# Converting log odds coefficients to probabilities

Suppose we've ran a logistic regression on some data where all predictors are nominal. With dummy coding the coefficients are ratios of log odds to the reference levels. A colleague claims that we can "solve for $p$" or "apply the inverse logit to the estimated parameters" interpret the result as relative probabilities, but my intuition tells me that the logit function isn't so well behaved. See the more concrete example below:

mydata<-data.frame(outcome = rbinom(1000, 1, 0.3),
Race=sample(c("White", "Black", "Hispanic", "Other"), 1000, replace=TRUE),
Gender=sample(c("M", "F"), 1000, replace=TRUE))
myglm <- glm(outcome ~ Race + Gender, family=binomial(), data=mydata)
x <- summary(myglm)\$coefficients[,1]
cbind(coef_log_odds = x, coef_prob = exp(x) / (1 + exp(x)) * 2)

#              coef_log_odds coef_prob
# (Intercept)    -1.09760895 0.5003763
# RaceHispanic    0.18458782 1.0920327
# RaceOther       0.08886623 1.0444039
# RaceWhite       0.04471730 1.0223549
# GenderM         0.40350628 1.1990596


We see that coef_prob for GenderM is 1.199. Can this be interpreted as "holding all else fixed, males are 1.2 times as likely as females to have outcome1"?

• You've applied twice the inverse logit. Apr 24, 2014 at 10:42

$$\frac{\frac{\exp (-1.0976 + 0.4035)}{1 + \exp (-1.0976 + 0.4035)}}{ \frac{\exp (-1.0976)}{1 + \exp (-1.098)}}\approx 1.331$$
$$\frac{\frac{\exp (-1.0976 + 0.1846 + 0.4035)}{1 + \exp (-1.0976 + 0.1846 + 0.4035)}}{ \frac{\exp (-1.0976+ 0.1846 )}{1 + \exp (-1.098+ 0.1846)}}\approx 1.311$$