"convergence not achieved" in multilevel logit model

[Gayoung Choi]

Hello, I have been working on running a random slope model, on longitudinal data (in long form) using the logit model. However, I have encountered an issue that I cannot resolve, so I wanted to reach out to you for guidance.

Here is the description of the model:
Focal variable (outcome): Work Assessment (binary, n=1 for 682 cases, 0 for 8,701 cases)
Main predictor: Early Retirement Pressure (binary)
Total number of observations in the dataset: 57,976
Number of observations when running this model: 3,463 with 1,363 individuals

My question is as follows: I attempted to implement the random slope model as you taught us last time.
The random intercept code works well, but when I include the random slope for centered age, I encounter a "convergence" issue, and the maximum log-likelihood is reported as "not concave."

(Random Intercept)
melogit workassess i.female edu lg_hinc i.urban $health i.earlyret##c.c_age || pid:, cov(un)

(Random Slope Model)
melogit workassess i.female edu lg_hinc i.urban $health i.earlyret##c.c_age || pid: c_age, cov(un)

Is there any way to modify the model settings in this case? And Here is my result.

Iteration 296: log likelihood = -921.96692 (not concave)
Iteration 297: log likelihood = -921.96692 (not concave)
Iteration 298: log likelihood = -921.96692 (not concave)
Iteration 299: log likelihood = -921.96692 (not concave)
Iteration 300: log likelihood = -921.96692 (not concave)
convergence not achieved

Mixed-effects logistic regression Number of obs = 3,463
Group variable: pid Number of groups = 1,363

Obs per group:
min = 1
avg = 2.5
max = 4

Integration method: mvaghermite Integration pts. = 7

Wald chi2(8) = 80.77
Log likelihood = -921.96692 Prob > chi2 = 0.0000
----------------------------------------------------------------------------------
workassess | Coefficient Std. err. z P>|z| [95% conf. interval]
-----------------+----------------------------------------------------------------
c_age | -.0023769 .017445 -0.14 0.892 -.0365685 .0318147
1.female | .4622551 .1717717 2.69 0.007 .1255889 .7989214
edu | -.2801203 .138019 -2.03 0.042 -.5506326 -.0096081
lg_hinc | -.3565346 .1255152 -2.84 0.005 -.6025399 -.1105292
1.urban | .2396985 .2315582 1.04 0.301 -.2141473 .6935443
1.firselfhealth | .8681222 .1615322 5.37 0.000 .5515249 1.184719
chronic | .195936 .0959162 2.04 0.041 .0079438 .3839283
1.earlyret | .463443 .1886819 2.46 0.014 .0936333 .8332527
_cons | -.7321033 1.005356 -0.73 0.466 -2.702565 1.238358
-----------------+----------------------------------------------------------------
pid |
var(c_age)| .0051863 3.91e-07 .0051855 .005187
var(_cons)| .9579369 .0000659 .9578077 .9580661
-----------------+----------------------------------------------------------------
pid |
cov(c_age,_cons)| .070485 . . . . .
----------------------------------------------------------------------------------
convergence not achieved
r(430);



Additionally, I have come across the use of the "gsem" command for multilevel models. Since "gsem" is a structural equation modeling command, would it be appropriate to use it as an alternative to the current code? Thank you for your assistance.

[Joseph Coveney]

Is there any way to modify the model settings in this case?

A couple of things to try:

1. In order to help with numerical stability, divide the centered age by 100 or even by 1000

2. Simplify the model. You can start with changing the random effects covariance: cov(un) → covariance(independent)

And melogit sees 3463 out of 57 976 total observations, 1363 out of 682 + 8701 total cases. Maybe consider triage on predictors that have a lot of missing values.

(Random Slope Model)
melogit workassess i.female edu lg_hinc i.urban $health i.earlyret##c.c_age || pid: c_age, cov(un)

The output that you show doesn't correspond to that model specification. The output has no interaction term in it. Not that age is very impressive in its association with the outcome variable.

. . . as you taught us last time

I took that as just an expression and not as an indication that this is homework.

[Joseph Coveney]

Focal variable (outcome): Work Assessment (binary, n=1 for 682 cases, 0 for 8,701 cases)

So the outcome doesn't change within individual? If you're just exploring the association between the outcome and the two predictors and their interaction, then, given the attrition from missing data, you might as well consider something along the following lines at least as a first-pass assessment.

generate double adc = age / 10 // uncentered age
generate byte mco = 0
foreach var of varlist workassess female edu lg_hinc urban irselfhealth chronic earlyret adc {
    quietly replace mco = mco + missing(`var')
}
gsort +pid +mco -adc
by pid: generate byte use = _n == 1

logit workassess i.earlyret##c.adc ///
    i.(female urban irselfhealth) c.(edu lg_hinc chronic) if use, nolog // -vce()- to taste
quietly centile adc, centile(25 50 75) // you need only two, but the graph is cuter with three
quietly margins earlyret, at(adc = (`r(c_1)' `r(c_2)' `r(c_3)')) predict(xb)
marginsplot , ///
    title("") xtitle(Employee Age (Decades)) ///
    plotopts(lcolor(black) msymbol(O) mcolor(black) mfcolor(white)) ///
    level(50) ciopts(lcolor(black)) ///
    scheme(s2color) ylabel( , angle(horizontal) nogrid) legend(off)

The code plots the predictions in the estimation metric, but you could opt to plot predicted proportions.

[Gayoung Choi]

A couple of things to try:

1. In order to help with numerical stability, divide the centered age by 100 or even by 1000

2. Simplify the model. You can start with changing the random effects covariance: cov(un) → covariance(independent)

And melogit sees 3463 out of 57 976 total observations, 1363 out of 682 + 8701 total cases. Maybe consider triage on predictors that have a lot of missing values.

The output that you show doesn't correspond to that model specification. The output has no interaction term in it. Not that age is very impressive in its association with the outcome variable.

I took that as just an expression and not as an indication that this is homework.

Thanks for your advice!!
It really helpful for me!
After I divide the centered age by 100, the model works!

However, I have another problem.

Based on the suggestions you all provided, I divided the values by 100 and reported the odds ratio.
However, the numbers are being reported as excessively large. In this case, how should I proceed with the interpretation?

the code is...

gen c_age100 = c_age/100
melogit loneliness ($indi $health i.earlyret##ib1.sparti_tie)##c.c_age100 || pid: c_age100, cov(un) or

. melogit loneliness ($indi $health i.earlyret##ib1.sparti_tie)##c.c_age100 || pid: c_age100, cov(un) or

Fitting fixed-effects model:

Iteration 0:   log likelihood = -4219.2826  
Iteration 1:   log likelihood = -4213.4815  
Iteration 2:   log likelihood =  -4213.479  
Iteration 3:   log likelihood =  -4213.479  

Refining starting values:

Grid node 0:   log likelihood = -4035.9539

Fitting full model:

Iteration 0:   log likelihood = -4035.9539  
Iteration 1:   log likelihood = -4001.8698  
Iteration 2:   log likelihood = -3998.4615  
Iteration 3:   log likelihood = -3998.4105  
Iteration 4:   log likelihood = -3998.4105  

Mixed-effects logistic regression               Number of obs     =      7,798
Group variable: pid                             Number of groups  =      1,815

                                                Obs per group:
                                                              min =          1
                                                              avg =        4.3
                                                              max =          8

Integration method: mvaghermite                 Integration pts.  =          7

                                                Wald chi2(22)     =     173.64
Log likelihood = -3998.4105                     Prob > chi2       =     0.0000
------------------------------------------------------------------------------------------------
                    loneliness | Odds ratio   Std. err.      z    P>|z|     [95% conf. interval]
-------------------------------+----------------------------------------------------------------
                      c_age100 |    1425814    8821996     2.29   0.022     7.715937    2.63e+11
                      1.female |   1.262223   .1254929     2.34   0.019     1.038742    1.533784
                               |
                           edu |
                            2  |   .8987175    .097272    -0.99   0.324     .7269325    1.111098
                            3  |   .8137522   .1246497    -1.35   0.178     .6027066    1.098698
                               |
                       lg_hinc |   .7933476   .0402326    -4.56   0.000     .7182853     .876254
                       0.urban |   1.043351   .1315593     0.34   0.736     .8148913     1.33586
               1.firselfhealth |   1.546035   .1178667     5.71   0.000     1.331452    1.795202
                     c_chronic |   1.151301   .0725386     2.24   0.025     1.017556    1.302625
                    1.earlyret |   1.277575   .2477085     1.26   0.206     .8736693    1.868209
                  0.sparti_tie |   1.174979   .1169125     1.62   0.105      .966794    1.427993
                               |
           earlyret#sparti_tie |
                          1 0  |   1.609173   .3474376     2.20   0.028     1.053948    2.456893
                               |
         c.c_age100#c.c_age100 |   2.88e-06   .0000296    -1.24   0.215     4.98e-15    1659.099
                               |
             female#c.c_age100 |
                            1  |   .1733652   .2587977    -1.17   0.240      .009296    3.233154
                               |
                edu#c.c_age100 |
                            2  |   4.640404   7.620424     0.93   0.350     .1856613    115.9819
                            3  |   416.0817   965.8807     2.60   0.009     4.397669    39367.21
                               |
          c.lg_hinc#c.c_age100 |    .159505   .1197276    -2.45   0.014     .0366307    .6945488
                               |
              urban#c.c_age100 |
                            0  |   .0673028   .1292882    -1.40   0.160     .0015591    2.905258
                               |
      firselfhealth#c.c_age100 |
                            1  |   .2638762   .3436184    -1.02   0.306     .0205575    3.387124
                               |
        c.c_chronic#c.c_age100 |   1.934838   1.919119     0.67   0.506     .2769227    13.51857
                               |
           earlyret#c.c_age100 |
                            1  |   390.8541   1305.431     1.79   0.074     .5611099    272258.5
                               |
         sparti_tie#c.c_age100 |
                            0  |     9.5464   15.49126     1.39   0.164     .3967957    229.6742
                               |
earlyret#sparti_tie#c.c_age100 |
                          1 0  |   .0009308   .0035178    -1.85   0.065     5.65e-07    1.533925
                               |
                         _cons |   1.024956    .426968     0.06   0.953     .4530211    2.318953
-------------------------------+----------------------------------------------------------------
pid                            |
                  var(c_age100)|   103.4066   31.49621                      56.92253    187.8505
                     var(_cons)|   1.595473   .1645452                      1.303475    1.952882
-------------------------------+----------------------------------------------------------------
pid                            |
            cov(c_age100,_cons)|   3.163299   1.668592     1.90   0.058    -.1070801    6.433679
------------------------------------------------------------------------------------------------
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(3) = 430.14              Prob > chi2 = 0.0000

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

**In my opinion, the reason for the odds ratio being extremely large is likely due to the division of age by 100. Is there a way to report the table in its original state, without dividing by 100?
***I'd like to justify dividing centered age by 100. Is there a statistical paper that discusses this?