Stata’s estat icc command is a postestimation command that can be used after linear, logistic, or probit random-effects models. It estimates intraclass correlations for multilevel models.
We fit a three-level mixed model for gross state product using mixed. Fixed-effects covariates include the state unemployment rate and different categories of public capital stock: hwy, water, and other. Random intercepts are present at both the region and state levels. Seventeen years of annual data are used. We use estat icc to estimate the intraclass correlations for this model.
. webuse productivity
(Public Capital Productivity)
. mixed gsp private emp hwy water other unemp || region: || state:
Performing EM optimization ...
Performing gradient-based optimization:
Iteration 0: log likelihood = 1430.5017
Iteration 1: log likelihood = 1430.5017
Computing standard errors ...
Mixed-effects ML regression Number of obs = 816
Grouping information
No. of Observations per group Group variable groups Minimum Average Maximum region 9 51 90.7 136 state 48 17 17.0 17
Wald chi2(6) = 18829.06
Log likelihood = 1430.5017 Prob > chi2 = 0.0000
| gsp | Coefficient Std. err. z P>|z| [95% conf. interval] | |
| private | .2671484 .0212591 12.57 0.000 .2254814 .3088154 | |
| emp | .754072 .0261868 28.80 0.000 .7027468 .8053973 | |
| hwy | .0709767 .023041 3.08 0.002 .0258172 .1161363 | |
| water | .0761187 .0139248 5.47 0.000 .0488266 .1034109 | |
| other | -.0999955 .0169366 -5.90 0.000 -.1331906 -.0668004 | |
| unemp | -.0058983 .0009031 -6.53 0.000 -.0076684 -.0041282 | |
| _cons | 2.128823 .1543854 13.79 0.000 1.826233 2.431413 | |
| Random-effects parameters | Estimate Std. err. [95% conf. interval] | |
| region: Identity | ||
| var(_cons) | .0014506 .0012995 .0002506 .0083957 | |
| state: Identity | ||
| var(_cons) | .0062757 .0014871 .0039442 .0099855 | |
| var(Residual) | .0013461 .0000689 .0012176 .0014882 | |
| Level | ICC Std. err. [95% conf. interval] | |
| region | .159893 .127627 .0287143 .5506202 | |
| state|region | .8516265 .0301733 .7823466 .9016272 | |
estat icc reports two intraclass correlations for this three-level nested model. The first is the level-3 intraclass correlation at the region level, the correlation between productivity years in the same region. The second is the level-2 intraclass correlation at the state-within-region level, the correlation between productivity years in the same state and region.
Conditional on the fixed-effects covariates, we find that annual productivity is only slightly correlated within the same region, but it is highly correlated within the same state and region. We estimate that state and region random effects compose approximately 85% of the total residual variance.
Now we fit a three-level logistic model for successful completion of the Tower of London computerized task. The variable group is used to classify individuals as controls (1), relatives of a schizophrenic (2), or schizophrenic (3). The difficulty level of the task and separate indicators for the different values of group are fixed-effect covariates. Random intercepts are present at both the family and subject levels.
. webuse towerlondon
(Tower of London data)
. melogit dtlm difficulty i.group || family: || subject:, or nolog
Mixed-effects logistic regression Number of obs = 677
Grouping information
No. of Observations per group Group variable groups Minimum Average Maximum family 118 2 5.7 27 subject 226 2 3.0 3
Integration method: mvaghermite Integration points = 7
Wald chi2(3) = 74.90
Log likelihood = -305.12041 Prob > chi2 = 0.0000
| dtlm | Odds ratio Std. err. z P>|z| [95% conf. interval] | |
| difficulty | .1923372 .037161 -8.53 0.000 .1317057 .2808806 | |
| group | ||
| 2 | .7798263 .2763763 -0.70 0.483 .3893369 1.561961 | |
| 3 | .3491318 .13965 -2.63 0.009 .15941 .764651 | |
| _cons | .226307 .0644625 -5.22 0.000 .1294902 .3955112 | |
| family | ||
| var(_cons) | .5692105 .5215654 .0944757 3.429459 | |
| family> | ||
| subject | ||
| var(_cons) | 1.137917 .6854853 .3494165 3.705762 | |
We use estat icc to estimate the intraclass correlations for this model.
. estat icc Residual intraclass correlation
| Level | ICC Std. err. [95% conf. interval] | |
| family | .1139105 .0997727 .0181851 .4715289 | |
| subject|family | .3416307 .0889471 .192923 .5297291 | |
estat icc reports two intraclass correlations for this three-level nested model. The first is the level-3 intraclass correlation at the family level, the correlation between latent measurements of the cognitive ability in the same family. The second is the level-2 intraclass correlation at the subject-within-family level, the correlation between the latent measurements of cognitive ability in the same subject and family.
There is not a strong correlation between individual realizations of the latent response, even within the same subject.
