Hi Statalisters

I am finalizing analyses in a project where a cross-level interaction is of interest. I am wondering whether my -margins- specification and interpretation is correct. I have searched quite extensively without finding any definite recommendations, and since there is so much expertise on this forum, I hope I might get some help here.

I have a binary outcome and a binary exposure at individual level and a five-level categorical variable at country level. Since the outcome is common, I am using -mepoisson- to obtain risk ratios:
Code:
 mepoisson poorhlth i.fdiff##i.welfareregime immstat i.parentSES || country: fdiff, irr
After testing that there is support for an interaction effect, I examine the interaction with:
Code:
. margins fdiff##welfareregime, vsquish

Predictive margins                              Number of obs     =     28,637
Model VCE    : Robust

Expression   : Marginal predicted mean, predict()

----------------------------------------------------------------------------------------
                       |            Delta-method
                       |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-----------------------+----------------------------------------------------------------
                 fdiff |
                   No  |   .3047144   .0136302    22.36   0.000     .2779997    .3314291
                  Yes  |   .4587111    .013653    33.60   0.000     .4319518    .4854705
         welfareregime |
         Scandinavian  |   .2455998   .0172017    14.28   0.000     .2118851    .2793145
          Bismarckian  |   .3221907   .0265032    12.16   0.000     .2702453    .3741361
          Anglo-Saxon  |   .2606679   .0113356    23.00   0.000     .2384505    .2828854
     Eastern-European  |   .3893255   .0144292    26.98   0.000     .3610448    .4176062
    Southern-European  |   .3972321   .0348144    11.41   0.000     .3289972    .4654671
   fdiff#welfareregime |
      No#Scandinavian  |    .218395   .0165498    13.20   0.000      .185958    .2508319
       No#Bismarckian  |   .3023092   .0275353    10.98   0.000      .248341    .3562773
       No#Anglo-Saxon  |   .2326351   .0107442    21.65   0.000     .2115769    .2536933
  No#Eastern-European  |   .3603786   .0135944    26.51   0.000     .3337341    .3870231
 No#Southern-European  |   .3715952   .0312916    11.88   0.000     .3102648    .4329257
     Yes#Scandinavian  |   .3931677   .0209561    18.76   0.000     .3520945    .4342408
      Yes#Bismarckian  |   .4300347   .0237197    18.13   0.000      .383545    .4765243
      Yes#Anglo-Saxon  |   .4127271   .0149689    27.57   0.000     .3833887    .4420656
 Yes#Eastern-European  |    .546343   .0232894    23.46   0.000     .5006966    .5919895
Yes#Southern-European  |   .5362952   .0539636     9.94   0.000     .4305285    .6420618
----------------------------------------------------------------------------------------
I interpret this table as giving me the predicted probabilities of the outcome, conditional on included covariates in the model (I should perhaps include the -atmeans- options). I further examine the interaction to get at what I consider the main interaction results that I will report:
Code:
. contrast r.welfareregime@1.fdiff, nowald effects

Contrasts of marginal linear predictions

Margins      : asbalanced

---------------------------------------------------------------------------------------------------------
                                        |   Contrast   Std. Err.      z    P>|z|     [95% Conf. Interval]
----------------------------------------+----------------------------------------------------------------
poorhlth                                |
                    welfareregime@fdiff |
     (Bismarckian vs Scandinavian) Yes  |   .0896297    .079263     1.13   0.258     -.065723    .2449823
     (Anglo-Saxon vs Scandinavian) Yes  |   .0485506   .0637509     0.76   0.446    -.0763989       .1735
(Eastern-European vs Scandinavian) Yes  |   .3290109    .068269     4.82   0.000     .1952061    .4628156
   (Southern-European vs Scandinavian)  |
                                   Yes  |   .3104486   .1128225     2.75   0.006     .0893206    .5315765
---------------------------------------------------------------------------------------------------------
I interpret this the following way (using the third row as an example): The group with value 1 on the exposure ("Yes") in Eastern-European countries have a 33 percent increased probability of the outcome (in relative terms), compared to the group with value 1 on the exposure in Scandinavian countries.

I have two questions:
  1. Is the margins specification correct? I am somewhat unsure as I have a binary outcome and I am not using a logistic model. Specifically, I wonder whether the "marginal prediction mean"/"marginal linear prediction" expression is suitable.
  2. Given a correct specification, is my interpretation correct?