Understanding the Common Ground Between Linear and Logistic Regression in Data Science

When preparing for interviews in #DataScience or #MachineLearning, questions like this are common, but the depth of understanding they require often trips people up. On the surface, linear and logistic regressions seem quite different—linear predicts continuous outcomes, while logistic predicts probabilities for categorical outcomes. So, what’s the common ground? Let’s dig deeper, not by focusing on the mathematical formulas, but by understanding the underlying concept that unites them: Generalized Linear Models (GLM).

Generalized Linear Models (GLM): A Unifying Framework

Both linear regression and logistic regression are special cases of Generalized Linear Models (GLM). GLM offers a broad statistical framework that extends the linear model to allow for response variables that follow different types of distributions, not just a normal distribution.

1. The Core Structure of GLM:

GLM consists of three key components:

• Random Component: Defines the probability distribution of the response variable (e.g., normal distribution for linear regression, binomial distribution for logistic regression).
• Systematic Component: This refers to the linear combination of predictor variables, often represented as XβX\betaXβ, where XXX is the matrix of input features and β\betaβ is the vector of coefficients.
• Link Function: This function connects the random and systematic components, transforming the predicted value to fit the desired distribution of the response.

What Do They Have in Common?

2. Prediction of a Numerical Outcome:

At their core, both linear and logistic regression aim to predict a numerical outcome. In both cases, we model the expected value of a response variable based on a linear combination of predictor variables.

• In linear regression, the outcome is a continuous numerical value, such as predicting house prices or stock returns.
• In logistic regression, the outcome is the probability of an event occurring, such as whether a customer will churn or not. Although the final prediction is binary, what we are truly modeling is the log-odds (logarithm of the odds) of an event happening, which is a numerical value.

3. The "Kind of Thing" Being Predicted:

Both linear and logistic regressions predict the expected value of the response variable. In linear regression, this expected value is simply the continuous variable itself (e.g., the actual house price). In logistic regression, the expected value represents the probability of a categorical outcome (the likelihood that the event occurs).

While linear regression models the expected outcome directly, logistic regression models the expected probability through a transformation (the logit function), which converts the probability into a numerical format that can be linearly modeled.

4. A Focus on the Left Side of the Equation:

As the interview question hints, rather than focusing on the right side of the equation (the predictors, or XβX\betaXβ), the key insight lies on the left side, the outcome variable. In both linear and logistic regression, this outcome is numerical—either a direct value (in linear regression) or a transformed value like a probability or log-odds (in logistic regression).

Why Is This Important?

Understanding this commonality is crucial in data science because it highlights that the distinction between regression models isn't always about the types of data (numerical vs. categorical) but rather about how we choose to model the relationship between predictors and the outcome. Once you grasp that both linear and logistic regression predict an expected numerical value, you unlock a broader understanding of regression models and their applications.

Final Thoughts

In summary, both linear and logistic regressions are grounded in the concept of predicting a numerical outcome, whether that outcome is a continuous value or a probability transformed into a log-odds scale. Recognizing this shared foundation through the lens of Generalized Linear Models (GLM) helps unify these seemingly different techniques under a broader statistical framework.

Next time you’re in an interview or working on a data science project, remember: linear and logistic regressions aren’t so different after all—they are both about predicting the expected value of something meaningful.