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I am trying to interpret output from a 3 level HLM (city, school, individual). Would the following interpretation be accurate?

.0143922 is the variance explained between cities .006966 is the variance explained between schools .7206755 is the variance within schools/between subjects?

Output from Stata xtmixed:



------------------------------------------------------------------------------
 Random-effects Parameters  |   Estimate   Std. Err.     [95% Conf. Interval]
-----------------------------+------------------------------------------------
city_id: Identity            |
                 var(_cons) |   .0143922   .0088338      .0043218     .047928
-----------------------------+------------------------------------------------
school_id: Identity          |
                 var(_cons) |    .006966   .0026191      .0033338    .0145552
-----------------------------+------------------------------------------------
              var(Residual) |   .7206755   .0074094      .7062988    .7353449
------------------------------------------------------------------------------
LR test vs. linear model: chi2(2) = 250.45                Prob > chi2 = 0.0000
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You are on the right track, gzhang. Think of these variances as representing dispersion of the outcome values at a given level around some mean value. In a three level model, at level 1, you have individual outcome dispersion around a school mean (residual), at level 2, dispersion of school outcome values around a city mean, and then at level 3 you have dispersion of city outcome values around the overall or grand mean.

Accordingly, the variance estimates in the table are only explaining dispersion of outcome values around these three means. And the way that you have referred to them as within or between is also acceptable but can be confusing to those not used to these terms. Here is how I would label the variances, pulling from your own language:

.0143922 is the variance between cities, .006966 is the variance between schools within the same city, and .7206755 is the variance between subjects within the same school

The other thing to consider is whether these are conditional variances coming from a model in which you have included predictors or whether these are unconditional variances from a model without predictors. Often people calculate intraclass correlation coefficients from a model without predictors to get a sense of how much relative variation in the outcome is sitting at these different levels. In Stata, you can get ICCs using estat icc after your model.

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