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Logistic regression intercept is the point at which the logistic regression line crosses the y-axis. It is the value of the response variable when all the predictor variables are equal to zero. An example of this is when predicting whether a student will pass or fail a course, the intercept might be the probability of passing when all of the predictor variables such as the student’s GPA, course difficulty, and attendance are set to zero.
Logistic regression is a method we can use to fit a regression model when the 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 = eβ0 / (1 +eβ0)
The following example shows how to interpret a logistic regression intercept in practice.
Related:
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 , , , or ) 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 = eβ0 / (1 +eβ0)
- 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.