Interpret multiple logistic regression

Question

I have two groups of patients who were part of an intervention many years ago, and some covariates about their characteristics e.g., age, BMI, and years since the intervention. I have conducted a logistic regression in order to see the association of the intervention (yes/no) with a clinical outcome (yes/no), controlling for these known individual factors such that

$Outcome = \beta_0 + \beta_1 Intervention + \beta_2 Age + \beta_3 BMI + \beta_3 YearsSinceInt$

I have obtained some coefficients from my logistic regression but now when it comes to interpret the results I am being asked by a physician to interpret each coefficient in order to see the effect of each variable on having the binary outcome. I am a bit afraid of running into the Table 2 Fallacy so I argue that we should only interpret the Intervention variable. Nonetheless, it is a cross-sectional study so I am a bit torn whether we can actually give interpretation to each coefficient and only talk about association and never about causal effects.

What do you suggest? Can I give interpretation to each coefficient or I should only interpret the Intervention coefficient? Is there a way to see the effect of each variable on the outcome without running the risk of falling into Table 2 Fallacy and over-interpretating control variables?

Answer 1

The interpretation of fitted regression coefficients $\beta_i$ as causal effect requires some additional insight into your scenario. E.g., if you know for sure that your variable Intervention is not causally affecting any of the other regressors Age, BMI, or YearsSinceInt, then you can interpret $\beta_1$ as the total causal effect of Intervention on the (log-odds of) Outcome. Otherwise, it is only the direct causal effect.

For your variables other than Intervention you first have to convince yourself, probably through domain knowledge, that they are causing Outcome and not the other way around, if at all. The fitted values don't tell you that. Once you have done that, it is similar to the interpretation of the fitted coefficients of your Intervention variable: If you know, through domain knowledge, that any variable is causally influencing another variable, its coefficient is only the direct causal effect, not the total. At least, you don't have to bother about your variables having a causal effect on the variable Intervention because, being intervened on, it is not causally affected by anything.

In general, if you don't know anything about the "causal directions", the fitted coefficients cannot be interpreted as causal effects. They only help you to obtain the covariance between the pertaining variable and the dependent variable if all the other variables are fixed. And, of course, you can use them for prediction: predict the value of Outcome for given values of your independent variables.