# Interpreting Logit Results

I have a logit model where the explanatory variable is the percent of total a total. As this is not a binary model I am unsure how to read the regression results.

I understand that the coefficient normally equates to the log odds which I can then convert to the probability using:

exp(coef) / (1 + exp(coef))

but I am still unsure how to interpret this in terms of the share of total explanatory.

for example: b_1 = 0.7166

so after converting to a probability: b_0 = exp(0.7166) / (1 + exp(0.7166)) = 0.6718

If I were interpreting this like an OLS (which I do not believe makes sense) I would say that a 1 point increase in the explanatory variable is associated with a 0.6718 percentage point increase in the dependent variable.

However, I do not know how to construct similar statement with the logit.

To make matters more complicated (at least to me) my explanatory variable is expressed as a decimal percent (i.e 10% = 0.10).

Should I first divide my coefficient by 100 so that initial part of my statement is correct "A one percentage point increase in.."?

Ultimately my goal is to be able to make a statement about how a change in the explanatory variable affects the share value of my dependent variable.

Let's look at some particular toy data in R which also gives a coefficient of about $$0.71$$:

datf <- data.frame(x = c(0.19, 0.21, 0.37, 0.40, 0.55, 0.56),
success = c(  0,    1,    1,    1,    0,    1))
fit <- glm(success ~ x, data=datf, family="binomial")
fit
#    Coefficients:
# (Intercept)            x
#      0.4251       0.7102


What this is saying is the maximum likelihood logistic curve has

• the log-odds as $$0.4251+0.7102x$$
• the odds as $$e^{0.4251+0.7102x}= 1.52968 \times 2.0343^x$$
• the probability as $$\frac{1.52968 \times 2.0343^x}{1+1.52968 \times 2.0343^x}$$

So when $$x=0$$ you would get predicted log-odds of $$0.4251$$, odds of $$1.5297$$ and a probability of $$0.6047$$; when $$x=1$$ you would get predicted log-odds of $$1.1352$$ ($$0.7102$$ more), odds of $$3.1118$$ (multiplying by $$e^{0.7102}=2.0343$$) and a probability of $$0.7568$$

A $$1$$ percentage point increase in $$x$$ increases the predicted log-odds by $$0.004251$$ and multiplies the predicted odds by $$2.0343^{0.01}=1.00713$$. You cannot make such a simple statement about the predicted probabilities.

• Thank you this is tremendously helpful!!! Commented Sep 17, 2021 at 16:35