Binary Logistic Regression

Binary logistic regression is a statistical method to model the relationship between the binary outcome variable and one or more predictor variables. It is a fundamental technique in statistics and data analysis with wide-ranging applications in various fields such as healthcare, finance, marketing and social sciences.

In this article, we will learn about binary logistic regression discussing its definition, importance, methodology, interpretation, practical applications, and others in detail.

Regression analysis is a statistical method to investigate the relationship between the dependent variable and one or more independent variables. It aims to understand how the value of the dependent variable changes when one or more independent variables are varied.

Binary logistic regression is a type of regression analysis used when the dependent variable is binary. The goal of binary logistic regression is to predict the probability that an observation falls into one of the two categories based on one or more independent variables.

Logistic Regression

Logistic regression is a statistical model that uses the logistic function to model the probability of the binary outcome. Unlike linear regression which predicts continuous outcomes logistic regression predicts the probability of the categorical outcome.

Binary logistic regression uses the logistic function known as the sigmoid curve to model the relationship between the independent variables and the probability of the binary outcome. The logistic function is defined as:

where,

• P(Y = 1∣X) is Probability of Outcome Variable
• Y is equal to 1 given Values of Independent Variables X
• e is Base of Natural Logarithm
• z is Linear Combination of Independent Variables and their Coefficients

Logistic regression models the probability of the event occurring using the odds ratio. Odd ratio compares the probability of the success to the probability of the failure, providing insight into the relationship between variables and outcomes. Odds ratios greater than 1 indicate higher odds of the event occurring while those less than 1 suggest lower odds. The logistic function transforms linear regression output into the probabilities bounded between the 0 and 1.

For example, Predicting the likelihood of the customer buying a product based on the demographic variables.

Application: Widely used in the fields like medicine, finance and social sciences to the predict binary outcomes.

• Parameter Estimation: Fitting a binary logistic regression model involves estimating coefficients for the independent variables.

• Maximum Likelihood Estimation (MLE): Common method used to the find parameter estimates that maximize the likelihood of the observed data.

• Gradient Descent: Optimization algorithm used when MLE is computationally expensive or infeasible.

• Iterative Process: Model fitting is an iterative process where coefficients are adjusted until the model converges.

• Goodness of Fit: Measures like AIC and BIC help assess the fit of the model to the data.

• Overfitting and Regularization: Techniques like ridge and lasso regression are employed to the prevent overfitting and improve model generalization.

Model Evaluation and Validation

• Cross-Validation: Technique to the assess how well a model generalizes to the new data by splitting the dataset into the training and testing subsets.

• ROC Curve Analysis: Receiver Operating Characteristic curve evaluates the trade-off between the sensitivity and specificity.

• Area Under Curve (AUC): AUC measures the overall performance of the model in the distinguishing between the classes.

• Confusion Matrix Analysis: Evaluates the performance of the classification model by the comparing predicted and actual values.

• Precision, Recall, and F1 Score: Metrics used to the evaluate the performance of the binary classification models.

• Validation Set Approach: Divides data into the training, validation and test sets to the tune model hyperparameters and assess performance.

Differences between binary logistic regression and multinomial logistic regression is shown in the table added below:

Binary logistic regression finds applications in the various fields such as:

• Predicting the likelihood of the disease occurrence based on the risk factors in the medical research.

• Assessing the probability of the default on the loan in financial analysis.

• Determining customer churn in the marketing analytics.

• Identifying sentiment polarity in the text classification.

Q1. Predicting the probability of the customer buying a product based on the demographic information.

Q2. Estimating the likelihood of the patient having a specific disease based on the medical test results.

Q3. Analyzing the factors influencing employee attrition in a company.

Q4. Assessing the risk of credit card fraud based on the transaction patterns.

Binary logistic regression is a powerful statistical tool for the analyzing binary outcome variables and identifying the predictors associated with them. By understanding its methodology, interpretation and practical applications researchers and analysts can make informed the decisions and draw meaningful conclusions from the their data.

What is a binomial logistic regression?

Binomial logistic regression, predicts the probability that an observation falls into one of two categories of a dichotomous dependent variable based on one or more independent variables that can be either continuous or categorical.

What are the three types of logistic regression?

Three main types of logistic regression are:

• Binary Logistic Regression
• Multinomial Logistic Regression
• Ordinal Logistic Regression

What is the target of binary logistic regression?

Target variable has two possible outcomes such as Spam or Not Spam, Cancer or No Cancer, etc.

How are categorical independent variables used in binary logistic regression?

Categorical variables are typically encoded using the dummy variables before being included in the logistic regression model.

How multicollinearity used in logistic regression?

Multicollinearity among independent variables can lead to the unstable estimates in the logistic regression. The Regularization techniques such as the L1 or L2 regularization can be used to the mitigate multicollinearity.

What is odds ratio in logistic regression?

Odds ratio represents the change in the odds of the dependent variable being ‘1’ for the one-unit change in the independent variable.

What is goodness of fit of a logistic regression model?

Goodness of fit in the logistic regression can be assessed using the various metrics such as the Hosmer-Lemeshow test, deviance, AIC , BIC and ROC curve analysis.