Interpreting two cross-level interaction effects at the same time in MLM

[Philipp Zintl]

Hi Statalists,

I am running a multilevel model which predicts individual-level public support for the EU using Stata 14.1. My main independent variable of interest measures to what extent people have benefitted from EU integration on a scale of 0-8. I am examining if this effect is constant across EU member states or if the effect sizes vary between Eastern/Western Europe and/or countries that benefit from EU fiscal transfers. All variables (except for the country-level variables) are group-mean centered.

cbenefit = IV of interest
east = dummy for Eastern Europe
contribution = EU budgetary balance as % of GDP
eastbenefit = east*cbenefit
conbenefit = contribution*cbenefit

When adding the first interaction term, I receive the following result:

. mixed image ceducation13 cclass cfinancehh csomebill cmostbill ceuroidentity cknowledge cage cgender
> cbenefit east contribution eastbenefit, || country1: eastbenefit

Performing EM optimization:

Performing gradient-based optimization:

Iteration 0:   log likelihood = -27405.958  
Iteration 1:   log likelihood = -27405.958  

Computing standard errors:

Mixed-effects ML regression                     Number of obs     =     22,270
Group variable: country1                        Number of groups  =         28

                                                Obs per group:
                                                              min =        363
                                                              avg =      795.4
                                                              max =      1,231

                                                Wald chi2(13)     =    3523.28
Log likelihood = -27405.958                     Prob > chi2       =     0.0000

-------------------------------------------------------------------------------
        image |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
--------------+----------------------------------------------------------------
 ceducation13 |   .0058476   .0018642     3.14   0.002     .0021938    .0095015
       cclass |   .0102325   .0066986     1.53   0.127    -.0028966    .0233616
   cfinancehh |   .1878532   .0095551    19.66   0.000     .1691255     .206581
    csomebill |  -.0466971   .0143436    -3.26   0.001      -.07481   -.0185842
    cmostbill |  -.1877039    .022934    -8.18   0.000    -.2326536   -.1427541
ceuroidentity |   .4447555   .0125727    35.37   0.000     .4201135    .4693975
   cknowledge |   .0449204   .0066214     6.78   0.000     .0319428     .057898
         cage |  -.0019568   .0003392    -5.77   0.000    -.0026215    -.001292
      cgender |   -.054252   .0112298    -4.83   0.000    -.0762619   -.0322421
     cbenefit |   .0402523   .0031457    12.80   0.000     .0340868    .0464177
         east |   .0321025   .1468598     0.22   0.827    -.2557374    .3199423
 contribution |   .0313233   .0492996     0.64   0.525    -.0653022    .1279487
  eastbenefit |  -.0068053   .0112885    -0.60   0.547    -.0289303    .0153198
        _cons |    2.20782   .0557533    39.60   0.000     2.098546    2.317095
-------------------------------------------------------------------------------

------------------------------------------------------------------------------
  Random-effects Parameters  |   Estimate   Std. Err.     [95% Conf. Interval]
-----------------------------+------------------------------------------------
country1: Independent        |
               var(eastbe~t) |   .0011693   .0005506      .0004646    .0029426
                  var(_cons) |   .0510652   .0140429      .0297883    .0875397
-----------------------------+------------------------------------------------
               var(Residual) |   .6818768   .0064678      .6693172    .6946721
------------------------------------------------------------------------------
LR test vs. linear model: chi2(2) = 1174.41               Prob > chi2 = 0.0000

The interaction term eastbenefit is not significant. When adding the second interaction term I receive this result:

. mixed image ceducation13 cclass cfinancehh csomebill cmostbill ceuroidentity cknowledge cage cgender
> cbenefit east contribution eastbenefit conbenefit, || country1: eastbenefit conbenefit

Performing EM optimization:

Performing gradient-based optimization:

Iteration 0:   log likelihood = -27380.361  
Iteration 1:   log likelihood = -27380.207  
Iteration 2:   log likelihood = -27380.158  
Iteration 3:   log likelihood = -27380.157  

Computing standard errors:

Mixed-effects ML regression                     Number of obs     =     22,270
Group variable: country1                        Number of groups  =         28

                                                Obs per group:
                                                              min =        363
                                                              avg =      795.4
                                                              max =      1,231

                                                Wald chi2(14)     =    2943.20
Log likelihood = -27380.157                     Prob > chi2       =     0.0000

-------------------------------------------------------------------------------
        image |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
--------------+----------------------------------------------------------------
 ceducation13 |   .0055638   .0018612     2.99   0.003     .0019159    .0092117
       cclass |   .0106923   .0066924     1.60   0.110    -.0024246    .0238091
   cfinancehh |   .1867956   .0095406    19.58   0.000     .1680964    .2054949
    csomebill |   -.045531   .0143277    -3.18   0.001    -.0736129   -.0174492
    cmostbill |  -.1927679   .0229465    -8.40   0.000    -.2377422   -.1477937
ceuroidentity |   .4428502   .0125575    35.27   0.000     .4182379    .4674625
   cknowledge |   .0452663   .0066071     6.85   0.000     .0323167    .0582158
         cage |  -.0018657    .000339    -5.50   0.000    -.0025301   -.0012013
      cgender |  -.0549059   .0112049    -4.90   0.000    -.0768671   -.0329447
     cbenefit |   .0289173   .0054457     5.31   0.000      .018244    .0395906
         east |  -.0254096   .1620855    -0.16   0.875    -.3430913    .2922721
 contribution |   .0562929   .0545108     1.03   0.302    -.0505463    .1631321
  eastbenefit |   .0945455   .0407223     2.32   0.020     .0147312    .1743597
   conbenefit |  -.0656482   .0175314    -3.74   0.000    -.1000091   -.0312874
        _cons |   2.170424   .0625917    34.68   0.000     2.047746    2.293101
-------------------------------------------------------------------------------

------------------------------------------------------------------------------
  Random-effects Parameters  |   Estimate   Std. Err.     [95% Conf. Interval]
-----------------------------+------------------------------------------------
country1: Independent        |
               var(eastbe~t) |   1.52e-12   1.91e-11      3.33e-23    .0695549
               var(conben~t) |   .0040515   .0016208      .0018497    .0088743
                  var(_cons) |   .0622493    .017163      .0362616    .1068619
-----------------------------+------------------------------------------------
               var(Residual) |   .6783882   .0064391      .6658844    .6911268
------------------------------------------------------------------------------
LR test vs. linear model: chi2(3) = 1226.00               Prob > chi2 = 0.0000

When adding the second interaction term, eastbenefit becomes positively (and marginally) significant. Conbenefit has a negative coefficient and is significant as well. Following my interpretation, this can be traced back to the fact that most Eastern European countries strongly benefit from EU fiscal transfers. If only eastbenefit is part of the model, the positive interaction effect of east and the negative interaction effect of contribution eliminate each other. Only if conbenefit is added, it becomes possible to isolate both interaction effects.

My questions are:

1. Is my interpretation plausible? Does the 'suppressor variable' logic also translate to moderating variables?
2. How do I exactly interpret the coefficients of two interaction effects? 'The effect of cbenefit is 0.095 points higher in Eastern Europe for countries with the mean value on contribution and keeping all individual-level variables constant.' I am insecure about how to interpret two cross-level interaction effects at the same time.

I am looking forward to your input!

Philipp

[Marcos Almeida]

Just as a reflection about the models, I recommend a) to leave Stata to deal with interactions, I mean, using # and ##; b) not to use the interaction terms as random slopes.

[Philipp Zintl]

Thanks for you suggestions Marcos. Does that mean you would not include any random slope when you test interaction effects or are there cases in which including random slopes for interaction terms is appropriate? Do you also have ideas regarding the two questions that I formulated?

Best,
Philipp

[Marcos Almeida]

Well, I’m not sure whether there’s a rule of thumb concerning this topic. But I suggest to a) start simply with the null model, i.e., only the DV and the levels (aka random interceps; b) then, include the main IV; then include other predictors; then check whether including interaction terms (in the left parcel of the equation) would improve the model; finally, check if adding a random slope for a given predictor would improve the model. That being said, I personally never needed nor found any reason to add an interaction term as a random slope.

[Philipp Zintl]

Oh I obviously took all of these steps. I just didn't want to include more information in my post than was necessary in order to understand my question. Thank you for your input, it's appreciated.