# Interpreting logit coefficients that have been converted to probability

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?

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.

• Thank you so much! I have just added in intercept and covariate information. Based on this post, I assume I would do: logit2prob(-3.75205 + .290) = 0.0304 Comparing that to the probability of being in the control group logit2prob(-3.75205) yields 0.022, or a difference of 0.0074, which is very close to the output from the margins command. Does this look right? I interpret this as essentially <1 percentage point difference? Is it also fair to say that the treatment has about 33% higher odds of the outcome than the control? exp(.290) = 1.336427 Sep 15, 2019 at 18:19
• @Parseltongue the interpretations in your comment seem to be OK. I would caution you, however, that many people (myself included) don't have a good instinctive understanding of changes in odds or odds ratios. I'm much more comfortable with changes in probabilities. One example: the same 33% higher odds of the outcome in the treatment group would mean a 7 percentage-point difference if your intercept had been 0, versus a 0.7% percentage-point difference with your intercept of -3.75.
– EdM
Sep 16, 2019 at 14:20
• Gotcha, thanks Ed. I suppose that's the reason I've been cautioned against using odds ratios. I'm wondering if it's valid to interpret a "33% higher odds of the outcome" as a "33% higher likelihood of the outcome." Are those compatible? They strike me as slightly different (the latter seems to connote probability), but I've seen % changes in odds ratios interpreted in this fashion Sep 16, 2019 at 16:11
• @Parseltongue "likelihood" has an important specific technical meaning in statistics, so I would recommend against using that word loosely when other words are available. But that word is certainly more closely related to probabilities than to odds, as you note..
– EdM
Sep 16, 2019 at 16:42