the relationship between a. the ICC for surgeons and the b. the variation due to surgeons.
In Udyavar,2018 (The impact of individual physicians on outcomes after trauma: is it the system or the surgeon?") both the ICC ("Surgeons with higher mortality rates were not clustered at specific hospitals, as the intraclass correlation for surgeon level mortality rates was 0.02") and the quote in the subject above were given. I am doing a systematic review and wonder how the ICC and this particular variation are related (the relationship is not that the ICC is the square of the variation).

The reason is that some papers list the variation due to the surgeon while other papers show the ICC (Intra-class correlation coefficient).

(The ICC is important as even a small ICC can have a substantial design effect - if you cluster a randomized controlled trial by practitioner - surgeon for example - you will need substantially more patients to gain sufficient statistical power than if the ICC is nil).

Does anybody know how the ICC for a practitioner and the variation due to a practitioner are related?

[daniel klein]

In general, the (one-way random-effects) ICC is defined as \(ICC = \frac{{\sigma_r}^2}{{\sigma_r}^2+{\sigma_e}^2}\), where \({\sigma_r}^2\) is the variation due to raters (or surgeons) and \({\sigma_e}^2\) is the unexplained variation.The relation between variation due to raters (surgeons) and the ICC will depend on the specific ICC (e.g., one-way, two-way, mixed), which, in turn, varies in how the nominator and the denominator are defined. Some definitions of the ICC are documented in the Methods and formulas section of [R] icc.

[Christoph Schnelle]

Thank you, Daniel. I am looking for a relationship between the one-way random-effects ICC and the patient outcome variation due to the surgeon in a multilevel (two levels or three levels with hospital, surgeon and patient) model.