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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 ...

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

LR test vs. linear model: chi2(2) = 1154.73                Prob > chi2 = 0.0000

Note: LR test is conservative and provided only for reference.

. estat icc

Residual intraclass correlation

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

Note: Estimates are transformed only in the first equation to odds ratios.
Note: _cons estimates baseline odds (conditional on zero random effects).
LR test vs. logistic model: chi2(2) = 17.54               Prob > chi2 = 0.0002

Note: LR test is conservative and provided only for reference.

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.