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Multilevel probit models


Stata allows you to fit multilevel mixed-effects probit models with meprobit. A multilevel mixed-effects probit model is an example of a multilevel mixed-effects generalized linear model (GLM). You can fit the latter in Stata using meglm.

Let's fit a crossed-effects probit model. A crossed-effects model is a multilevel model in which the levels of random effects are not nested. We investigate the extent to which two salamander populations, whiteside and roughbutt, cross-breed. We label whiteside males wsm, whiteside females wsf, roughbutt males rbm, and roughbutt females rbf. Our dependent variable y is coded 1 if there was a successful mating and 0 otherwise. Let's fit our model:

. webuse salamander

. meprobit y wsm##wsf || _all: R.male || female:

note: crossed random-effects model specified; option intmethod(laplace) implied

Fitting fixed-effects model:

Iteration 0:   log likelihood = -223.01026  
Iteration 1:   log likelihood = -222.78736  
Iteration 2:   log likelihood = -222.78735  

Refining starting values:

Grid node 0:   log likelihood = -216.49485

Fitting full model:

Iteration 0:   log likelihood = -216.49485  (not concave)
Iteration 1:   log likelihood = -214.34477               
Iteration 2:   log likelihood = -209.90745
Iteration 3:   log likelihood = -208.25543
Iteration 4:   log likelihood = -208.11848  
Iteration 5:   log likelihood = -208.11715  
Iteration 6:   log likelihood = -208.11259               
Iteration 7:   log likelihood = -208.11187  
Iteration 8:   log likelihood = -208.11182  
Iteration 9:   log likelihood = -208.11182  

Mixed-effects probit regression                 Number of obs      =       360

  No. of Observations per Group
Group Variable Groups Minimum Average Maximum
_all 1 360 360.0 360
female 60 6 6.0 6
Integration method: laplace Wald chi2(3) = 41.50 Log likelihood = -208.11182 Prob > chi2 = 0.0000
y Coef. Std. Err. z P>|z| [95% Conf. Interval]
 
1.wsm -.4121977 .2735675 -1.51 0.132 -.9483802 .1239847
1.wsf -1.720323 .3223052 -5.34 0.000 -2.35203 -1.088617
 
wsm#wsf
1 1 2.121115 .3611665 5.87 0.000 1.413242 2.828989
 
_cons .5950942 .2350714 2.53 0.011 .1343628 1.055826
_all>male
var(_cons) .3867491 .1789793 .156139 .95796
female
var(_cons) .4464111 .1976024 .1874794 1.062959
LR test vs. probit model: chi2(2) = 29.35 Prob > chi2 = 0.0000 Note: LR test is conservative and provided only for reference.

Our model has two random-effects equations, separated by ||. We use the _all notation that identifies all the observations as one big group. We use the R. notation to tell Stata to treat male as an indicator variable.

The output table includes the fixed-effect portion of our model and the estimated variance components. The estimates of the random intercepts suggest that the heterogeneity among the female salamanders is larger than the heterogeneity among the male salamanders.

If we wish, we can constrain the two random intercepts to be equal.

. constraint 1 _b[/var(_cons[_all>male])] = _b[/var(_cons[female])]

. meprobit y wsm##wsf || _all: R.male || female:, constraint(1) nolog

note: crossed random-effects model specified; option intmethod(laplace) implied

Mixed-effects probit regression                 Number of obs      =       360

  No. of Observations per Group
Group Variable Groups Minimum Average Maximum
_all 1 360 360.0 360
female 60 6 6.0 6
Integration method: laplace Wald chi2(3) = 41.78 Log likelihood = -208.14477 Prob > chi2 = 0.0000 ( 1) [var(_cons[_all>male])]_cons - [var(_cons[female])]_cons = 0
y Coef. Std. Err. z P>|z| [95% Conf. Interval]
 
1.wsm -.4131745 .2697393 -1.53 0.126 -.9418539 .1155048
1.wsf -1.720811 .3141051 -5.48 0.000 -2.336446 -1.105177
 
wsm#wsf
1 1 2.119462 .3570329 5.94 0.000 1.41969 2.819234
 
_cons .5965032 .2237045 2.67 0.008 .1580504 1.034956
_all>male
var(_cons) .4156343 .1488829 .2059706 .8387207
female
var(_cons) .4156343 .1488829 .2059706 .8387207

You can also fit our model using a logit model (see melogit) or a complementary log-log model (see mecloglog).

Also watch A tour of multilevel GLMs.

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