# Summary

Logistic regression and linear regression are two of the most popular machine learning algorithms used for predictive modelling. Both algorithms are supervised learning algorithms, so we train them on labeled data. However, there are some key differences between the two algorithms.

# Demystifying Logistic Regression: A Simple Guide

Logistic regression is a popular statistical technique used to predict the probability of binary outcomes. They widely used it in various fields such as finance, marketing, healthcare, and social sciences. In this article, we will demystify logistic regression by providing a simple guide to help you understand its concepts, uses, and interpretation.

## 1. Introduction to Logistic Regression

Logistic regression is a statistical technique used to model the relationship between a set of independent variables (features) and a binary outcome variable. It is a supervised learning algorithm that is mainly used for classification tasks, where the outcome variable can take two values, such as “Yes” or “No,” “True” or “False,” or 1 or 0. The goal of logistic regression is to estimate the probabilities of the binary outcomes based on the values of the input features.

## 2. Understanding Binary Classification

### 3. Features vs. Labels

In logistic regression, we have a set of features, also known as independent variables or predictors, which are used to predict the binary outcome variable. These features can be numerical or categorical. The outcome variable, also known as the label or dependent variable, is the variable we want to predict based on the features.

### 4. Log-Odds and Probability

Logistic regression works by transforming the linear combination of the input features into a probability value between 0 and 1 using the logistic function, also known as the sigmoid function. The logistic function models the relationship between the probability of the event happening and the input features. It takes the form:

*p = 1 / (1 + e^(-z))*

where *p* represents the probability of the event happening and *z* represents the linear combination of the input features weighted by their coefficients.

## 3. Logistic Regression Model

### 4. Mathematical Formulation

We can mathematically formulate the logistic regression model as follows:

*logit(p) = Î˛â + Î˛âxâ + Î˛âxâ + … + Î˛âxâ*

where *logit(p)* is the natural logarithm of the odds ratio of the event happening, *Î˛â* is the intercept term, *Î˛â*, *Î˛â*, …, *Î˛â* are the coefficients corresponding to the input features, and *xâ*, *xâ*, …, *xâ* are the values of the input features.

### 5. Estimating Model Parameters

The model parameters, including the intercept term and coefficients, are estimated using maximum likelihood estimation. The goal is to find the values of the parameters that maximize the likelihood of observing the data. This involves finding the optimal values of the parameters that minimize the difference between the predicted probabilities and the actual binary outcomes.

## 4. Hypothesis Testing and Model Evaluation

### 6. Likelihood Ratio Test

In logistic regression, we commonly conducted hypothesis testing using the likelihood ratio test. This test compares the likelihood of the data under the null hypothesis (a model without a specific predictor) to the likelihood of the data under the alternative hypothesis (a model with the specific predictor). The test helps determine whether adding or removing a predictor significantly improves the model’s fit.

### 7. Model Evaluation Metrics

To evaluate the performance of a logistic regression model, various metrics can be used. These include accuracy, precision, recall, F1 score, and the area under the receiver operating characteristic curve (AUC-ROC). This metrics provide insights into how well the model can classify the binary outcomes.

## 5. Interpreting Logistic Regression Coefficients

### 7. Odds Ratio

One of the key advantages of logistic regression is the ability to interpret the coefficients as odds ratios. The odds ratio represents the change in odds of the event happening associated with a unit increase in the corresponding predictor variable, all else being equal. A value greater than 1 suggests a positive effect on the odds, while a value less than 1 suggests a negative effect.

### 8. Importance of Feature Scaling

When interpreting logistic regression coefficients, it is important to consider the scale of the input features. If the features are on different scales, the coefficients may not accurately represent the relative importance of the predictors. Therefore, it is often recommended to scale the features before fitting the logistic regression model.

## 6. Handling Categorical Variables

### 10. Dummy Variable Encoding

Categorical variables need to be encoded as numeric variables before being used in the logistic regression model. One common approach is to use dummy variable encoding, where each category of the variable is represented by a binary variable (0 or 1). We can then include these binary variables as predictors in the model.

### 11. Multicollinearity Issue

Multicollinearity occurs when two or more predictors in a logistic regression model are highly correlated. This can lead to unstable coefficient estimates and difficulties in interpreting the model. To address multicollinearity, one can remove one of the highly correlated predictors or use advanced techniques such as ridge regression or lasso regression.

## 7. Dealing with Imbalanced Data

### 13. Sampling Techniques

In real-world datasets, often one outcome is more prevalent than the other, resulting in imbalanced data. This can lead to biased models that are biased towards predicting the majority class. To overcome this issue, various sampling techniques such as over-sampling the minority class or under-sampling the majority class can balance the dataset.

### 14. Performance Metrics for Imbalanced Data

When working with imbalanced data, accuracy alone may not be a reliable measure of model performance. Other metrics such as precision, recall, and F1 score are more appropriate for evaluating the performance of a logistic regression model on imbalanced data.

## 8. Assumptions of Logistic Regression

### 16. Linearity Assumption

Logistic regression assumes a linear relationship between the log-odds of the binary outcome and the input features. This assumption implies that the change in the log-odds is constant for a one-unit increase in the predictors. Violation of this assumption may lead to biased coefficient estimates and inaccurate predictions.

### 17. Independence of Observations

Logistic regression assumes that observations are independent of each other. The data points should be unrelated to each other in terms of the outcome variable. Violation of this assumption, such as in time series or longitudinal data, may require alternative models that account for the dependence between observations.

## 9. Conclusion

In conclusion, logistic regression is a powerful tool for binary classification tasks. It allows us to model the relationship between a set of input features and the probability of a binary outcome. By understanding the concepts, assumptions, and interpretation of logistic regression, we can leverage this technique to make accurate predictions and gain valuable insights in various domains.

**Now, Let’s Examine Linear Regression**

Linear regression is a statistical method that predicts a dependent variable from one or more independent variables. The dependent variable is the variable that we want to predict, and the independent variables are the variables that we used to make the prediction.

Linear regression assumes that there is a linear relationship between the dependent variable and the independent variables. This means that we can model the dependent variable as a straight line function of the independent variables.

**The following equation defines the linear regression model:**

```
y = mx + b
```

where

- y is the dependent variable
- m is the slope of the line
- b is the y-intercept
- x is the independent variable

The slope of the line, m, tells us how much the dependent variable changes for a unit change in the independent variable. The y-intercept, b, tells us the value of the dependent variable when the independent variable is 0.

##### Linear regression can solve many problems. Some common applications of linear regression include:

- Predicting the price of a house based on its square footage and number of bedrooms
- Predicting the number of sales made based on the amount of advertising spent
- Predicting the risk of a customer defaulting on a loan based on their credit score

Linear regression is a powerful tool that can make predictions about the real world. However, it is important to remember that linear regression is only a model, and it is not perfect. The accuracy of the model depends on the quality of the data that is used to train it.

##### Here are some assumptions of linear regression:

- The dependent variable is a continuous variable.
- The independent variables are independent of each other.
- The relationship between the dependent variable and the independent variables is linear.
- The residuals are normally distributed.

If it violated any of these assumptions, then the accuracy of the linear regression model may be reduced.

Linear regression is a versatile and powerful tool that can solve many problems. However, it is important to understand the assumptions of the model and to use it carefully.

## Key differences between Logistic Distribution and Linear Distribution

**Logistic Regression**

Logistic regression is a classification algorithm. This means that it is used to predict a categorical outcome, such as whether a customer will churn, or whether a tumor is malignant or benign. Logistic regression works by fitting a logistic curve to the data. The logistic curve is a sigmoid function that maps the input values to a probability of the output being 1.

**Linear Regression**

Linear regression is a regression algorithm. This means that it is used to predict a continuous outcome, such as the price of a house or the number of sales made. Linear regression works by fitting a linear line to the data. The linear line is a straight line that maps the input values to the output values.

**Comparison of Logistic Regression and Linear Regression**

The following table summarizes the key differences between logistic regression and linear regression:

Feature | Logistic Regression | Linear Regression |
---|---|---|

Type of algorithm | Classification | Regression |

Outcome variable | Categorical | Continuous |

Model | Logistic curve | Linear line |

Assumptions | The dependent variable follows a binomial distribution | The dependent variable follows a normal distribution |

Applications | Predicting customer churn, classifying tumors as malignant or benign, predicting whether it will repay a loan | Predicting house prices, predicting the number of sales made, predicting the amount of money spent by a customer |

We should use logistic regression when the outcome variable is categorical. For example, you could use logistic regression to predict whether a customer will churn, whether a tumor is malignant or benign, or whether it will repay a loan.

**When to Use Linear Regression**

We should use linear regression when the outcome variable is continuous. For example, you could use linear regression to predict house prices, the number of sales made, or the amount of money spent by a customer.

**Conclusion**

Logistic regression and linear regression are both powerful machine learning algorithms that can be used for predictive modelling. The choice of which algorithm to use depends on the type of outcome variable that you are trying to predict. If the outcome variable is categorical, then logistic regression is the better choice. If the outcome variable is continuous, then linear regression is the better choice.

**References**

- Logistic Regression vs Linear Regression: https://www.analyticsvidhya.com/blog/2020/12/beginners-take-how-logistic-regression-is-related-to-linear-regression/
- Linear and Logistic Regression Models: When to Use and Why: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9747134/
- The Differences Between Linear Regression and Logistic Regression: https://www.geeksforgeeks.org/ml-linear-regression-vs-logistic-regression/