Interpreting logit coefficients that have been converted to probability

Question

Let's say I run the following specification:

fit <- glm(outcome ~ treatment + gender + language + age, data = date_restriction, family = "binomial")

Coefficients:
(Intercept)    treatment       gender      language        age  
   -3.75205      0.29006      0.13278     -0.44440     -0.00104  

The resulting coefficient on treatment is .290. I would like to convert this to probability, so I run the following function:

logit2prob <- function(logit){
  odds <- exp(logit)
  prob <- odds / (1 + odds)
  return(prob)
}

logit2prob(.290)

>> 0.5719961

I am interpreting this to mean that individuals in the treatment condition are 57.1% more likely to do (outcome) than individuals in the control condition. But this seems highly unusual. Is this "%" interpretation consistent with the probability conversion?

Furthermore, I have been given advice to never convert logit coefficients to probabilities, and to instead compute average marginal effects. Doing so yields the following:

>margins(fit) 
 treatment 
 0.007139 

Why does this appear so small relative to the .57 probability estimate above? Is it because marginal effects are interpreted in terms of percentage points, and not percent? What is the most appropriate way to interpret this effect?

Answer 1

The problem here is that the coefficient for a predictor in a logistic regression is not quite what you take it to be. It's the change in logit of the outcome for a one-unit change in the predictor. How much that changes the probability of the outcome depends on the intercept of the model (for the case when treatment = 0 in your example) and any other predictors that might be included.

So when you use logit2prob to convert to a probability you have to include all those other coefficients. For example, if the intercept in your model is 0, the probability of outcome in the control group is logit2prob(0+0) or 0.5; being in the treatment group increases that probability to logit2prob(0+0.29) or your value of 0.57. That's only a 7 percentage-point difference.

If the control group was unlikely to have the outcome (say, intercept = -2 for about a 12% probability of outcome) then your coefficient for treatment changes that to logit2prob(-2+0.29) = 0.153, or about a 15% probability for about a 3 percentage-point change.

This dependence of the outcome probability on the intercept (and on values of other predictors and their coefficients) is a major reason why you have been warned, quite rightly, not to try to convert logistic regression coefficients to probabilities. Converting logistic regression predictions from the logit value returned by the model to a probability, on the other hand, is fine.