Interpreting an interaction term in the context of study

Question

I developed the ordinal model where the outcome (high, middle low) is predicted from variables socioeconomic status (low, middle or high), child/adult relationship (family type A, B or C), and some other variables.

I fitted the interaction between the status of the family and the adult-child relationship, but I am not sure how to interpret the effects of this interaction.

If there were no interaction I would say that the middle family status increases the odds of the outcome by 50% and the high by 130% and family_B has no significant effect, while family_C increases the odds by 24%.

I think that in the presence of interaction, the above odds are only interpretable as above if the other variable is at the reference level and at the other levels of both variables there is the effect modification.
But considering that the effects for one family_B level and many levels of interaction are not significant, I do not know by how much, and if at all the effects are modified.

For example:

1. In the case of the high socioeconomic status and family_C would the actual effect be: exp(beta(high) + beta(family_C) + beta(high*family_C)

exp( log(2.32) + log(1.24) + log(0.98))

or, should I ignore beta(high*family_C) as it is not significant?

2. In the case of the middle socioeconomic status and family_B would the actual effect be: exp(beta(middle) + beta(family_B) + beta(middle*family_B),

but in this case, both family_B and middle*family_B are not significant. Are they included or are they both ignored?

I want to write two sentences: The odds of the outcome high vs middle or low when growing up with middle SES and family_B are XX higher/ lower than of growing up in the family with low SES and family_A.

The odds of the outcome high vs middle or low when growing up with high SES and family_C are XXX higher/lower than of growing up in a family with low SES and family_A.

How to calculate what the XX and XXX are equal to?

The problem of interaction and not significant terms have been discussed many times, but I have not found the answer where there was an explanation on how to provide the actual size of modification when dealing with not significant variables.

Thanks for any help

I used the emmeans package as reccomended in the answer and this are the results:

The interaction between high SES and family C is insignificant but pairwise comparisons of high ses family_c to high ses family_a and high ses family_c to middle ses family_a and high ses family_c to low ses family_a are all significant. Does it mean that there is an interaction on some levels but not the other? But why is the interaction in main output in the model not significant? Are these pairwise effect spurious? Additionally when comering the model with interaction to one without, adjusted R2 are the same, but Wald test is significant. I expected the interaction to be significant based on the previous evidence from literature review, and I would not want to misinterpret the results I got.

Answer 1

If you included the interactions in your model to start with, you should do your post-modeling calculations with all of the coefficients in your model, even if some aren't individually "statistically significant." In particular, the apparent "significance" (in terms of difference from 0) of lower-level coefficients of predictors involved in interactions (like Family_B) can depend on how the interacting predictors (like SES) are coded.

Your general approach seems to be a good start, but it doesn't provide confidence intervals for the specific scenarios you are examining. For confidence intervals you need to use not only the variances of the individual coefficient estimates but also the covariances among them. Regression models typically contain the needed variance-covariance matrix but only directly report the variances of the individual coefficients. You then can use the formula for the variance of a weighted sum to get the corresponding variances (and thus standard errors and confidence intervals) for the sums of coefficients like those you are calculating. You should do that work in the original log-odds scales of the coefficients, followed by exponentiation at the end to report results in terms of odds ratios.

Instead of doing this by hand yourself, it's less error-prone to employ well-vetted post-modeling tools. In R one good choice is the emmeans package.

In response to comment and edited question

The list of pairwise comparisons helps to document which differences between scenarios are statistically significant. If both 95% confidence limits are on the same side of 0 (either both positive or both negative) then that suggests a significant difference. It's not clear whether this display takes into account the multiple comparisons, however, so check the documentation. The emmeans package can do a Tukey correction for all pairwise multiple comparisons.

If you want overall estimates of the statistical significance of predictors and interactions, then you need to do joint tests on all the corresponding model coefficients. The default of the Anova() function (note the capital "A") in the R car package is a good choice. That will indicate overall significance of SES, family type, and their interaction across all their levels at once. Even if the overall interaction isn't "significant" on that basis, however, the individual comparisons shown in the table (based on the interaction terms) are still OK provided that the multiple-comparison issue is addressed appropriately.