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## Multilevel generalized linear models

Stata fits multilevel mixed-effects generalized linear models (GLMs) with meglm. GLMs for cross-sectional data have been a workhorse of statistics because of their flexibility and ease of use. Stata’s xtgee command extends GLMs to the use of longitudinal/panel data by the method of generalized estimating equations.

Now you can use meglm to fit GLMs to hierarchical multilevel datasets with normally distributed random effects. Seven distributions for the response variable are supported (Gaussian, Bernoulli, binomial, gamma, negative binomial, ordinal, and Poisson); and five link functions are possible (identity, log, logit, probit, and complementary log-log).

Let's fit a three-level model.

We have student-level data, where students are nested in classes, and classes are nested in schools. Our dependent variable thk is an ordered categorical variable that takes on the values 1, 2, 3, or 4; and we have three explanatory variables: prethk, cc, and tv. We will treat prethk as continuous. cc and tv are binary, and we want to include their interaction the model. Let's fit an ordered logit model:

. webuse tvsfpors

. meglm thk prethk cc##tv || school: || class:, family(ordinal) link(logit)

Fitting fixed-effects model:

Iteration 0:   log likelihood =  -2212.775
Iteration 1:   log likelihood =  -2125.509
Iteration 2:   log likelihood = -2125.1034
Iteration 3:   log likelihood = -2125.1032

Refining starting values:

Grid node 0:   log likelihood = -2152.1514

Fitting full model:

Iteration 0:   log likelihood = -2152.1514  (not concave)
Iteration 1:   log likelihood = -2125.9213  (not concave)
Iteration 2:   log likelihood = -2120.1861
Iteration 3:   log likelihood = -2115.6177
Iteration 4:   log likelihood = -2114.5896
Iteration 5:   log likelihood = -2114.5881
Iteration 6:   log likelihood = -2114.5881

Mixed-effects GLM                               Number of obs      =     1,600
Family: 	      ordinal

No. of       Observations per Group
Group Variable     Groups    Minimum    Average    Maximum

school         28         18       57.1        137
class        135          1       11.9         28

Integration method: mvaghermite                 Integration points =         7

Wald chi2(4)       =    124.39
Log likelihood = -2114.5881                     Prob > chi2        =    0.0000

thk        Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]

prethk     .4085273    .039616    10.31   0.000     .3308814    .4861731
1.cc     .8844369   .2099124     4.21   0.000     .4730161    1.295858
1.tv      .236448   .2049065     1.15   0.249    -.1651614    .6380575

cc#tv
1 1     -.3717699   .2958887    -1.26   0.209     -.951701    .2081612

/cut1    -.0959459   .1688988    -0.57   0.570    -.4269815    .2350896
/cut2     1.177478   .1704946     6.91   0.000     .8433151    1.511642
/cut3     2.383672   .1786736    13.34   0.000     2.033478    2.733865

school
var(_cons)    .0448735   .0425387                      .0069997    .2876749

school>class
var(_cons)    .1482157   .0637521                       .063792    .3443674

LR test vs. ologit regression: chi2(2) = 21.03            Prob > chi2 = 0.0000

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


Our model has two random-effects equations, separated by ||. Our first is a random intercept at the school level, and the second is a random intercept at the class level. The order we listed them matters: class comes after school, meaning that classes are nested within schools. Using the same logic, we could include more levels of nesting. meglm also allows crossed-effects models.

The output table includes the fixed-effect portion of our model, the estimated cutpoints (because this is an ordered logit model), and the estimated variance components.

This model can alternatively be fit with meologit, which is a convenient use for meglm with an ordinal family and a logit link. See the example fit with meologit.

Also watch A tour of multilevel GLMs.