# Intraclass Correlation Coefficient in mixed model with random slopes

I have the following model m_plot fitted with lme4::lmer with crossed random effects for participants (lfdn) and items (content):

Random effects:
Groups   Name             Variance Std.Dev. Corr
lfdn     (Intercept)      172.173  13.121
role1             62.351   7.896    0.03
inference1        24.640   4.964    0.08 -0.30
inference2        52.366   7.236   -0.05  0.17 -0.83
inference3        21.295   4.615   -0.03  0.22  0.86 -0.77
content  (Intercept)       23.872   4.886
role1              2.497   1.580   -1.00
inference1        18.929   4.351    0.52 -0.52
inference2        14.716   3.836   -0.16  0.16 -0.08
inference3        17.782   4.217   -0.17  0.17  0.25 -0.79
role1:inference1   9.041   3.007    0.10 -0.10 -0.10 -0.21  0.16
role1:inference2   5.968   2.443   -0.60  0.60 -0.11  0.78 -0.48 -0.50
role1:inference3   4.420   2.102    0.30 -0.30  0.05 -0.97  0.71  0.37 -0.90
Residual                  553.987  23.537
Number of obs: 3480, groups:  lfdn, 435 content, 20


I want to know the Intraclass Correlation Coefficients (ICC) for participants and items. Thanks to this great answer I in principle know how to get the ICC for my model. However, I am unsure on whether or not to include the random slopes or not:

vars <- lapply(summary(m_plot)$varcor, diag) resid_var <- attr(summary(m_plot)$varcor, "sc")^2
total_var <- sum(sapply(vars, sum), resid_var)

# with random slopes
sapply(vars, sum)/total_var
##       lfdn    content
## 0.33822396 0.09880349

# only random intercepts:
sapply(vars, function(x) x[1]) / total_var
##   lfdn.(Intercept) content.(Intercept)
##         0.17496587          0.02425948


What is the appropriate measure for the correlation between two responses from the same participant respective to the same item?

• Merlo et al 2005 "A brief conceptual tutorial on multilevel analysis in social epidemiology: investigating contextual phenomena in different groups of people" might be a useful reference. – N Brouwer Nov 13 '14 at 15:26
• @Henrik did you ever find an answer to this question? I'm interested as well. – Patrick S. Forscher Dec 8 '17 at 19:22
• @PatrickS.Forscher As far as I understand, ICC does not make sense with random slopes. I have learned this from Jake Westfall. – Henrik Dec 10 '17 at 22:20
• Got a link to a reading relevant reading by chance? – Patrick S. Forscher Dec 11 '17 at 3:23
• @PatrickS.Forscher As you can see, Jake Westfall now provided a great answer. – Henrik Dec 12 '17 at 15:31