What is Logistic Regression? A Comprehensive Guide

Logistic regression is a statistical technique used to determine the relationship between two data factors to make a binary prediction. In business, this categorization takes myriad forms, from predicting whether or not a customer will stop purchasing a company’s products to determining whether or not to approve a loan based on a lender’s attributes. Logistic regression is also a fundamental algorithm in machine learning and statistics. Understanding these main applications and how logistic regression works can help your organization learn how to use this powerful technique. Here’s what you need to know.

What Is Logistic Regression?

Logistic regression is a statistical technique that uses a set of independent variables and a single binary dependent variable to estimate the likelihood of a particular event occurring. Since the predicted outcome is a probability, the dependent variable is either 0 or 1. This is referred to as a logit transformation. Logistic regression, therefore, makes a prediction about two possible scenarios:

• An event doesn’t happen (0)
• An event happens (1)

For example, logistic regression is commonly used to predict whether or not customers will default on their loans as a measure of creditworthiness.

How Logistic Regression Works

Logistic regression employs a logistic function with a sigmoid (S-shaped) curve to map linear combinations of predictions and their probabilities. Sigmoid functions map any real value into probability values between 0 and 1. Understanding the components and concepts that underlie logistic regression can help you understand how the technique works overall.

Dependent and Independent Variables

Logistic regression models have one dependent variable and several independent categorical or continuous predictor variables. Unlike standard linear regression models, logistic regression does not require a linear relationship between the independent and dependent variables. Homoscedasticity—a constant variance of error terms across all independent variables—is not required. However, other requirements do apply, depending on the type of logistic regression. For binary logistic regression, dependent variables must be binary, while ordinal logistic regression requires ordinal dependent variables—variables that occur in natural, ordered categories.

Log-Odds and the Logit Function

In logistic regression, the logit function assigns a number to a probability. So, in the case of a binary logistic regression model, the dependent variable is a logit of p, with p being the probability that the dependent variables have a value of 1. Log-odds are a way to represent odds—the likelihood of an event occurring—in logarithmic function. Log-odds are simply the logarithm of the odds.

Maximum Likelihood Estimation

Maximum likelihood estimation (MLE) is a widely used probabilistic method for estimating the parameters of a logistic regression model. MLE selects the model parameter values that maximize the likelihood function—essentially the parameters that best fit the data.

Odds Ratio and Interpretation

In logistic regression, the odds ratio is the constant effect of an independent predictor variable on the likelihood that a particular dependent outcome will occur. A ratio greater than 1 denotes a positive association or higher odds of the outcome, whereas a ratio less than 1 denotes a negative association or lower odds.

Assumptions of Logistic Regression

Logistic regression relies on the following underlying assumptions and requirements:

• The dependent variable must be binary for binary logistic regression and ordinal for ordinal logistic regression.
• Observations must be independent of each other and not originate from repeated measurements.
• Independent variables must not have multicollinearity, meaning two or more independent variables must not be correlated.
• Independent variables must be linearly related to the dependent variable’s log-odds.
• The logistic regression model was created with large sample sizes.

Three Types of Logistic Regression

The following are three of the most commonly used types of logistic regression:

• Binary Logistic Regression: Binary logistic regression is employed when the dependent variable has only two outcomes—in this case, the dependent variable is referred to as a dichotomous variable. A common binary logistic regression scenario involves predicting a positive or negative value or yes/no value.
• Multinomial Logistic Regression: Multinomial logistic regression is used when the dependent variable is nominal with more than two categories without a rank or order. In other words, it’s used to predict whether the dependent variable belongs to more than two categories.
• Ordinal Logistic Regression: Ordinal logistic regression is used to make predictions when three or more categories exist with a natural ordering but not necessarily with even intervals. Ranking tasks commonly employ ordinal logistic regression, such as classifying a student’s performance as above average, average, or below average.

Logistic Regression Model Applications

Most logistic regression use cases involve binary logistic regression, determining whether an example belongs to a particular class. Many practical problems require a straightforward yes-or-no prediction, and logistic regression provides fast and accurate predictions that are simpler to interpret and computationally efficient. Additionally, binary outcomes are often easy to measure and collect and align with many binary-targeted business, healthcare, and technology goals.

Logistic regression serves a range of applications in healthcare, finance, and marketing. Using logistic models with independent predictor variables like weight, exercise habits, and age, for example, cardiologists can determine the likelihood of patients suffering first-time or repeat heart attacks. Logistic regression models are also widely used in finance to assess the credit risk of individuals and organizations applying for loans. Marketers use logistic regression to predict customer purchasing habits based on predictors like age, location, income, and education level.

Advantages and Limitations of Logistic Regression

Logistic regression is highly versatile and applicable across a wide array of fields and disciplines due to several key advantages, the most important being its ease of use and explainability. Logistic models are straightforward to implement, easy to interpret, and can be efficiently trained in a short amount of time. They can easily be adapted to take on multiple classes and probabilistic models and can use model coefficients to show which features are most important. Logistic regression predictive models are also less prone to overfitting, which is when the number of observations exceeds the number of features.

However, logistic regression also has some limitations. Logistic models are predicated on the assumption of linearity between the independent variables and the log-odds of the dependent variable. This assumption can hinder model performance in highly nonlinear scenarios. Overfitting may also occur if the number of features exceeds the number of observations, and logistic regression only works when there is low or no multicollinearity between independent variables.

Logistic Regression Model Evaluation: Assessing Performance and Accuracy

Data professionals use various statistical methods to assess the performance and accuracy of logistic regression models. These measures are often incorporated into artificial intelligence/machine learning (AI/ML) platforms as explainable AI (XAI) tools to comprehend the results of ML algorithms and bolster trust in their predictions.

Confusion Matrices

Despite its name, a confusion matrix summarizes a classification model’s performance straightforwardly. Its purpose is to reveal the types of errors a model makes—where it might be “confusing” classes. Since logistic regression is often applied to binary or multiclass classification tasks, the confusion matrix breaks down these predictions into counts of true positives (TP), true negatives (TN), false positives (FP), and false negatives (FN).

Accuracy, Precision, and F1 Score

Data practitioners can use the numbers derived from a confusion matrix to calculate their logistic regression models’ accuracy, recall, and F1 score.

• Accuracy: Accuracy or precision is measured by dividing the number of correct predictions—true positives and true negatives—by the total number of predictions.

(TP + TN) / (TP + FP + TN + FN)

• Precision: Also referred to as recall, sensitivity, or true positive rate, this measures the proportion of actual positives, calculated by dividing the number of true positives (TP) by the number of true positives and false positives.

TP / (TP + FP)

• F1 Score: This score provides a single, balanced measure of accuracy and recall by taking the harmonic mean of both values.

2 * (precision + recall / precision * recall)

ROC Curve and AUC Score

The receiver operating characteristic (ROC) curve and area under the ROC curve (AUC) represent a logistic regression classifier’s model performance by depicting the trade−off rates between TPs and FPs for given categorization criteria. Data professionals can visually observe a model’s ROC Curve and calculate its AUC score to gauge its accuracy and reliability. A better-performing model has a higher AUC score.

Steps and Considerations for Training a Logistic Regression Model

The procedure to train a logistic regression model is similar to that of other ML predictive models. The following comprise the three main steps:

1. Data Preparation: Create the labeled dataset with input features (independent variables) and output labels (dependent variables), preprocess the data, and split it into training and test sets.
2. Model Initialization and Training: Using the training data, determine each feature’s coefficients or weights that minimize the cost function and minimize using gradient descent.
3. Model Evaluation: Using the test data, gauge the model’s accuracy by applying the previously discussed evaluation methods, such as confusion matrices or accuracy/precision/F1 score.
4. Making Predictions: Once the logistic regression model is trained and the results are validated, it can predict the probability of a binary outcome for new, unseen data. At this stage, models are typically deployed to make real-time predictions in production environments and serialized and saved for future use.

Eight Alternatives to Logistic Regression

Logistic regression is not the only statistical model you can use for predictions. Here are eight of the most popular alternatives:

• Linear Regression: If predicting a continuous value, simple linear regression using a straight line may be more appropriate for estimating the relationship between one independent predictor variable and one dependent outcome variable.
• Polynomial Regression: Polynomial regression can estimate the relationship between a predictor and an outcome variable using the predictor variable’s nth-degree polynomial for more complex variable relationships.
• Ridge Regression: A ridge regression is a statistical regularization technique that corrects overfitting on training data by minimizing the sum of squared differences between observed and predicted values.
• Lasso Regression: Lasso regression, or L1 regularization, is another statistical regularization technique that corrects overfitting by applying a penalty to enhance model accuracy.
• Elastic Net Regression: Elastic net linear regression combines lasso and ridge techniques to regularize regression models for better accuracy and is especially effective at handling multicollinearity and overfitting.
• Decision Trees Regression: Decision tree regression uses a tree-like model to predict continuous numerical values and is ideal for use over logistic regression when categorical outcomes are not required, data sets are large, and feature/target variable relationships are complex and non-linear.
• Random Forest Regression: Random forest regression brings together multiple decision trees to create a single predictive model. In a random forest, each “tree” makes its own unique predictions from a different subset of data, and final predictions are made using the average or weighted average of the combined trees’ predictions.
• Support Vector Machine: Support vector machine (SVM ) is an ideal algorithm for learning complex, non-linear functions. SVM works by creating a hyperplane or line/decision boundary that separates data into classes and is widely used in compound classification, ranking, and regression problems.

Popular Software for Logistic Regression

While a number of tools can be used to help perform logistic regression and other statistical analysis, RStudio, JMP, and Minitab stand out.

Frequently Asked Questions (FAQs)

Bottom Line: Mastering Logistic Regression

Many data professionals regard logistic regression as their preferred statistical method, and for good reason: it is a powerful tool for modeling binary outcomes, with applications across diverse fields like medicine, finance, and marketing. Mastering logistic regression equips you with an invaluable tool for risk assessment, diagnosis, and decision-making tasks, making it an essential component of any data professional’s toolkit.

Explore our list of the top AI companies defining the industry to learn more about the tools, apps, and platforms at the forefront of this dynamic technology.

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