Interpretation of Fixed Effects from Mixed Effect Logistic Regression

I am confused by statements at a UCLA webpage about mixed effects logistic regression. They show a table of fixed effects coefficients from fitting such a model and the first paragraph belows seems to interpret the coefficients exactly like a normal logistic regression. But then when they talk about odds ratios, they say you have to interpret them conditional on the random effects. What would make the interpretation of the log-odds different than their exponentiated values?

1. Wouldn't either require "holding everything else constant"?
2. What is the proper way to interpret fixed effect coefficients from this model? I was always under the impression nothing changed from the "normal" logistic regression because the random effects have expectation zero. So you interpreted log-odds and odds ratios exactly the same with or without random effects - only the SE changed.

The estimates can be interpreted essentially as always. For example, for IL6, a one unit increase in IL6 is associated with a .053 unit decrease in the expected log odds of remission. Similarly, people who are married or living as married are expected to have .26 higher log odds of being in remission than people who are single.

Many people prefer to interpret odds ratios. However, these take on a more nuanced meaning when there are mixed effects. In regular logistic regression, the odds ratios the expected odds ratio holding all the other predictors fixed. This makes sense as we are often interested in statistically adjusting for other effects, such as age, to get the “pure” effect of being married or whatever the primary predictor of interest is. The same is true with mixed effects logistic models, with the addition that holding everything else fixed includes holding the random effect fixed. that is, the odds ratio here is the conditional odds ratio for someone holding age and IL6 constant as well as for someone with either the same doctor, or doctors with identical random effects

• I may be wrong but doubt it. There is no special consideration for odds ratios over differences in log odds. Holding everything else constant means conditional on both remaining fixed and random effects. "people who are married or living as married are expected to have .26 higher log odds of being in remission than people who are single" should have "if they have same age, ILS and random intercept value" added to it. It's a plain old equation. – Heteroskedastic Jim Sep 8 '18 at 3:42

An easy example to think of is the following: Say that you have a multi-center clinical trial in which patients in each hospital are randomized to two treatments, A or B. Say also that the outcome of interest is binary (e.g., did the patient require an operation, yes or no). To account for the multi-center nature of the trial we fit a mixed effects logistic regression with a random effect per hospital (i.e., a random intercepts model). From this model we get the regression coefficient for the treatment variable, say $$\beta$$. This $$\beta$$ is the logs odds ratio between the two treatments for patients coming from the same hospital. Now, if you had analyzed the same data with a generalized estimating equations (GEE) approach, then you would get coefficients with a marginal interpretation. Continuing in the example above, the estimated coefficient $$\beta$$ from a GEE would be the log odds ratio between the two treatments for patients across hospitals - in other words the log odds ratio averaged over the hospitals.
There are ways to obtain coefficients with a marginal interpretation from a mixed effects logistic regression. For more details on this, you can have a look in Section 5.2 of my course notes. For an implementation in R of this approach to obtain coefficients with a marginal interpretation from a GLMM, check function marginal_coefs() in the GLMMadaptive package; more info is also available here.
• No, even if you do not exponentiate, the log odds will still have a subject-specific interpretation. I.e., in a mixed effects logistic regression you model $\log \frac{\mbox{Pr}(Y = 1 | b)}{1 - \mbox{Pr}(Y = 1 | b)}$. If you take the expectation w.r.t. the distribution of the random effects you get $X\beta$, the fixed-effects part. But $E_b\{\log \frac{\mbox{Pr}(Y = 1 | b)}{1 - \mbox{Pr}(Y = 1 | b)}\} = X\beta \neq \log \frac{E_b\{\mbox{Pr}(Y = 1 | b)\}}{1 - E_b\{\mbox{Pr}(Y = 1 | b)\}}$, which are the marginal log odds. – Dimitris Rizopoulos Sep 9 '18 at 19:03