Test of difference for variance components of random slopes using multilevel mixed-effects models (melogit)

[Jessica Herzing]

Hello all,
I am using melogit in Stata 14.2. In the following I use an example, which is based on example 3, section on meqrlogit, page 261 in the Stata multilevel mixed-effects reference manual, release 14 (https://www.stata.com/manuals14/me.pdf). Thus, I am using the Bangladesh fertility survey, with binary variable c_use as independent variable, urban as one covariate in the fixed part, and district in the random part of the model. As I am interested whether the size of district effects is different for rural and urban areas, I introduce separate random effects, within each district, for the urban and rural areas of that district.

Now I want to know whether the difference in size of district effects is significantly different for rural and urban areas. As rule of thumb I was looking at the overlap of the confidence intervals of the variance components. Nevertheless, I would prefer using a test of difference. I came up with the Wald test (test) or the linear combination test (lincom). However, neither do I know whether these tests are appropriate nor am I able to implement the tests. I tried several functions to assess the estimates correctly, but Stata did not find the equations. I hope somebody has a solution for my problem.

Thanks in advance!
Jessica

use http://www.stata-press.com/data/r13/bangladesh

*Coding the dummy for the random slope
generate byte rural = 1 - urban

**Fixed part: contrast coding; Random part: separate coding
melogit c_use urban age child*, noconstant || district: rural urban, noconstant cov(unstr)

*Assessing
mat list e(b)

*Wald test
test (_b[var(rural~):_cons])=(_b[var(urban~):_cons])

And here my output

. use http://www.stata-press.com/data/r13/bangladesh
(Bangladesh Fertility Survey, 1989)

.
. *Coding the dummy for the slope
. generate byte rural = 1 - urban

.
. **Fixed part: contrast coding; Random part: separate coding
. melogit c_use urban age child*, noconstant || district: rural urban, noconstant cov(unstr)

Fitting fixed-effects model:

Iteration 0:   log likelihood = -1319.0486  
Iteration 1:   log likelihood = -1318.2334  
Iteration 2:   log likelihood = -1318.2333  

Refining starting values:

Grid node 0:   log likelihood = -1281.9182

Fitting full model:

Iteration 0:   log likelihood = -1281.9182  (not concave)
Iteration 1:   log likelihood = -1278.7665  
Iteration 2:   log likelihood = -1255.4999  
Iteration 3:   log likelihood =  -1243.247  
Iteration 4:   log likelihood = -1242.2767  
Iteration 5:   log likelihood = -1242.1427  
Iteration 6:   log likelihood = -1242.1407  
Iteration 7:   log likelihood = -1242.1407  

Mixed-effects logistic regression               Number of obs     =      1,934
Group variable:        district                 Number of groups  =         60

                                                Obs per group:
                                                              min =          2
                                                              avg =       32.2
                                                              max =        118

Integration method: mvaghermite                 Integration pts.  =          7

                                                Wald chi2(5)      =      35.37
Log likelihood = -1242.1407                     Prob > chi2       =     0.0000
 ( 1)  [c_use]_cons = 0
---------------------------------------------------------------------------------
          c_use |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
----------------+----------------------------------------------------------------
          urban |  -.7432387   .2431403    -3.06   0.002    -1.219785   -.2666925
            age |  -.0129732   .0081923    -1.58   0.113    -.0290298    .0030833
         child1 |   .7860069   .1653889     4.75   0.000     .4618505    1.110163
         child2 |   .9398673   .1839723     5.11   0.000     .5792882    1.300446
         child3 |   .8591355   .1929315     4.45   0.000     .4809967    1.237274
          _cons |          0  (omitted)
----------------+----------------------------------------------------------------
district        |
      var(rural)|   2.604838   .7142998                      1.521822     4.45859
      var(urban)|   .2849728   .1908777                      .0766761    1.059125
----------------+----------------------------------------------------------------
district        |
cov(urban,rural)|  -.3807802   .4236954    -0.90   0.369    -1.211208    .4496474
---------------------------------------------------------------------------------
LR test vs. logistic model: chi2(3) = 152.19              Prob > chi2 = 0.0000

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

.
. *Assessing
. mat list e(b)

e(b)[1,9]
           c_use:        c_use:        c_use:        c_use:        c_use:        c_use:  var(rural~):  var(urban~):  cov(urban~i:
           urban           age        child1        child2        child3         _cons         _cons         _cons         _cons
y1    -.74323873    -.01297323     .78600689      .9398673     .85913554             0     2.6048377     .28497281    -.38078023

.
. *Wald test
. test (_b[var(rural~):_cons])=(_b[var(urban~):_cons])
equation var(rural~) not found

[Joseph Coveney]

Try

test _b[var(rural[district]):_cons] = _b[var(urban[district]):_cons]

although in general I would use likelihood-ratio tests for variance components.