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I am trying to work out how to express the results of interaction contrasts between two categorical variables in a hierarchical logistic regression, one a two level variable Studygroup and the other a three-level variable assessment_cat. Thanks to this excellent question and answer I have been able to work out how to put together the interactions by adding the coefficients estimates from the glmer table. However interpreting the output once the log odds have been exponentiated to odds ratios is more confusing.

This is the glmer table of coefficients

                                          Estimate Std..Error    z.value     Pr...z..
(Intercept)                             0.72369704  0.5638152  1.2835712 0.1992920165
Studygroupintervention                 -0.06223667  0.1335099 -0.4661579 0.6411025172
assessment_cat2                         1.68103031  0.4424892  3.7990310 0.0001452629
assessment_cat3                        -0.39206346  0.3181081 -1.2324851 0.2177679252
Studygroupintervention:assessment_cat2 -0.80185985  0.6182360 -1.2970124 0.1946269037
Studygroupintervention:assessment_cat3  1.03723190  0.4367688  2.3747849 0.0175591776

I have attempted to draw up a log-odds table in the same manner as in the answer above (applied to regular regression), so that each cell has a corresponding estimate comprised of the additive combination of the values from the glmer output. The int-cont column is the difference in log-odds between the intervention and control groups for each level of assessment_cat

              Control   Intervention    Int-Cont
Category 1     0.7237        0.66146    -0.06224
Category 2     2.4047        1.54063    -0.86407
Category 3     0.3332        1.30663     0.97343

My question is, can I just exponentiate these values to get estimates of odds-ratios, like so?

1. Cont/Cat1 vs Cont/Cat2: 0.724 vs 2.405 = +1.68; Odds-Ratio 5.365556 
2. Cont/Cat1 vs Cont/Cat3: 0.724 vs 0.322 = -0.392; Odds-Ratio 0.6757041
3. Cont/Cat2 vs Cont/Cat3: 2.405 vs 0.322 = -2.07; Odds-Ratio 0.1261858
4. Cont/Cat1 vs Int/Cat1: 0.724 vs 0.662 = -0.064; Odds-Ratio 0.938005
5. Cont/Cat2 vs Int/Cat2: 2.405 vs 1.541 = -0.864; Odds-Ratio 0.4214728
6. Cont/Cat3 vs Int/Cat3: 0.333 vs 1.307 = 0.973; Odds-Ratio 2.64587

and like so for the interaction contrasts?

7. 4 vs 5: -0.064 vs -0.864 = -0.801; Odds-Ratio 0.4488799 
8. 4 vs 6: -0.064 vs +0.973 = 1.03; Odds-Ratio 2.820742 
9. 5 vs 6: -0.864 vs +0.973 = 1.839; Odds-Ratio 6.290245

Or do the sort of additions used to derive interaction contrasts with standard regression tables break-down when converting from log-odds to odds-ratios?

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