Interpretation of interaction term in mixed effects logistic regression model

[Ijeoma Itanyi]

Hello,

I am analysing data for predictors of cigarette smoking among adolescents using mixed effects logistic regression with random effects at school level. Schools were selected from urban and rural locations (strata), then classes were selected within schools and all students in selected classes were studied. The following command was used for the logistic regression analysis using Stata 11:

Please see the output below:

. xtmelogit cursmkcig stratum agecat sex cwkspend cparentswork sesfat strasesfat
>  livetog rsmkparents smkfriends smkclass rcignearsch advert antitob shs schcur
> r if adol==1, || sampschid:, covariance(unstructured) or variance
Note: single-variable random-effects specification; covariance structure set to
      identity

Refining starting values: 

Iteration 0:   log likelihood = -1437.3631  (not concave)
Iteration 1:   log likelihood = -1425.2512  
Iteration 2:   log likelihood =  -1423.267  

Performing gradient-based optimization: 

Iteration 0:   log likelihood =  -1423.267  
Iteration 1:   log likelihood = -1423.2267  
Iteration 2:   log likelihood = -1423.2265  

Mixed-effects logistic regression               Number of obs      =      4332
Group variable: sampschid                       Number of groups   =        49

                                                Obs per group: min =        50
                                                               avg =      88.4
                                                               max =       115

Integration points =   7                        Wald chi2(16)      =    371.02
Log likelihood = -1423.2265                     Prob > chi2        =    0.0000

------------------------------------------------------------------------------
   cursmkcig | Odds Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
     stratum |   1.237299   .2309897     1.14   0.254     .8581532    1.783957
      agecat |    .906214   .0860734    -1.04   0.300     .7522852    1.091639
         sex |   .7272792   .0804917    -2.88   0.004     .5854562    .9034579
    cwkspend |   1.270641    .085474     3.56   0.000     1.113689    1.449712
cparentswork |   .6930044   .0564859    -4.50   0.000     .5906846    .8130483
      sesfat |   .7812136   .1303561    -1.48   0.139     .5632942    1.083439
  strasesfat |   1.821569   .4241586     2.58   0.010     1.154092    2.875086
     livetog |    1.27858   .1680938     1.87   0.062     .9881464    1.654378
 rsmkparents |   1.502987   .1489246     4.11   0.000     1.237694    1.825144
  smkfriends |   1.922712   .1552835     8.09   0.000     1.641228    2.252473
    smkclass |   1.397217   .1085893     4.30   0.000     1.199803    1.627113
 rcignearsch |   1.779504   .2280784     4.50   0.000     1.384206     2.28769
      advert |   1.372885   .1807741     2.41   0.016     1.060602    1.777116
     antitob |    1.12164   .1596499     0.81   0.420     .8485873    1.482553
         shs |   2.002395   .2363955     5.88   0.000     1.588766    2.523711
     schcurr |   .5912504    .069033    -4.50   0.000     .4703131    .7432857
------------------------------------------------------------------------------

------------------------------------------------------------------------------
  Random-effects Parameters  |   Estimate   Std. Err.     [95% Conf. Interval]
-----------------------------+------------------------------------------------
sampschid: Identity          |
                  var(_cons) |   .2302386   .0713563      .1254213    .4226541
------------------------------------------------------------------------------
LR test vs. logistic regression: chibar2(01) =    50.32 Prob>=chibar2 = 0.0000

The dependent variable is current cigarette smoking (cursmkcig). The main independent variable is school location (stratum) labelled as urban or rural. The variable strasesfat is an interaction term between socioeconomic status (sesfat) and school location (stratum). Since there is evidence of interaction from the above output, I would like to know how to present the results reporting the association between practice of current smoking and school location within each level of socioeconomic status (low SES and high SES).

I tried reporting coefficients and using lincom command and it didn't work.

. xtmelogit cursmkcig stratum agecat sex cwkspend cparentswork sesfat strasesfat
> livetog rsmkparents smkfriends smkclass rcignearsch advert antitob shs schcur
> r if adol==1, || sampschid:, covariance(unstructured) variance
Note: single-variable random-effects specification; covariance structure set to
      identity

Refining starting values: 

Iteration 0:   log likelihood = -1437.3631  (not concave)
Iteration 1:   log likelihood = -1425.2512  
Iteration 2:   log likelihood =  -1423.267  

Performing gradient-based optimization: 

Iteration 0:   log likelihood =  -1423.267  
Iteration 1:   log likelihood = -1423.2267  
Iteration 2:   log likelihood = -1423.2265  

Mixed-effects logistic regression               Number of obs      =      4332
Group variable: sampschid                       Number of groups   =        49

                                                Obs per group: min =        50
                                                               avg =      88.4
                                                               max =       115

Integration points =   7                        Wald chi2(16)      =    371.02
Log likelihood = -1423.2265                     Prob > chi2        =    0.0000

------------------------------------------------------------------------------
   cursmkcig |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
     stratum |   .2129306   .1866887     1.14   0.254    -.1529726    .5788338
      agecat |  -.0984798   .0949813    -1.04   0.300    -.2846397    .0876802
         sex |  -.3184449   .1106751    -2.88   0.004     -.535364   -.1015257
    cwkspend |   .2395212   .0672684     3.56   0.000     .1076775    .3713649
cparentswork |   -.366719   .0815087    -4.50   0.000    -.5264731   -.2069648
      sesfat |  -.2469067   .1668636    -1.48   0.139    -.5739533    .0801398
  strasesfat |   .5996982   .2328534     2.58   0.010     .1433138    1.056083
     livetog |   .2457503   .1314691     1.87   0.062    -.0119244    .5034249
 rsmkparents |   .4074544   .0990858     4.11   0.000     .2132498    .6016589
  smkfriends |   .6537365   .0807628     8.09   0.000     .4954445    .8120286
    smkclass |   .3344821   .0777183     4.30   0.000     .1821571    .4868072
 rcignearsch |   .5763345   .1281697     4.50   0.000     .3251265    .8275424
      advert |   .3169141   .1316746     2.41   0.016     .0588366    .5749917
     antitob |   .1147916   .1423362     0.81   0.420    -.1641823    .3937654
         shs |   .6943441   .1180564     5.88   0.000     .4629579    .9257303
     schcurr |  -.5255157   .1167577    -4.50   0.000    -.7543566   -.2966748
       _cons |  -4.025837   .4399083    -9.15   0.000    -4.888042   -3.163633
------------------------------------------------------------------------------

------------------------------------------------------------------------------
  Random-effects Parameters  |   Estimate   Std. Err.     [95% Conf. Interval]
-----------------------------+------------------------------------------------
sampschid: Identity          |
                  var(_cons) |   .2302386   .0713563      .1254213    .4226541
------------------------------------------------------------------------------
LR test vs. logistic regression: chibar2(01) =    50.32 Prob>=chibar2 = 0.0000

. *Use lincom command to estimate combination of odds ratios

. lincom _cons + sesfat, or

 ( 1)  [eq1]sesfat + [eq1]_cons = 0

------------------------------------------------------------------------------
   cursmkcig | Odds Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         (1) |   .0139435   .0062817    -9.48   0.000     .0057663    .0337167
------------------------------------------------------------------------------

. *OR for student with low father's education living in urban area

. lincom _cons, or

 ( 1)  [eq1]_cons = 0

------------------------------------------------------------------------------
   cursmkcig | Odds Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         (1) |   .0178485   .0078517    -9.15   0.000     .0075362    .0422719
------------------------------------------------------------------------------

. *OR for student with high father's education living in urban area

. lincom _cons + sesfat, or

 ( 1)  [eq1]sesfat + [eq1]_cons = 0

------------------------------------------------------------------------------
   cursmkcig | Odds Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         (1) |   .0139435   .0062817    -9.48   0.000     .0057663    .0337167
------------------------------------------------------------------------------

. *OR for student with low father's education living in rural area

. lincom _cons + stratum, or

 ( 1)  [eq1]stratum + [eq1]_cons = 0

------------------------------------------------------------------------------
   cursmkcig | Odds Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         (1) |   .0220839   .0096375    -8.74   0.000     .0093887    .0519452
------------------------------------------------------------------------------

. *OR for student with high father's education living in rural area

. lincom _cons + stratum + sesfat + strasesfat, or

 ( 1)  [eq1]stratum + [eq1]sesfat + [eq1]strasesfat + [eq1]_cons = 0

------------------------------------------------------------------------------
   cursmkcig | Odds Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         (1) |   .0314261   .0142652    -7.62   0.000     .0129094    .0765026
------------------------------------------------------------------------------

I also tried performing the analysis to report stratum-specific odds ratios using the variable 'strases' as the interaction term, and the output showed evidence of collinearity

 . xtmelogit cursmkcig stratum agecat sex cwkspend cparentswork sesfat strases* l
> ivetog rsmkparents smkfriends smkclass rcignearsch advert antitob shs schcurr 
> if adol==1, || sampschid:, covariance(unstructured) or variance
Note: single-variable random-effects specification; covariance structure set to
      identity
note: strases1 omitted because of collinearity

Refining starting values: 

Iteration 0:   log likelihood = -1437.3631  (not concave)
Iteration 1:   log likelihood = -1425.2494  
Iteration 2:   log likelihood = -1423.2669  

Performing gradient-based optimization: 

Iteration 0:   log likelihood = -1423.2669  
Iteration 1:   log likelihood = -1423.2267  
Iteration 2:   log likelihood = -1423.2265  

Mixed-effects logistic regression               Number of obs      =      4332
Group variable: sampschid                       Number of groups   =        49

                                                Obs per group: min =        50
                                                               avg =      88.4
                                                               max =       115

Integration points =   7                        Wald chi2(16)      =    371.02
Log likelihood = -1423.2265                     Prob > chi2        =    0.0000

------------------------------------------------------------------------------
   cursmkcig | Odds Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
     stratum |   2.253825   .5379541     3.40   0.001     1.411729    3.598233
      agecat |    .906214   .0860734    -1.04   0.300     .7522852    1.091639
         sex |   .7272792   .0804917    -2.88   0.004     .5854561    .9034579
    cwkspend |   1.270641    .085474     3.56   0.000     1.113689    1.449712
cparentswork |   .6930044   .0564859    -4.50   0.000     .5906846    .8130483
      sesfat |   .7812134   .1303561    -1.48   0.139     .5632941    1.083438
    strases0 |   .5489772   .1278312    -2.58   0.010     .3478156    .8664819
    strases1 |  (omitted)
     livetog |    1.27858   .1680938     1.87   0.062     .9881464    1.654378
 rsmkparents |   1.502987   .1489246     4.11   0.000     1.237694    1.825144
  smkfriends |   1.922712   .1552835     8.09   0.000     1.641228    2.252473
    smkclass |   1.397217   .1085893     4.30   0.000     1.199803    1.627113
 rcignearsch |   1.779504   .2280784     4.50   0.000     1.384206     2.28769
      advert |   1.372885   .1807741     2.41   0.016     1.060602    1.777116
     antitob |    1.12164   .1596499     0.81   0.420     .8485873    1.482553
         shs |   2.002395   .2363955     5.88   0.000     1.588766    2.523711
     schcurr |   .5912504    .069033    -4.50   0.000     .4703131    .7432856
------------------------------------------------------------------------------

------------------------------------------------------------------------------
  Random-effects Parameters  |   Estimate   Std. Err.     [95% Conf. Interval]
-----------------------------+------------------------------------------------
sampschid: Identity          |
                  var(_cons) |   .2302386   .0713563      .1254213    .4226542
------------------------------------------------------------------------------
LR test vs. logistic regression: chibar2(01) =    50.32 Prob>=chibar2 = 0.0000

Thank you.

[Clyde Schechter]

So, the -lincom- approach gives odds ratios that are conditional on all other variables being zero, which given the large odds ratios associated with several of your other variables means these are probably not statistics of interest for practical purposes. You don't actually explain the other variables (and I'm not asking you to) but often it isn't even possible for them to be zero, or being zero would be an unusual case. You need to use the -margins- command. But you can't, because you didn't use factor variable notation in your regression command. So read -help fvvarlist- to learn about factor variable notation, and then revise your command. I will apply it here to stratum and sesfat, which I assume are discrete variables. You should also apply it to other variables as appropriate:

xtmelogit cursmkcig i.stratum##i.sesfat agecat sex cwkspend cparentswork ///
livetog rsmkparents smkfriends smkclass rcignearsch advert antitob shs schcur ///
r if adol==1, || sampschid:, or variance

margins stratum#sesfat

The output of -margins- will show you the predicted probabilities of smoking in each combination of stratum and sesfat, adjusted to the distribution of other variables. There is much more you can get from -margins-. The best introduction to the -margins- command around is the excellent Richard Williams' https://www3.nd.edu/~rwilliam/stats/Margins01.pdf. It includes worked examples, some of which are interaction models like yours.

Other comments:

1. If you are running a recent version of Stata, the command is now called -melogit-. The old name still works, but is no longer preferred.

2. It seems to me that this analysis is not well matched to your data. You describe a complex survey design with strata and PSU's, but you are not using a design-based analysis. While the use of random effects will account for some of the departures from simple random sampling, it certainly does not reflect the stratification, nor does it appear to fully reflect selection either. You might want to look into doing this with -svy:-.

[Ijeoma Itanyi]

Thank you, Clyde! I will try the margins command and 'logistic regression using 'svy'.

[Ijeoma Itanyi]

Hi Clyde,

I tried the margins command and received the following output. I don't know what is wrong..

. xtmelogit cursmkcig i.stratum##i.sesfat agecat sex cwkspend cparentswork livet > og rsmkparents smkfriends smkclass rcignearsch advert antitob shs schcurr if a > dol==1, || sampschid:, or variance Refining starting values: Iteration 0: log likelihood = -1437.3631 (not concave) Iteration 1: log likelihood = -1425.2512 Iteration 2: log likelihood = -1423.267 Performing gradient-based optimization: Iteration 0: log likelihood = -1423.267 Iteration 1: log likelihood = -1423.2267 Iteration 2: log likelihood = -1423.2265 Mixed-effects logistic regression Number of obs = 4332 Group variable: sampschid Number of groups = 49 Obs per group: min = 50 avg = 88.4 max = 115 Integration points = 7 Wald chi2(16) = 371.02 Log likelihood = -1423.2265 Prob > chi2 = 0.0000 ------------------------------------------------------------------------------ cursmkcig | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- 1.stratum | 1.237299 .2309897 1.14 0.254 .8581533 1.783957 1.sesfat | .7812136 .1303561 -1.48 0.139 .5632942 1.083439 | stratum#| sesfat | 1 1 | 1.821569 .4241586 2.58 0.010 1.154092 2.875086 | agecat | .906214 .0860734 -1.04 0.300 .7522852 1.091639 sex | .7272792 .0804917 -2.88 0.004 .5854562 .9034579 cwkspend | 1.270641 .085474 3.56 0.000 1.113689 1.449712 cparentswork | .6930044 .0564859 -4.50 0.000 .5906846 .8130483 livetog | 1.27858 .1680938 1.87 0.062 .9881464 1.654378 rsmkparents | 1.502987 .1489246 4.11 0.000 1.237694 1.825144 smkfriends | 1.922712 .1552835 8.09 0.000 1.641228 2.252473 smkclass | 1.397217 .1085893 4.30 0.000 1.199803 1.627113 rcignearsch | 1.779504 .2280784 4.50 0.000 1.384206 2.28769 advert | 1.372885 .1807741 2.41 0.016 1.060602 1.777116 antitob | 1.12164 .1596499 0.81 0.420 .8485873 1.482553 shs | 2.002395 .2363955 5.88 0.000 1.588766 2.523711 schcurr | .5912504 .069033 -4.50 0.000 .4703131 .7432857 ------------------------------------------------------------------------------ ------------------------------------------------------------------------------ Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval] -----------------------------+------------------------------------------------ sampschid: Identity | var(_cons) | .2302386 .0713562 .1254213 .422654 ------------------------------------------------------------------------------ LR test vs. logistic regression: chibar2(01) = 50.32 Prob>=chibar2 = 0.0000 . margins stratum#sesfat default prediction is a function of possibly stochastic quantities other than e(b) r(498).

[Clyde Schechter]

What version of Stata are you using? I'm running the current version, 15.1, and I have no problem getting -margins- to work after -melogit-. But I vaguely recall that in earlier versions, it would not.

[Ijeoma Itanyi]

Oh! Running Stata 11. I will try 'svy' . Thank you