[R-sig-ME] ICC from lmer with back transform

[Pierre de Villemereuil]

Sorry, I was a bit confused during my previous response. Thinking now more clearly about this.

- If you are using a log-transform in the formula as per your first email, then Eq. 35 is actually much simpler and does not depend on Eq. 36, it becomes:
ICC = (exp(var_effect) - 1) / (exp(var_tot) -1)
where var_effect is the variance of the effect you want to measure the ICC of and var_tot is the sum of the random variance components. Note that this ICC will measure how much of the variance you can predict knowing the effect, but it does not work as a "variance decomposition" framework, because the ICCs from all variance components (including "residual") will not sum up to 1 as they do for a classical LMM.

- Using a log-link as I suggested actually makes everything more complicated and I am still unsure what would the impact of Jensen's equality be. I'll need to think more about this in terms of the impact on these back-transformations... unless someone already has a clue on this mailing list?

Cheers,
Pierre.

Le vendredi 30 octobre 2020, 11:35:57 CET Pierre de Villemereuil a écrit :
> Hi,
> 
> I am not sure I understand your calculation proposal, but if you want to compute the ICC from the original scale before a log-transformation, you will need to also account for the intercept and the formula is a bit more complex. You can see equations 35 and 36 of:
> Nakagawa, S. & Schielzeth, H. Repeatability for Gaussian and non Gaussian data: a practical guide for biologists. Biological Reviews 85, 935–956 (2010).
> 
> Note that, due to Jensen's inequality, I believe that, to use these equations, you'd need your to use a log-link rather than a log-transform in the formula (although in practice, the difference might be subtle). Something like:
> 
> model <- lmer(VARIABLE ~ 1 +(1|Side)+(1|Asessor)+(1|ID), data = data, family = gaussian(link = "log"), REML=FALSE)
> 
> Hope this helps,
> Pierre
> 
> Le vendredi 30 octobre 2020, 11:08:30 CET fabien leboeuf a écrit :
> > Hello
> > what a nice idea to have a forum dedidated to lmer question :-). i came
> > acros it from cross-validated.
> > 
> > Here is my question:
> > 
> > I want to calculate the ICC from a mixed model coded with lmer as follow.
> > 
> > model <- lmer(formula = log(VARIABLE) ~ 1
> > +(1|Side)+(1|Asessor)+(1|ID), data = data,REML=FALSE)
> > 
> > am i wrong if i compute the iCC from back transform , like that
> > 
> > vc <- as.data.frame((VarCorr(model)))
> > ICC_log = sum(exp(vc$vcov[1]),exp(vc$vcov[3]))/(sum(exp(vc$vcov)))
> > 
> > I appreciate any replies.
> > 
> > Fabien
> > 
> > 
> 
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