Does the ICC require that my data are normally distributed?
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Solved – Intraclass correlation coefficient, non parametric data
intraclass-correlationnonparametric
Related Solutions
You need to work with a Generalized Linear Mixed-effects Model (GLMM). The repeated measures ANOVA is actually a special case of the Linear Mixed-effects Model (LMM), so GLMM:LMM as GLM:LM. From there, just recognize that the ICC is a descriptive statistic that assesses how distinctive the units are relative to the total spread of the data. The standard formula is:
$$
ICC = \frac{\sigma_{\bar{x_i}}^2}{\sigma_{\bar{x_i}}^2+\sigma_\varepsilon^2}
$$
In the context of a mixed-effects model, the distinctiveness of the individuals is the variance of the random effects. Since variances add, the total is the variance of the random effects plus the residual variance.
To illustrate, here is a simple example, coded in R:
library(lme4) # we'll use this package to fit the GLMM
set.seed(1189) # this makes the example exactly reproducible
N = 20 # there are 20 people in your study
reps = 3 # w/ 3 observation each
id = rep(1:N, each=reps) # this is the ID indicator
sub.lambda = rnorm(N, mean=1.6, sd=.3) # here we generate each unit's mean
sub.lambda = rep(sub.lambda, each=reps) # & copy them for each observation
y = rpois(reps*N, lambda=exp(sub.lambda)) # here we generate the counts
mod = glmer(y~(1|id), family=poisson) # this fits the Poisson GLMM
summary(mod) # here is the output:
# Generalized linear mixed model fit by maximum likelihood
# (Laplace Approximation) [glmerMod]
# Family: poisson ( log )
# ...
#
# Random effects:
# Groups Name Variance Std.Dev.
# id (Intercept) 0.1066 0.3265
# Number of obs: 60, groups: id, 20
#
# Fixed effects:
# Estimate Std. Error z value Pr(>|z|)
# (Intercept) 1.45853 0.09745 14.97 <2e-16 ***
## now we get the variance of the random effects, the residuals, & calculate the ICC:
v.RE = summary(mod)$varcor$id[1]; v.RE # 0.1066211
v.res = var(resid(mod)); v.res # 1.141968
ICC = v.RE/(v.RE+v.res); ICC # 0.08539324
You can easily get the ICC from the sjstats-package, which also offers bootstrapping. I would try bootstrapping to obtain se- and p-values (you can also take a look at this discussion in the r-sig-mailinglist for other approaches, but bootstrapping is one recommendation, two).
library(sjstats) # for ICC and bootstrapping
library(dplyr) # for the pipe
library(lme4)
# compute ICC
fit <- lmer(Reaction ~ Days + (Days | Subject), sleepstudy)
icc(fit)
> Linear mixed model
> Family: gaussian (identity)
> Formula: Reaction ~ Days + (Days | Subject)
>
> ICC (Subject): 0.483090
# compute SE of ICC
dummy <- sleepstudy %>%
# generate 100 bootstrap replicates of dataset
bootstrap(100) %>%
# run mixed effects regression on each bootstrap replicate
mutate(models = lapply(.$strap, function(x) {
lmer(Reaction ~ Days + (Days | Subject), data = x)
})) %>%
# compute ICC for each "bootstrapped" regression
mutate(icc = unlist(lapply(.$models, icc)))
# now compute SE and p-values for the bootstrapped ICC
boot_se(dummy, icc)
> term std.err
> 1 icc 0.1024704
boot_p(dummy, icc)
> term p.value
> 1 icc 1.936526e-08
The above code takes the
sleepstudydataset and creates 100 bootstrap-samples, which are saved as "list-variable" namedstrap, in a new data frame.This list-variable (i.e. the data frame with this variable), containing the 100 bootstrap-samples, is then "piped" into the
lapply-command to run 100lmer-commands (which are saved in another list-variable namedmodels, seemutate-command).This list-variable is then used to compute the ICC with the
sjstats::iccfunction, so you have 100 ICC-values (for all models applied to the 100 bootstrap-samples) in the variable namedicc.
Now boot_se and boot_p compute the standard error and p-value of any bootstrapped values, in this case the ICC-values.
Best Answer
Depends on the method - if you're using the ANOVA (general linear) method, then yes. But you can also calculate the ICC using generalized linear methods (I'm not sure about non-parametric methods - anyone else?).
Nakagawa and Schielzeth wrote the user-friendly rptR package for R, which uses general or generalized linear mixed effects modelling (for normal, binomial and count data) to calculate repeatability within classes using ANOVA, REML or MCMC methods. I think this is what you want to do?
http://rptr.r-forge.r-project.org/