Odds ratio, what is the change in percentage? I have been asked to describe what percentage-change OR 0.4 and 6.8 mean for the participants of my study, related to categories of nurtrition and level of hospital care. I have performed mixed models, for categorical outcomes the model was a multiple logistic mixed model.
After surching the whole web, notting but the web and the one and only web, I am confused as to whether this is an approach I should even try for?
If the change in percent is something I should provide, then what do I do?
It is important for me that the consept of odds ratio is not lost in my paper, but I struggle with how to explain the findings in a way that will please the reviewers and the common reader (if ever published)
Thankfull for any answers - I am absolutely a novise in this...
 A: In (mixed effect) logistic regression, which I expect you have conducted, the odds ratio is a way to express the effect of your predictor of interest conditional on the other predictors that is invariant to the levels of the other predictors. This why the OR is widely used in discussions of logistic regression, even though it is far less intuitive to understand than relative risks or absolute changes in probabilities.
It is unfortunately not trivial to transform a statement like, for example, females have three times the odds of men for reacting positively to treatment X, adjusting for important risk factors into a probability-based statement [you could phrase it as "300% the odds" if you want to use percentages, as your question title suggests, but I guess this is not really what you want], because in a non-linear probability model, the probabilities will depend on other values of your predictors. What you can do is arrive at predicted probabiltities for fixed values of the other predictors. This is easy to grasp. Look at the following logistic function and you will see that a fixed increase in x will have a different meaning (in probability terms) depending on where you are on the x axis:

So you could arrive at a statement like: females at age Z with underlying diseases Y will have a 10 percentage points higher probability of reacting positively to treatment X than men. This can be easier to grasp for the reader and can easily be plotted/tabulated for some important or typical combinations of predictor values (nurtrition and level of hospital care, in your case).
You can also pick some informative probability pairs, as for example in Table 2 of this paper:

*

*Liberman, A. M. (2005). How much more likely? The implications of odds ratios for probabilities. American Journal of Evaluation, 26(2), 253-266.

This is a very accessible and open access introduction to the interpretation and transformation of odds ratios:

*

*Norton, E. C., Dowd, B. E., & Maciejewski, M. L. (2018). Odds ratios—current best practice and use. Jama, 320(1), 84-85.

In line with what you have been asked to do, the authors also propose to communicate outcomes from logistic regression in several ways to make interpretation easier:
"The reader should understand odds ratios in the context of other information, such as the underlying probability. When the probabilities are small, odds ratios and relative risk ratios are nearly identical, but they can diverge widely for large probabilities. The magnitude of the odds ratio is hard to interpret because of the arbitrary scaling factor and cannot be compared with odds ratios from other studies. It is best to examine study results presented in several ways to better understand the true meaning of study findings."
