# Logistic regression model

Logistic regression model |
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**Logistic regression** is a statistical method used in predictive analytics to estimate the probability of a target variable based on one or more predictor variables. It is used to classify data into binary categories such as yes/no, pass/fail, true/false, etc. Logistic regression is a form of supervised learning that models the relationship between a set of dependent variables and one or more independent variables in order to make predictions about future outcomes. It is a powerful tool for managers to make strategic decisions, such as predicting customer churn, or the probability of an event occurring.

## Example of logistic regression model

- Logistic regression is commonly used in medical research to predict the likelihood of a patient developing a certain disease. For example, researchers may use logistic regression to predict the probability of a patient developing diabetes based on their age, genetics, lifestyle, and other factors.
- Logistic regression is also used in credit scoring and loan approval. Credit agencies use logistic regression to determine the probability of an individual defaulting on a loan. This is done by analyzing the borrower's financial information and other factors such as credit history, employment history, and living situation.
- Logistic regression can also be used in marketing research. Companies use logistic regression to predict the probability of a customer purchasing a product or service. This is done by analyzing the customer's demographic information, past purchases, and other factors.
- Logistic regression can also be used in fraud detection. Banks and other financial institutions use logistic regression to determine the probability of a transaction being fraudulent. This is done by analyzing the transaction data and other factors such as customer information, location, time of day, etc.

## Formula of logistic regression model

Logistic regression is a type of generalized linear model (GLM) used to model a binary response variable. The logistic regression model is expressed as a linear equation of the form:

$$\begin{equation} \hat{p} = \frac{e^{\beta_0 + \beta_1X_1 + \beta_2X_2 + \dots + \beta_nX_n}}{1+e^{\beta_0 + \beta_1X_1 + \beta_2X_2 + \dots + \beta_nX_n}} \end{equation}$$

where $$\hat{p}$$ is the probability of a particular outcome occurring given the independent variables $$X_1, X_2, ... X_n$$, and the coefficients $$\beta_0, \beta_1, \beta_2, ... \beta_n.$$

The $$\beta_0$$ is called the intercept, and represents the estimated probability of the outcome occurring when all the independent variables are 0. The $$\beta_1, \beta_2, ... \beta_n$$ coefficients each represent the estimated change in the probability of the outcome occurring for a unit increase in the corresponding independent variable.

In order to estimate the coefficients, the logistic regression model needs to be fitted to a set of data. This is done by maximizing the likelihood of the observed data given the model. Mathematically, this is expressed as:

$$\begin{equation} L(\beta_0, \beta_1,...,\beta_n) = \prod_{i=1}^N \hat{p_i}^{y_i}(1-\hat{p_i})^{1-y_i} \end{equation}$$

where N is the number of observations, $$\hat{p_i}$$ is the estimated probability of the outcome occurring for the ith observation, and $$y_i$$ is the observed binary outcome for the ith observation (either 0 or 1). The coefficients $$\beta_0, \beta_1, ... \beta_n$$ are then estimated by finding the values that maximize the likelihood of the observed data.

## When to use logistic regression model

Logistic regression can be used in a variety of applications, including but not limited to:

- Classification/prediction of binary outcomes, such as whether a customer will churn or not;
- Estimation of probability of an event occurring;
- Assessing the impact of variables on a given outcome;
- Modeling the probability of a given outcome of a categorical response variable;
- Assessing the influence of multiple independent variables on a dependent variable;
- Identifying the most important independent variables in a dataset;
- Determining the relationship between a binary response variable and a set of predictor variables;
- Modeling complex nonlinear relationships between dependent and independent variables.

## Types of logistic regression model

Logistic regression is a statistical method used in predictive analytics to model the relationship between a set of dependent variables and one or more independent variables in order to make predictions about future outcomes. The types of logistic regression models include:

- Binary logistic regression: This is the most basic form of logistic regression which is used to predict the probability of a binary outcome (i.e. yes/no, pass/fail, true/false, etc.).
- Multinomial logistic regression: This is used to predict the probability of a categorical outcome with more than two levels.
- Ordinal logistic regression: This is used to predict the probability of an ordinal outcome (i.e. an outcome that has an order, such as low, medium, high).
- Logistic regression with interaction terms: This adds a layer of complexity to the model by allowing for interaction terms, which can capture the effect of two or more variables on the outcome.
- Logistic regression with polynomial terms: This is used when the data is non-linear and allows the model to capture more complex relationships.
- Logistic regression with splines: This is used to fit a smooth curve to the data and allows for more flexibility when modeling non-linear relationships.

## Steps of logistic regression model

Logistic regression is a powerful tool for predictive analytics that estimates the probability of a target variable based on one or more predictor variables. The following are the steps in building a logistic regression model:

- Collect data: The first step is to collect data on the dependent and independent variables. This data should include information on each of the participants, as well as the outcome of interest.
- Clean data: It is important to clean the data so that any outliers or incorrect values are removed.
- Explore data: Exploring the data helps to identify any patterns and relationships that may exist between the variables.
- Select a model: Depending on the data, the analyst can select a linear or logistic regression model.
- Fit the model: The model must be fitted to the data in order to estimate the coefficients of the predictors.
- Validate the model: The model must be validated to ensure that it is accurate and reliable.
- Interpret the results: The results of the analysis must be interpreted in order to make meaningful conclusions.

## Advantages of logistic regression model

Logistic regression is a powerful tool for predictive analytics which has numerous advantages. The main advantages of logistic regression are as follows:

- It is simple to implement and interpret. Logistic regression provides a straightforward way to predict the probability of an event occurring based on one or more predictor variables. It is also easy to interpret the results as the coefficients of each independent variable can be evaluated to determine the effect of that variable on the outcome.
- It is robust to outliers and can handle non-linear relationships. Logistic regression is resistant to outliers, which is useful when the data set contains outliers that could affect the predictive power of the model. Additionally, the model can handle non-linear relationships between the predictor variables and the target variable.
- It allows for multivariate analysis. Logistic regression can be used to analyze and predict the probability of an event occurring based on multiple predictor variables. This is useful for managers to gain insights into the factors that influence a particular outcome.

## Limitations of logistic regression model

Logistic regression is a powerful tool for making predictions and classifying data, but it has its limitations. The following are some of the limitations of logistic regression:

- Logistic regression is limited to binary classification tasks and cannot be used for multi-class classification.
- Its performance is affected by the presence of outliers in the data, as well as by the number of independent variables.
- It is not suitable for data with non-linear relationships.
- It cannot capture complex relationships between variables.
- Logistic regression is a linear model and is sensitive to the scaling of the data.
- It is also prone to overfitting, which can lead to inaccurate predictions.

Logistic regression is a powerful tool for managers to make strategic decisions, such as predicting customer churn, or the probability of an event occurring. Other approaches related to logistic regression model include:

- Decision Trees: Decision trees are a type of supervised learning algorithm used for classification and regression tasks. They are constructed by repeatedly splitting the data along a given attribute until the data is divided into homogeneous subgroups.
- Naive Bayes Classifier: Naive Bayes classifiers are a type of supervised learning algorithm used to classify data points by predicting the probability of an event based on the values of other features. It uses Bayes' theorem to calculate the probability of an event given the values of the other features.
- Support Vector Machines: Support vector machines are a type of supervised learning algorithm used for classification and regression tasks. They are based on the idea of creating a hyperplane that separates the data into two classes.
- K-Nearest Neighbors: K-nearest neighbors is a type of supervised learning algorithm used for both classification and regression tasks. It is based on the idea of predicting the outcome of a data point based on its similarity to other data points.

In summary, logistic regression is a powerful tool for making predictions about future outcomes and is often used in predictive analytics. Other approaches related to logistic regression model include decision trees, naive Bayes classifiers, support vector machines, and K-nearest neighbors. Each of these approaches has its own strengths and weaknesses, and should be considered in combination with logistic regression to develop a comprehensive predictive model.

## Suggested literature

- Hilbe, J. M. (2009).
*Logistic regression models*. Chapman and hall/CRC. - Pregibon, D. (1981).
*Logistic regression diagnostics*. The annals of statistics, 9(4), 705-724.