Can delta method be applied for determining the between subject variability (random variance) of a function of X? Say, for example, I square root transformed X such that it follows normal distribution, fitted a linear mixed effects model and obtained between subject variability (BSV) of sqrt(X). How do I now calculate the BSV of X?
 A: I do not think this is possible. The delta method is most often an approximation for calculating the variance of transformed quantities when you know the formula for doing the transformation. So you have an X (and know its variance), you transform it using an equation to Xtrans. What is then the variance of Xtrans?
In your situation, the original quantity is the between-subject variability (BSV) calculated using $\sqrt{X}$. Your question is how does one transform this quantity to the scale of $X$? The delta method would apply if you already had some transformation for BSV (and had a variance for BSV) to make it BSVTrans and wanted to calculate the variance of BSVTrans.

EDIT in response to comment
If you really want the ICC on the scale of X, then I think you should run your mixed model on the scale of X. If you use a transformation, it is akin to claiming that it is better/more natural to study the relationships on the transformed scale. If that is true (and I believe it is what using a transformation implies), then you should be okay with this ICC you have already calculated. Hopefully, the field you are in agrees with you and everyone else uses the square root scale. If transformations are arbitrary within each application to promote model-data fit, then the contribute-to-the-literature role of the ICC is defeated.
EDIT 2
At the end of the day, the ICC is just an R-squared, so:


*

*You could use your current random-intercept only mixed effects model to calculate the predictions for all your cases

*Then transform these predictions to the original scale by squaring them

*Calculate the correlation between these predictions and your original untransformed outcome

*Square this correlation to obtain an R-squared


If I read this, I would find it dubious. I generally find transformations dubious, especially when they are done simply to obtain a "normal distribution".
