How to Interpret the weights in Logistic Regression

Logistic regression, also known as binary logit and binary logistic regression, is a particularly useful predictive modeling technique. It is used to predict outcomes involving two options, whether you voted or didn’t vote for example. Below I will try to just explain what the weights mean when you using them for interpretations.

The interpretation of the weights in logistic regression differs from the interpretation of the weights in linear regression, since the outcome in logistic regression is a probability between 0 and 1. The weights do not influence the probability linearly any longer. The weighted sum is transformed by the logistic function to a probability. Therefore we need to reformulate the equation for the interpretation so that only the linear term is on the right side of the formula.

We call the term in the log() function “odds” (probability of event divided by probability of no event) and wrapped in the logarithm it is called log odds.

This formula shows that the logistic regression model is a linear model for the log odds. Great! That does not sound helpful! With a little shuffling of the terms, you can figure out how the prediction changes when one of the features xjxj is changed by 1 unit. To do this, we can first apply the exp() function to both sides of the equation:

Then we compare what happens when we increase one of the feature values by 1. But instead of looking at the difference, we look at the ratio of the two predictions:

We apply the following rule:

And we remove many terms:

In the end, we have something as simple as exp() of a feature weight. A change in a feature by one unit changes the odds ratio (multiplicative) by a factor of exp(βj)exp⁡(βj). We could also interpret it this way: A change in xjxj by one unit increases the log odds ratio by the value of the corresponding weight. Most people interpret the odds ratio because thinking about the log() of something is known to be hard on the brain. Interpreting the odds ratio already requires some getting used to. For example, if you have odds of 2, it means that the probability for y=1 is twice as high as y=0. If you have a weight (= log odds ratio) of 0.7, then increasing the respective feature by one unit multiplies the odds by exp(0.7) (approximately 2) and the odds change to 4. But usually you do not deal with the odds and interpret the weights only as the odds ratios. Because for actually calculating the odds you would need to set a value for each feature, which only makes sense if you want to look at one specific instance of your dataset.

These are the interpretations for the logistic regression model with different feature types:

• Numerical feature: If you increase the value of feature xjxj by one unit, the estimated odds change by a factor of exp(βj)exp⁡(βj)
• Binary categorical feature: One of the two values of the feature is the reference category (in some languages, the one encoded in 0). Changing the feature xjxj from the reference category to the other category changes the estimated odds by a factor of exp(βj)exp⁡(βj).
• Categorical feature with more than two categories: One solution to deal with multiple categories is one-hot-encoding, meaning that each category has its own column. You only need L-1 columns for a categorical feature with L categories, otherwise it is over-parameterized. The L-th category is then the reference category. You can use any other encoding that can be used in linear regression. The interpretation for each category then is equivalent to the interpretation of binary features.
• Intercept β0β0: When all numerical features are zero and the categorical features are at the reference category, the estimated odds are exp(β0)exp⁡(β0). The interpretation of the intercept weight is usually not relevant.

We use the logistic regression model to predict cervical cancer based on some risk factors. The following table shows the estimate weights, the associated odds ratios, and the standard error of the estimates.

Interpretation of a numerical feature (“Num. of diagnosed STDs”): An increase in the number of diagnosed STDs (sexually transmitted diseases) changes (decreases) the odds of cancer vs. no cancer by a factor of 0.44, when all other features remain the same. Keep in mind that correlation does not imply causation. No recommendation here to get STDs.

Interpretation of a categorical feature (“Hormonal contraceptives y/n”): For women using hormonal contraceptives, the odds for cancer vs. no cancer are by a factor of 1.12 higher, compared to women without hormonal contraceptives, given all other features stay the same.

Like in the linear model, the interpretations always come with the clause that ‘all other features stay the same’.