Logistic regression is a method we can use to fit a regression model when the response variable is binary.
When we fit a logistic regression model, the intercept term in the model output represent the log odds of the response variable occurring when all predictor variables are equal to zero.
However, since log odds is hard to interpret we usually frame the intercept in terms of probability.
We can use the following formula to understand the probability that the response variable occurs, given that each predictor variable in the model is equal to zero:
P = e^{β0} / (1 +e^{β0})
The following example shows how to interpret a logistic regression intercept in practice.
Related: How to Interpret Logistic Regression Coefficients
Example: How to Interpret Logistic Regression Intercept
Suppose we would like to fit a logistic regression model using gender and number of practice exams taken to predict whether or not a student will pass a final exam in some class.
Suppose we fit the model using statistical software (such as R, Python, Excel, or SAS) and receive the following output:
Coefficient Estimate | Standard Error | Z-Value | P-value | |
---|---|---|---|---|
Intercept | -1.34 | 0.23 | 5.83 | <0.001 |
Gender (Male = 1) | -0.56 | 0.25 | 2.24 | 0.03 |
Practice Exams | 1.13 | 0.43 | 2.63 | 0.01 |
We can see that the intercept term has a value of -1.34.
This means that when Gender is equal to zero (i.e. the student is female) and when Practice Exams is equal to zero (the student took no practice exams in preparation for the final exam) the log odds of the student passing the exam are -1.34.
Since log odds are hard to understand, we can instead rewrite things in terms of probability:
- Probability of Passing = e^{β0} / (1 +e^{β0})
- Probability of Passing = e^{-1.34} / (1 +e^{-1.34})
- Probability of Passing = 0.208
When both predictor variables are equal to zero (i.e. a female student who took no prep exams), the probability that the student passes the final exam is 0.208.
Additional Resources
The following tutorials provide additional information about logistic regression:
How to Report Logistic Regression Results
Understanding the Null Hypothesis for Logistic Regression
The Difference Between Logistic Regression vs. Linear Regression