What is linear regression? Function and use explained simply

Machine learning has become an integral part of data analysis for many companies. The challenge here is the interpretation of the complex models used. Depending on the model, algorithm and target, companies are faced with analysis results that are difficult to understand. In contrast, linear regression is a statistical method that delivers robust results - and therefore simplifies decision-making.

We will show you how linear regression works and how you can use it profitably in your company.

The most Important in a Nutshell

• The two basic types of linear regression are simple and multiple linear regression.
• The prerequisites for the use of linear regression include a linear relationship between the dependent variable and the independent variables as well as a constant variance of the errors.
• Practical applications of linear regression analysis include quality control in manufacturing and price optimization in retail.
• In the context of machine learning, linear regression acts as an algorithm that companies can use to optimize their processes and make well-founded decisions.

Linear regression Definition

Linear regression is a statistical method for modeling the relationship between a dependent variable and one or more independent variables. The aim is to find a straight line relationship that best describes the data. The method estimates the coefficients of the line in order to make predictions for the dependent variable. In doing so, it minimizes the sum of squared differences between the observed and predicted values. This enables precise predictions based on existing data.

Linear regression types

There are a number of different types of linear regression. However, the two basic types are simple linear regression and multiple linear regression:

Simple linear regression

Simple linear regression is a method for modeling relationships between two variables. There is an independent variable that influences the dependent variable. The model searches for a linear relationship and estimates the parameters to enable predictions to be made. Users use it, for example, to understand the linear relationship between the living space of a house and its price.

Multiple linear regression

In contrast to simple regression, multiple linear regression takes several independent variables into account. This enables more complex modeling of correlations. A practical example is the prediction of a building's energy consumption, taking into account factors such as size, insulation and outside temperature.

Depending on the specific requirements of the data and the research question, users rely on different variants of multiple linear regression. Important characteristics include the following:

Stepwise regression (stepwise regression)

In stepwise regression, variables are gradually added to or removed from the model based on statistical criteria such as the significance level. This makes it possible to select the most relevant variables for the model.

In medicine, researchers use stepwise regression to identify the relevant factors that influence blood pressure. By gradually adding or removing variables, doctors develop a model that contains the most important predictors of blood pressure.

Forward selection

In forward selection, users gradually add variables to the model, starting with the one that makes the greatest contribution to explaining the dependent variable. This is continued until no further significant improvements are achieved.

In the finance the forward selection is used to identify the factors that best explain the share price of a particular company. Starting with the most fundamental variables such as sales and earnings, analysts gradually add other factors to improve the model.

Backward elimination

In contrast to forward selection, with backward selection users first include all variables in the model and then gradually remove them, starting with the one that makes the smallest contribution. Here too, the process ends if no further significant improvements are achieved.

In environmental science, researchers use backward selection to identify relevant environmental factors that influence CO2 emissions. By gradually removing variables from the model, they identify those that have the least influence.

All-subset regression

This method looks at all possible combinations of independent variables and selects the one that provides the best fit. This approach is computationally expensive and is often used for smaller data sets.

In market research, companies use all-subset regression to carry out comprehensive analyses of product development.

By considering all possible combinations of product features, they identify the one that shows the best fit to sales data to make an optimal product selection.

Requirements for linear regression

In order for the results of linear regression to be reliable and interpretable, various prerequisites should be met. If these are not met, users should consider alternative methods. The most important assumptions for a meaningful linear regression are as follows:

Linear relationship

The relationship between the dependent variable and the independent variables should be linear. This means that there is a straight-line relationship between the variables. In business, for example, companies use linear regression to examine the functional relationship between the number of advertising measures and sales figures. A linear relationship would mean that sales figures increase linearly with increasing advertising.

Independence of errors

The residuals (differences between the observed and predicted values) should be independent of each other. This means that the error for one observation does not depend on the errors of others. In educational research, linear regression is useful for analyzing the influence of learning time on exam results. Independent errors would mean that a student's exam result is not influenced by the errors of other students.

Homoscedasticity

The variance of the errors should be constant. This means that the distribution of errors across all observed values of the independent variables is constant. Let us consider the prediction of the energy consumption of a building based on different factors: Homoscedasticity would mean that the variance of the errors remains constant across different energy consumption levels.

No multicollinearity

The independent variables should not be strongly correlated with each other. Multicollinearity can lead to instability in the estimate and make it difficult to interpret the coefficients. In financial analysis, for example, banks use linear regression to analyze the influence of income and debt on a person's creditworthiness. Multicollinearity would be problematic if income and debt were strongly correlated.

No outliers

Outliers can have a significant impact on the results of linear regression. It is important to identify these and remove them if necessary. When predicting the growth of plants based on various environmental factors, for example, the identification and treatment of outliers is important to ensure reliable modeling.

Practical examples of linear regression

To better understand the statistical method, let's take a look at 3 specific examples of linear regression from business, industry and medicine:

Price optimization in retail

A retailer wants to set the optimal prices for its products in order to maximize profits and ensure competitiveness.

Solution with linear regression

By applying linear regression to historical sales data and considering factors such as product features, promotions and competitor prices, the company develops pricing models to identify optimal pricing.

Added value

Linear regression enables retailers to understand price elasticity and make informed decisions about how price changes will affect demand and overall sales.

Quality control in the manufacturing industry

A manufacturer of electronic components wants to ensure that its products meet quality standards and that the reject rate is minimized.

Solution with linear regression

By applying linear regression to measurement data from production processes, the company creates models to understand the relationship between process parameters and product quality.

Added value

Linear regression supports the company in identifying critical process parameters that influence product quality. This enables it to optimize production in order to reduce waste and increase efficiency.

Machine learning model for medical diagnoses

A medical research institute wants to develop a linear model to predict the risk of certain diseases based on patient data.

Solution with linear regression

In a machine learning model linear regression can be used as part of an algorithm to model the influence of various health indicators on disease risk.

Added value

Linear regression helps to understand the relationships between health indicators and disease risk. The developed model enables early detection of risk factors and supports the development of preventive measures to improve patient health.

Importance of machine learning and linear regression for business and industry

As the third example suggests, linear regression is a crucial tool for Machine Learning. This is because it makes it possible to model relationships between variables and make predictions based on data. It plays a key role as it is one of the simplest yet most powerful algorithms. In practice, more and more companies from business and industry are therefore using machine learning and algorithms such as linear regression analysis. The advantages are obvious:

Precise predictions and forecasts

By modeling historical data, machine learning and linear regression are suitable for tasks such as the precise prediction of future events. Companies can use linear regression models to informed decisions based on forecasts - be it for sales figures, energy consumption or production output.

Optimization of processes

By analysing relationships between explanatory variables, both simple and multiple linear regression help companies to optimize their processes. In industry, for example, it is used to identify critical factors that improve product quality and make processes more efficient.

Price optimization and risk management

The linear regression algorithm is suitable for tasks such as identifying factors influencing prices and risks. In the retail sector, it enables the optimization of prices, while in the financial sector it is used to assess risks and determine insurance premiums.

Personnel management and resource allocation

Linear regression enables data-based analyses in HR management. This enables companies to better understand employee qualifications and performance, which leads to targeted training, efficient resource allocation and successful career planning.

Effective decision-making

Linear regression provides evidence-based insights that support decision-making. By using linear regression models, companies make informed decisions and gain a competitive advantage.