Logistic Regression: A Comprehensive Introduction

10 min readFeb 6, 2023

Logistic regression is a powerful statistical tool used in predictive analytics to identify relationships between a set of independent and dependent variables. It is used in various applications, such as predicting the probability of an event occurring, forecasting customer behaviour, and understanding the impact of certain variables on a process.

This article will explore the fundamentals of logistic regression, its advantages and disadvantages, how it can be used, and the tools available for conducting logistic regression analyses.

What is Logistic Regression?

Logistic regression is a type of regression analysis used to predict the outcome of a categorical dependent variable based on one or more independent variables. It is a statistical technique that uses a logistic function to model a binary dependent variable, where the outcome can be either 0 (for example, no) or 1 (for example, yes).

For example, logistic regression would be a suitable tool to predict whether a customer will purchase a product.

The logistic function is an S-shaped curve that takes an input value and transforms it into a value between 0 and 1.

This output value can then be interpreted as the probability of a certain event occurring.

When used to model a binary dependent variable, logistic regression allows us to calculate the probability of the dependent variable being 0 or 1 for any given set of independent variables.

Interpreting Logistic Regression

Logistic regression is an interpretable model because the coefficients of the independent variables are easy to interpret.

For example, the coefficient associated with an independent variable can be interpreted as the increase in the odds of the dependent variable being 1 for a unit increase in the independent variable.

This makes it easy to assess the strength of the relationship between a given independent variable and the dependent variable.

Benefits of Using Logistic Regression

Logistic regression is widely used in predictive analytics due to its simplicity and interpretability. It is also relatively easy to implement, meaning it can be used even by people with minimal technical knowledge. Additionally, logistic regression output can be used to make decisions in various contexts.

Logistic regression can also handle non-linear relationships between independent variables and the dependent variable. This makes it a powerful tool for uncovering complex relationships between variables.

Linear Regression vs Logistic Regression

Linear and logistic regression are both commonly used statistical models for predictive analysis. However, they have different objectives and use cases:

Linear Regression is used to predict a continuous dependent variable from one or more independent variables. It models the relationship between the independent and dependent variables as a linear equation.

Logistic Regression, on the other hand, is used for classification tasks where the dependent variable is categorical. It models the probability of the dependent variable in a particular category given the independent variables, and outputs a binary result (yes/no, true/false, etc.).

In summary, Linear Regression is used to predict a continuous variable, while Logistic Regression is used to predict a categorical (binary) dependent variable.

Types of Logistic Regression

There are a few different types of logistic regression that can be used in different contexts:

a. Binary logistic regression

The first type is binary logistic regression, which is used to model the relationship between a binary dependent variable and one or more independent variables. This type of logistic regression is used when the outcome of interest is a yes or no (for example, will a customer purchase a product).

b. Multinomial logistic regression

The second type is multinomial logistic regression, which is used to model the relationship between a multinomial (more than two categories) dependent variable and one or more independent variables. This type of logistic regression is used when the outcome of interest has more than two categories (for example, will a customer purchase product A, B, or C).

c. Ordinal logistic regression

The third type is ordinal logistic regression, which is used to model the relationship between an ordinal (ranked) dependent variable and one or more independent variables. This type of logistic regression is used when the outcome of interest is on a scale (for example, how likely is a customer to purchase a product on a scale of 1 to 5).

Understanding Independent and Dependent Variables

In logistic regression, the independent variables are the factors that are used to predict the outcome of the dependent variable. The dependent variable is the outcome that is being predicted.

When building a logistic regression model, choosing independent variables that are highly correlated with the dependent variable is important. This means that the independent variables should be strongly related to the outcome that is being predicted. The independent variables should also be independent of each other, which means that they should not be highly correlated with each other.

It is important to note that having a strong relationship between independent variables and the outcome does not guarantee that a particular model will perform well. Other factors, such as model complexity, overfitting, and choice of variables, also play a role in model performance.

How to Build a Logistic Regression Model

Building a logistic regression model involves the following steps:

Define the Problem: Determine the problem you want to solve and the type of data you have available to solve it.
Data Preparation: Clean and prepare the data for analysis. Handle missing values and outliers, if any.
Exploratory Data Analysis (EDA): Analyze the data to gain insights and identify any relationships between the independent variables and the outcome.
Feature Selection: Select the independent variables to include in the model based on the EDA and domain knowledge results.
Model Building: Build the logistic regression model using a statistical software package or programming language of your choice.
Model Validation: Validate the model using techniques such as cross-validation or split sample validation to ensure the model generalizes well to new data.
Model Interpretation: Interpret the model results and understand the contribution of each independent variable to the predicted outcome.
Model Deployment: Deploy the model in production to predict new data.

It is important to note that building a logistic regression model involves making trade-offs between model accuracy and interpretability. Therefore, it is important to carefully consider the problem, data, and modelling techniques to build a model that fits your needs.

Analyzing Logistic Regression Results

The output of a logistic regression model can be analyzed in a few different ways. Here are some of them:

Model Fit: You evaluate this by measuring how well the model fits the data. Common goodness-of-fit measures include the deviance, Akaike Information Criterion (AIC), and Bayesian Information Criterion (BIC). A lower deviance or AIC/BIC value indicates a better fit.
Coefficient Estimates: The coefficient estimates represent the change in the log odds of the outcome for a one unit change in the independent variable, while holding all other variables constant. The magnitude of the coefficients can be used to rank the importance of the independent variables in predicting the outcome. The p-values associated with each coefficient can be used to assess the significance of each independent variable. A p-value less than 0.05 indicates that the independent variable is significantly associated with the outcome.
Odds Ratios: Odds ratios provide the ratio of the odds of the outcome for a one unit increase in the independent variable, while holding all other variables constant. An odds ratio greater than 1 indicates that the odds of the outcome increase with a one unit increase in the independent variable, while an odds ratio less than 1 indicates that the odds of the outcome decrease.
Classification Matrix: A classification matrix, also known as a confusion matrix, is a table that summarizes the performance of the model in terms of true positive (TP), false positive (FP), true negative (TN), and false negative (FN) predictions. It can calculate various performance metrics such as accuracy, precision, recall, and F1 score.
ROC Curve: The Receiver Operating Characteristic (ROC) curve is a graphical representation of the performance of the model, showing the trade-off between true positive rate (sensitivity) and false positive rate (1-specificity). The ROC curve is plotted by the true positive rate against the false positive rate at various classification thresholds. The area under the ROC curve (AUC) is a commonly used metric to evaluate the model's performance, with a value of 0.5 indicating a random model and a value of 1 indicating a perfect model.
Model Limitations: Analyzing the model's limitations can help identify areas for improvement. Common limitations include overfitting, underfitting, lack of predictive power, and violation of assumptions.

Applications of Logistic Regression

Logistic regression can be used in a variety of applications.

Binary classification: Logistic regression is commonly used for binary classification problems, where the goal is to predict one of two possible outcomes, such as yes/no, true/false, or positive/negative.
Predicting the likelihood of an event: Logistic regression can be used to predict the likelihood of an event happening based on a set of predictor variables. For example, it can be used to predict the likelihood of a customer buying a product or a patient developing a certain medical condition.
Modelling the relationship between variables: Logistic regression can be used to model the relationship between a binary outcome variable and one or more predictor variables. This allows us to identify which variables are most strongly associated with the outcome and how they influence the likelihood of the outcome.
Assessing the impact of changes in predictor variables: Logistic regression can be used to assess the impact of changes in predictor variables on the likelihood of the outcome. For example, it can be used to evaluate the impact of changes in marketing strategy on customer behaviour.
Exploring and testing hypotheses: Logistic regression can be used to explore and test hypotheses about the relationship between predictor variables and the outcome. This allows us to determine which variables are significant predictors of the outcome and which are not.

Logistic Regression Analysis Tools

There are a variety of tools available for conducting logistic regression analyses. Many of these tools are open source and free to use, making them accessible to people with any budget.

You can use the following tools for logistic regression:

R: It’s an open-source programming language and software environment for statistical computing and graphics which comes with a wide range of packages for logistic regression, including the base R package “stats” and specialized packages such as “glmnet” and “caret.”
SAS: Or Statistical Analysis Software, is a commercial software suite for data management and statistical analysis. It has a comprehensive suite of tools for logistic regression, including the PROC LOGISTIC procedure.
SPSS: Or Statistical Package for Social Sciences, is a commercial software package for statistical analysis. It provides a graphical user interface for conducting logistic regression and a syntax-based interface for more advanced users.
Stata: A commercial software package for data management and statistical analysis. It has a comprehensive suite of tools for logistic regression, including the logit and probit commands.
Excel: A widely used spreadsheet program that can be used for logistic regression through add-ins such as the Data Analysis ToolPak or the XLSTAT statistical software.
MATLAB: A high-level programming language and software environment for numerical computing and data analysis. It has a comprehensive suite of tools for logistic regression, including the Statistics and Machine Learning Toolbox.

These are just a few examples of the many tools available for conducting logistic regression analysis. The choice of tool depends on the specific requirements of the analysis, as well as personal preferences and skills.

Common Mistakes to Avoid When Conducting Logistic Regression

When conducting logistic regression, it is important to avoid some common mistakes. Here are some common mistakes to avoid when conducting logistic regression analysis:

Not transforming skewed variables: Logistic regression assumes that the predictor variables are linearly related to the log odds of the outcome. If this assumption is not met, the predictor variables should be transformed to meet this assumption.
Not handling collinearity: High levels of collinearity between predictor variables can lead to unstable and unreliable results in logistic regression. Identifying and addressing collinearity in the data is important before conducting the analysis.
Not using appropriate sample size: Logistic regression requires a sufficient sample size to produce reliable results. A general rule of thumb is to have at least 10–15 events per predictor variable in the sample.
Not checking model assumptions: Logistic regression has several underlying assumptions, including independence of errors, linearity of log odds, homoscedasticity, and absence of multicollinearity. It is important to check these assumptions and make necessary adjustments to the model if they are not met.
Overfitting the data: Logistic regression models can easily overfit the data, leading to too complex models performing poorly on new, unseen data. Regularization techniques, such as L1 or L2 regularization, can be used to prevent overfitting.
Ignoring interactions and non-linear relationships: Logistic regression models can only model linear relationships between predictor variables and the outcome. If interactions or non-linear relationships are present in the data, these should be considered in the model specification.
Need to interpret the results correctly: Logistic regression models provide estimates of the relationship between the predictor variables and the outcome. It is significant to interpret the results in a meaningful and appropriate context, considering the underlying assumptions and limitations of the model.

Conclusion

Logistic regression is a powerful tool for predictive analytics. It is relatively easy to implement, interpretable, and can be used to uncover complex relationships between variables. In this article, we have explored the fundamentals of logistic regression, its advantages, how it can be used, and the tools available for conducting logistic regression analyses.

I have also discussed some common mistakes to avoid when conducting logistic regression. With the right knowledge and tools, logistic regression can be a powerful tool for uncovering insights and making decisions.