Get ICC for levels in multilevel models

getICCfromMLM(modelOutput, percent = TRUE)

## Arguments

modelOutput output from nlme::lme(), lme4::lmer() or lmerTest::lmer() logical determining whether or to output percentage of variance as well as ICC

## Value

a tibble containing the ICCs for each level in the MLM model output and, if percent == TRUE, the percentages of variance too

## Background

6.iii.21: Do not use with nlme::lme(). This works for lme4::lmer() or lmerTest::lmer() but I cannot sort out issues about results in out of nlme::lme() differing from those of lme4::lmer()

The ICC gives the proportion of all variance on dependent variable in Multi-Level Models (MLMs). I my field multilevel models typically have levels of measurements points for individual participants and participants can be nested within therapist, service, organisation or perhaps area. Crucially the MLM handles the non-independence of observations within the "level". Typically, again, there may be fixed predictors on which all participants have a value, e.g. gender, age, social deprivation, previous therapy. Apart from handling non-independence of observations so that Standard Errors of estimates, or (sometimes, not with lme4::lmer()) p-values are robust to the non-independence, the other huge power of MLMs is that they can have random effects, so that instead of assuming that all participants have the same starting score on a measure they can be allowed to have different starting scores which the MLM algorithm will estimate. Perhaps even more importantly, they can fit a model in which it is not assumed that every participant's score will change at the same rate but that slope of score against time can be another random variable.

ICCs are important where you have multiple levels in your model and help you decide how much variance is shared within the level. A good way to think about the ICC, as described in help for the specr::icc_specs() help is that the ICC is the mean correlation across any possible pairs of scores on the dependent variable within the level. Typically, fitting MLMs to therapy change score trajectories the participant level will have a much higher ICC than, say, the service level. However, it may be clear that there is similarity between participants within services with a non-trivial ICC for the service level.

The specr::icc_specs() function, which I leant on heavily in writing my own function, only works for MLM output from lme4::lmer() so all this function adds really is that it handles output from nlme::lme() which actually structures its output substantially differently from that of lme4::lmer().

## Cautionary note

This is really not my expert area and I think there are quite varied views of what ICCs with multiple fixed predictors mean, and even more varied views even about whether ICCs can be meaningful for MLMs with random predictors so this function is only designed for intercept only MLMs, i.e. with no fixed predictors nor random predictors. For typical therapy change data I think this is probably all we need: it tells us about the partitioning of scores within participants and within whatever other levels you have: therapists, services etc.

On cursory testing the function gives the same results as specr::icc_specs() for two level and three level MLMs from lme4::lmer() and the same results when the same data are put through analysis using the same models in nlme::lme(). As ever, there is no warranty on this function!

## History/development log

Started before 5.iv.21

Chris Evans

## Examples

if (FALSE) {
getICCfromMLM(model2nlmer)
getICCfromMLM(model2lme4)
### check against specr::icc_specs()
specr::icc_specs(model2lme4)
}