intraclass coefficient (ICC) with random coefficients interpretation

[Don Vicendese]

I have used the following mixed effects model with random coefficients

 xtmixed outcome i.cat_var1 || L3: i.cat_var2 || L2: , reml difficult covariance(unstructured)

Stata gives the following output regarding the random effects parameters

Random-effects Parameters	Estimate	Std.Err.	95% CI
L3:Independent
sd(1.cat_var2)	3.03098	0.75935	1.854949	4.95261
sd(2.cat_var2)	4.335898	1.053712	2.692896	6.981335
sd(3.cat_var2)	5.246749	1.26292	3.27342	8.40967
sd(4.cat_var2)	6.66362	1.591003	4.173282	10.64003
sd(_cons)	3.593067	0.883755	2.218722	5.818724
L2:Identity
sd(_cons)	0.9140715	0.183185	0.617155	1.353835
sd(Residual)	0.9107009	0.074042	0.776554	1.068022

LR test vs. linear model: chi2(6) = 207.42 Prob > chi2 = 0.0000


All fine so far. I then get the following output when I run

 estat icc

Conditional intraclass correlation
Level	ICC	Std.Err.	95% CI
L3	0.869175	0.0620235	0.6952148	0.950863
L2|L3	0.943005	0.026437	0.8631898	0.977472
Note: ICC is conditional on zero values of random-effects covariates.

It is clear that the L3 ICC = 3.565361² / (3.565361² + 1.039124² + .9129913²)

and L2|L3 ICC = (3.565361² + 1.039124²) / (3.565361² + 1.039124² + .9129913²)

I presume that "Note: ICC is conditional on zero values of random-effects covariates" means that
the variances for the random coefficients are set to zero.

Is that correct?

If it is correct, why are the variances for the random coefficients not taken into account in calculating the ICCs?

How do we interpret and report the "conditional" aspect of the output in regard to the ICCs?

Can Stata re calculate ICC taking into account the variances for the random coefficients and if yes, how do we interpret and report them?

Thanks in advance,

Don

[Don Vicendese]

My apologies - I posted the incorrect results tables and calculations - but the questions stand just the same.

Here are the correct tables and calculations.

Random-effects Parameters	Estimate	Std.Err.	95% CI
L3:Independent
sd(1.cat_var2)	2.848369	0.684152	1.778876	4.560862
sd(2.cat_var2)	4.101485	0.950337	2.604431	6.459059
sd(3.cat_var2)	5.027052	1.151212	3.209115	7.874833
sd(4.cat_var2)	5.978903	1.43331	3.737357	9.564854
sd(_cons)	3.565002	0.878381	2.199542	5.77813
L2:Identity
sd(_cons)	0.9203229	0.185778	0.619606	1.366989
sd(Residual)	0.9231235	0.076697	0.784402	1.086379

Conditional intraclass correlation
Level	ICC	Std.Err.	95% CI
L3	0.882072	0.056373	0.7211176	0.955824
L2|L3	0.940857	0.027735	0.8569192	0.976881

It is clear that the L3 ICC = (3.5650022² ) / (3.5650022² + 0.92032292² + 0.92312352² ) = 0.882072

and L2|L3 ICC = (3.5650022² + 0.92032292² ) / (3.5650022² + 0.92032292² + 0.92312352² ) = 0.940857


Don