Interpreting cross-level interactions in logit model using odds ratios

[Straso Jovanovski]

Hello,
I am unsure on whether I am doing this correctly. I am attempting to interpret cross-level interactions after running a logit model using odds ratios (ORs). My individual level dependent variable is the binary "poor/fair health" (coded as 1 if a person has reported experiencing either poor or fair health, and 0 otherwise). The observations are individuals residing in (nested within) a number of regions. I am regressing on the main independent variable (at the region level): region-wide segregation (measured as an index assuming decimal values between 0 and 1). There are person-level controls: race, income, gender, married, and age.
Here is a sample logit regression result (specified with "or" option, as well as robust and clustered standard errors):

. eststo: logit poorfairhlth seg i.race ib4.income female mardum age age2 race#c.seg, or robust cluster(region)

Iteration 0:   log pseudolikelihood = -140784.52  
Iteration 1:   log pseudolikelihood = -123546.99  
Iteration 2:   log pseudolikelihood =  -121274.7  
Iteration 3:   log pseudolikelihood =  -121257.3  
Iteration 4:   log pseudolikelihood =  -121257.3  

Logistic regression                             Number of obs     =    317,018
                                                Wald chi2(21)     =          .
                                                Prob > chi2       =          .
Log pseudolikelihood =  -121257.3               Pseudo R2         =     0.1387

                                         (Std. Err. adjusted for 310 clusters in region)
-----------------------------------------------------------------------------------------
                        |               Robust
           poorfairhlth | Odds Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
------------------------+----------------------------------------------------------------
                    seg |   .4043839   .0981238    -3.73   0.000     .2513331    .6506357
                        |
                   race |
    Non-Hispanic Black  |   1.131217   .0999835     1.39   0.163     .9512872    1.345179
    Non-Hispanic Asian  |   .6750161   .2465322    -1.08   0.282     .3299398    1.380999
   Non-Hispanic Native  |   1.527686   .3368833     1.92   0.055      .991581    2.353639
              Hispanic  |   1.281861   .1856169     1.71   0.086      .965127    1.702539
                        |
                 income |
      Income < $25,000  |   8.051568   .1987019    84.52   0.000     7.671388    8.450589
Income $25,000-$49,999  |   2.989175   .0654866    49.98   0.000      2.86354    3.120322
Income $50,000-$74,999  |   1.706499   .0370476    24.62   0.000      1.63541    1.780677
                        |
                 female |   .8787638   .0108219   -10.49   0.000     .8578072    .9002324
                 mardum |   1.015183   .0120971     1.26   0.206     .9917474    1.039171
                    age |   1.102233   .0025901    41.42   0.000     1.097169    1.107322
                   age2 |   .9993602   .0000206   -31.09   0.000     .9993199    .9994006
                        |
             race#c.seg |
    Non-Hispanic Black  |   1.330019   .3434105     1.10   0.269     .8018219    2.206164
    Non-Hispanic Asian  |   1.474294   1.092774     0.52   0.600      .344872    6.302462
   Non-Hispanic Native  |    1.89186   1.282931     0.94   0.347     .5007977    7.146869
              Hispanic  |   3.111898   .8642645     4.09   0.000     1.805602    5.363259
                        |
                  _cons |   .0026307   .0003586   -43.59   0.000      .002014    .0034363
-----------------------------------------------------------------------------------------
Note: _cons estimates baseline odds.
(est1 stored)

The only (highly) statistically significant among the interaction terms is Hispanic#Segregation (which shows as very detrimental to health on its own). How can I arrive at the combined effect here? Since the main effect of segregation above is protective, odds ratio less then 1 (improving health), (that one being very significant as well), would the combined effect be obtained by: 3.112-(1-0.404)=2.516 (in terms of odds ratios) - in other words we would expect an increase in the odds of observing poor/fair health for the Hispanic group in segregated regions by 251.6%? These are all in terms of odds ratios for being able to make better sense of the effects/coefficients. Thank you so much for any help.

[Phil Bromiley]

You'll increase your chances of a helpful answer by following the FAQ on asking questions - provide Stata code in code delimiters, readable Stata output, and sample data using dataex.

I'm surprised that you have fair/poor health as 1 and good health as 0. You'll probably find it easier to explain if you make the better health the higher number.

After logit, you can use margins to examine the change in predicted value for a change in the iv including the interactions. This is particularly true with continuous variables like seq - you want to look at the influence of Hispanic at different levels of seg. I find probabilities easier to work with than odds ratios so I can't help you with the odds ratios per se.