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A Refresher on Regression Analysis

Understanding one of the most important types of data analysis.

You probably know by now that whenever possible you should be making data-driven decisions at work . But do you know how to parse through all the data available to you? The good news is that you probably don’t need to do the number crunching yourself (hallelujah!) but you do need to correctly understand and interpret the analysis created by your colleagues. One of the most important types of data analysis is called regression analysis.

• Amy Gallo is a contributing editor at Harvard Business Review, cohost of the Women at Work podcast , and the author of two books: Getting Along: How to Work with Anyone (Even Difficult People) and the HBR Guide to Dealing with Conflict . She writes and speaks about workplace dynamics. Watch her TEDx talk on conflict and follow her on LinkedIn . amyegallo

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Regression Analysis

Regression analysis is a quantitative research method which is used when the study involves modelling and analysing several variables, where the relationship includes a dependent variable and one or more independent variables. In simple terms, regression analysis is a quantitative method used to test the nature of relationships between a dependent variable and one or more independent variables.

The basic form of regression models includes unknown parameters (β), independent variables (X), and the dependent variable (Y).

Regression model, basically, specifies the relation of dependent variable (Y) to a function combination of independent variables (X) and unknown parameters (β)

                                    Y  ≈  f (X, β)   

Regression equation can be used to predict the values of ‘y’, if the value of ‘x’ is given, and both ‘y’ and ‘x’ are the two sets of measures of a sample size of ‘n’. The formulae for regression equation would be

Do not be intimidated by visual complexity of correlation and regression formulae above. You don’t have to apply the formula manually, and correlation and regression analyses can be run with the application of popular analytical software such as Microsoft Excel, Microsoft Access, SPSS and others.

Linear regression analysis is based on the following set of assumptions:

1. Assumption of linearity . There is a linear relationship between dependent and independent variables.

2. Assumption of homoscedasticity . Data values for dependent and independent variables have equal variances.

3. Assumption of absence of collinearity or multicollinearity . There is no correlation between two or more independent variables.

4. Assumption of normal distribution . The data for the independent variables and dependent variable are normally distributed

My e-book,  The Ultimate Guide to Writing a Dissertation in Business Studies: a step by step assistance  offers practical assistance to complete a dissertation with minimum or no stress. The e-book covers all stages of writing a dissertation starting from the selection to the research area to submitting the completed version of the work within the deadline. John Dudovskiy

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Home » Regression Analysis – Methods, Types and Examples

Regression Analysis – Methods, Types and Examples

Table of Contents

Regression Analysis

Regression analysis is a set of statistical processes for estimating the relationships among variables . It includes many techniques for modeling and analyzing several variables when the focus is on the relationship between a dependent variable and one or more independent variables (or ‘predictors’).

Regression Analysis Methodology

Here is a general methodology for performing regression analysis:

• Define the research question: Clearly state the research question or hypothesis you want to investigate. Identify the dependent variable (also called the response variable or outcome variable) and the independent variables (also called predictor variables or explanatory variables) that you believe are related to the dependent variable.
• Collect data: Gather the data for the dependent variable and independent variables. Ensure that the data is relevant, accurate, and representative of the population or phenomenon you are studying.
• Explore the data: Perform exploratory data analysis to understand the characteristics of the data, identify any missing values or outliers, and assess the relationships between variables through scatter plots, histograms, or summary statistics.
• Choose the regression model: Select an appropriate regression model based on the nature of the variables and the research question. Common regression models include linear regression, multiple regression, logistic regression, polynomial regression, and time series regression, among others.
• Assess assumptions: Check the assumptions of the regression model. Some common assumptions include linearity (the relationship between variables is linear), independence of errors, homoscedasticity (constant variance of errors), and normality of errors. Violation of these assumptions may require additional steps or alternative models.
• Estimate the model: Use a suitable method to estimate the parameters of the regression model. The most common method is ordinary least squares (OLS), which minimizes the sum of squared differences between the observed and predicted values of the dependent variable.
• I nterpret the results: Analyze the estimated coefficients, p-values, confidence intervals, and goodness-of-fit measures (e.g., R-squared) to interpret the results. Determine the significance and direction of the relationships between the independent variables and the dependent variable.
• Evaluate model performance: Assess the overall performance of the regression model using appropriate measures, such as R-squared, adjusted R-squared, and root mean squared error (RMSE). These measures indicate how well the model fits the data and how much of the variation in the dependent variable is explained by the independent variables.
• Test assumptions and diagnose problems: Check the residuals (the differences between observed and predicted values) for any patterns or deviations from assumptions. Conduct diagnostic tests, such as examining residual plots, testing for multicollinearity among independent variables, and assessing heteroscedasticity or autocorrelation, if applicable.
• Make predictions and draw conclusions: Once you have a satisfactory model, use it to make predictions on new or unseen data. Draw conclusions based on the results of the analysis, considering the limitations and potential implications of the findings.

Types of Regression Analysis

Types of Regression Analysis are as follows:

Linear Regression

Linear regression is the most basic and widely used form of regression analysis. It models the linear relationship between a dependent variable and one or more independent variables. The goal is to find the best-fitting line that minimizes the sum of squared differences between observed and predicted values.

Multiple Regression

Multiple regression extends linear regression by incorporating two or more independent variables to predict the dependent variable. It allows for examining the simultaneous effects of multiple predictors on the outcome variable.

Polynomial Regression

Polynomial regression models non-linear relationships between variables by adding polynomial terms (e.g., squared or cubic terms) to the regression equation. It can capture curved or nonlinear patterns in the data.

Logistic Regression

Logistic regression is used when the dependent variable is binary or categorical. It models the probability of the occurrence of a certain event or outcome based on the independent variables. Logistic regression estimates the coefficients using the logistic function, which transforms the linear combination of predictors into a probability.

Ridge Regression and Lasso Regression

Ridge regression and Lasso regression are techniques used for addressing multicollinearity (high correlation between independent variables) and variable selection. Both methods introduce a penalty term to the regression equation to shrink or eliminate less important variables. Ridge regression uses L2 regularization, while Lasso regression uses L1 regularization.

Time Series Regression

Time series regression analyzes the relationship between a dependent variable and independent variables when the data is collected over time. It accounts for autocorrelation and trends in the data and is used in forecasting and studying temporal relationships.

Nonlinear Regression

Nonlinear regression models are used when the relationship between the dependent variable and independent variables is not linear. These models can take various functional forms and require estimation techniques different from those used in linear regression.

Poisson Regression

Poisson regression is employed when the dependent variable represents count data. It models the relationship between the independent variables and the expected count, assuming a Poisson distribution for the dependent variable.

Generalized Linear Models (GLM)

GLMs are a flexible class of regression models that extend the linear regression framework to handle different types of dependent variables, including binary, count, and continuous variables. GLMs incorporate various probability distributions and link functions.

Regression Analysis Formulas

Regression analysis involves estimating the parameters of a regression model to describe the relationship between the dependent variable (Y) and one or more independent variables (X). Here are the basic formulas for linear regression, multiple regression, and logistic regression:

Linear Regression:

Simple Linear Regression Model: Y = β0 + β1X + ε

Multiple Linear Regression Model: Y = β0 + β1X1 + β2X2 + … + βnXn + ε

In both formulas:

• Y represents the dependent variable (response variable).
• X represents the independent variable(s) (predictor variable(s)).
• β0, β1, β2, …, βn are the regression coefficients or parameters that need to be estimated.
• ε represents the error term or residual (the difference between the observed and predicted values).

Multiple Regression:

Multiple regression extends the concept of simple linear regression by including multiple independent variables.

Multiple Regression Model: Y = β0 + β1X1 + β2X2 + … + βnXn + ε

The formulas are similar to those in linear regression, with the addition of more independent variables.

Logistic Regression:

Logistic regression is used when the dependent variable is binary or categorical. The logistic regression model applies a logistic or sigmoid function to the linear combination of the independent variables.

Logistic Regression Model: p = 1 / (1 + e^-(β0 + β1X1 + β2X2 + … + βnXn))

In the formula:

• p represents the probability of the event occurring (e.g., the probability of success or belonging to a certain category).
• X1, X2, …, Xn represent the independent variables.
• e is the base of the natural logarithm.

The logistic function ensures that the predicted probabilities lie between 0 and 1, allowing for binary classification.

Regression Analysis Examples

Regression Analysis Examples are as follows:

• Stock Market Prediction: Regression analysis can be used to predict stock prices based on various factors such as historical prices, trading volume, news sentiment, and economic indicators. Traders and investors can use this analysis to make informed decisions about buying or selling stocks.
• Demand Forecasting: In retail and e-commerce, real-time It can help forecast demand for products. By analyzing historical sales data along with real-time data such as website traffic, promotional activities, and market trends, businesses can adjust their inventory levels and production schedules to meet customer demand more effectively.
• Energy Load Forecasting: Utility companies often use real-time regression analysis to forecast electricity demand. By analyzing historical energy consumption data, weather conditions, and other relevant factors, they can predict future energy loads. This information helps them optimize power generation and distribution, ensuring a stable and efficient energy supply.
• Online Advertising Performance: It can be used to assess the performance of online advertising campaigns. By analyzing real-time data on ad impressions, click-through rates, conversion rates, and other metrics, advertisers can adjust their targeting, messaging, and ad placement strategies to maximize their return on investment.
• Predictive Maintenance: Regression analysis can be applied to predict equipment failures or maintenance needs. By continuously monitoring sensor data from machines or vehicles, regression models can identify patterns or anomalies that indicate potential failures. This enables proactive maintenance, reducing downtime and optimizing maintenance schedules.
• Financial Risk Assessment: Real-time regression analysis can help financial institutions assess the risk associated with lending or investment decisions. By analyzing real-time data on factors such as borrower financials, market conditions, and macroeconomic indicators, regression models can estimate the likelihood of default or assess the risk-return tradeoff for investment portfolios.

Importance of Regression Analysis

Importance of Regression Analysis is as follows:

• Relationship Identification: Regression analysis helps in identifying and quantifying the relationship between a dependent variable and one or more independent variables. It allows us to determine how changes in independent variables impact the dependent variable. This information is crucial for decision-making, planning, and forecasting.
• Prediction and Forecasting: Regression analysis enables us to make predictions and forecasts based on the relationships identified. By estimating the values of the dependent variable using known values of independent variables, regression models can provide valuable insights into future outcomes. This is particularly useful in business, economics, finance, and other fields where forecasting is vital for planning and strategy development.
• Causality Assessment: While correlation does not imply causation, regression analysis provides a framework for assessing causality by considering the direction and strength of the relationship between variables. It allows researchers to control for other factors and assess the impact of a specific independent variable on the dependent variable. This helps in determining the causal effect and identifying significant factors that influence outcomes.
• Model Building and Variable Selection: Regression analysis aids in model building by determining the most appropriate functional form of the relationship between variables. It helps researchers select relevant independent variables and eliminate irrelevant ones, reducing complexity and improving model accuracy. This process is crucial for creating robust and interpretable models.
• Hypothesis Testing: Regression analysis provides a statistical framework for hypothesis testing. Researchers can test the significance of individual coefficients, assess the overall model fit, and determine if the relationship between variables is statistically significant. This allows for rigorous analysis and validation of research hypotheses.
• Policy Evaluation and Decision-Making: Regression analysis plays a vital role in policy evaluation and decision-making processes. By analyzing historical data, researchers can evaluate the effectiveness of policy interventions and identify the key factors contributing to certain outcomes. This information helps policymakers make informed decisions, allocate resources effectively, and optimize policy implementation.
• Risk Assessment and Control: Regression analysis can be used for risk assessment and control purposes. By analyzing historical data, organizations can identify risk factors and develop models that predict the likelihood of certain outcomes, such as defaults, accidents, or failures. This enables proactive risk management, allowing organizations to take preventive measures and mitigate potential risks.

When to Use Regression Analysis

• Prediction : Regression analysis is often employed to predict the value of the dependent variable based on the values of independent variables. For example, you might use regression to predict sales based on advertising expenditure, or to predict a student’s academic performance based on variables like study time, attendance, and previous grades.
• Relationship analysis: Regression can help determine the strength and direction of the relationship between variables. It can be used to examine whether there is a linear association between variables, identify which independent variables have a significant impact on the dependent variable, and quantify the magnitude of those effects.
• Causal inference: Regression analysis can be used to explore cause-and-effect relationships by controlling for other variables. For example, in a medical study, you might use regression to determine the impact of a specific treatment while accounting for other factors like age, gender, and lifestyle.
• Forecasting : Regression models can be utilized to forecast future trends or outcomes. By fitting a regression model to historical data, you can make predictions about future values of the dependent variable based on changes in the independent variables.
• Model evaluation: Regression analysis can be used to evaluate the performance of a model or test the significance of variables. You can assess how well the model fits the data, determine if additional variables improve the model’s predictive power, or test the statistical significance of coefficients.
• Data exploration : Regression analysis can help uncover patterns and insights in the data. By examining the relationships between variables, you can gain a deeper understanding of the data set and identify potential patterns, outliers, or influential observations.

Applications of Regression Analysis

Here are some common applications of regression analysis:

• Economic Forecasting: Regression analysis is frequently employed in economics to forecast variables such as GDP growth, inflation rates, or stock market performance. By analyzing historical data and identifying the underlying relationships, economists can make predictions about future economic conditions.
• Financial Analysis: Regression analysis plays a crucial role in financial analysis, such as predicting stock prices or evaluating the impact of financial factors on company performance. It helps analysts understand how variables like interest rates, company earnings, or market indices influence financial outcomes.
• Marketing Research: Regression analysis helps marketers understand consumer behavior and make data-driven decisions. It can be used to predict sales based on advertising expenditures, pricing strategies, or demographic variables. Regression models provide insights into which marketing efforts are most effective and help optimize marketing campaigns.
• Health Sciences: Regression analysis is extensively used in medical research and public health studies. It helps examine the relationship between risk factors and health outcomes, such as the impact of smoking on lung cancer or the relationship between diet and heart disease. Regression analysis also helps in predicting health outcomes based on various factors like age, genetic markers, or lifestyle choices.
• Social Sciences: Regression analysis is widely used in social sciences like sociology, psychology, and education research. Researchers can investigate the impact of variables like income, education level, or social factors on various outcomes such as crime rates, academic performance, or job satisfaction.
• Operations Research: Regression analysis is applied in operations research to optimize processes and improve efficiency. For example, it can be used to predict demand based on historical sales data, determine the factors influencing production output, or optimize supply chain logistics.
• Environmental Studies: Regression analysis helps in understanding and predicting environmental phenomena. It can be used to analyze the impact of factors like temperature, pollution levels, or land use patterns on phenomena such as species diversity, water quality, or climate change.
• Sports Analytics: Regression analysis is increasingly used in sports analytics to gain insights into player performance, team strategies, and game outcomes. It helps analyze the relationship between various factors like player statistics, coaching strategies, or environmental conditions and their impact on game outcomes.

Advantages and Disadvantages of Regression Analysis

About the author.

Muhammad Hassan

Researcher, Academic Writer, Web developer

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The complete guide to regression analysis.

19 min read What is regression analysis and why is it useful? While most of us have heard the term, understanding regression analysis in detail may be something you need to brush up on. Here’s what you need to know about this popular method of analysis.

When you rely on data to drive and guide business decisions, as well as predict market trends, just gathering and analyzing what you find isn’t enough — you need to ensure it’s relevant and valuable.

The challenge, however, is that so many variables can influence business data: market conditions, economic disruption, even the weather! As such, it’s essential you know which variables are affecting your data and forecasts, and what data you can discard.

And one of the most effective ways to determine data value and monitor trends (and the relationships between them) is to use regression analysis, a set of statistical methods used for the estimation of relationships between independent and dependent variables.

In this guide, we’ll cover the fundamentals of regression analysis, from what it is and how it works to its benefits and practical applications.

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What is regression analysis?

Regression analysis is a statistical method. It’s used for analyzing different factors that might influence an objective – such as the success of a product launch, business growth, a new marketing campaign – and determining which factors are important and which ones can be ignored.

Regression analysis can also help leaders understand how different variables impact each other and what the outcomes are. For example, when forecasting financial performance, regression analysis can help leaders determine how changes in the business can influence revenue or expenses in the future.

Running an analysis of this kind, you might find that there’s a high correlation between the number of marketers employed by the company, the leads generated, and the opportunities closed.

This seems to suggest that a high number of marketers and a high number of leads generated influences sales success. But do you need both factors to close those sales? By analyzing the effects of these variables on your outcome,  you might learn that when leads increase but the number of marketers employed stays constant, there is no impact on the number of opportunities closed, but if the number of marketers increases, leads and closed opportunities both rise.

Regression analysis can help you tease out these complex relationships so you can determine which areas you need to focus on in order to get your desired results, and avoid wasting time with those that have little or no impact. In this example, that might mean hiring more marketers rather than trying to increase leads generated.

How does regression analysis work?

Regression analysis starts with variables that are categorized into two types: dependent and independent variables. The variables you select depend on the outcomes you’re analyzing.

Understanding variables:

1. dependent variable.

This is the main variable that you want to analyze and predict. For example, operational (O) data such as your quarterly or annual sales, or experience (X) data such as your net promoter score (NPS) or customer satisfaction score (CSAT) .

These variables are also called response variables, outcome variables, or left-hand-side variables (because they appear on the left-hand side of a regression equation).

There are three easy ways to identify them:

• Is the variable measured as an outcome of the study?
• Does the variable depend on another in the study?
• Do you measure the variable only after other variables are altered?

2. Independent variable

Independent variables are the factors that could affect your dependent variables. For example, a price rise in the second quarter could make an impact on your sales figures.

You can identify independent variables with the following list of questions:

• Is the variable manipulated, controlled, or used as a subject grouping method by the researcher?
• Does this variable come before the other variable in time?
• Are you trying to understand whether or how this variable affects another?

Independent variables are often referred to differently in regression depending on the purpose of the analysis. You might hear them called:

Explanatory variables

Explanatory variables are those which explain an event or an outcome in your study. For example, explaining why your sales dropped or increased.

Predictor variables

Predictor variables are used to predict the value of the dependent variable. For example, predicting how much sales will increase when new product features are rolled out .

Experimental variables

These are variables that can be manipulated or changed directly by researchers to assess the impact. For example, assessing how different product pricing ($10 vs $15 vs $20) will impact the likelihood to purchase.

Subject variables (also called fixed effects)

Subject variables can’t be changed directly, but vary across the sample. For example, age, gender, or income of consumers.

Unlike experimental variables, you can’t randomly assign or change subject variables, but you can design your regression analysis to determine the different outcomes of groups of participants with the same characteristics. For example, ‘how do price rises impact sales based on income?’

Carrying out regression analysis

So regression is about the relationships between dependent and independent variables. But how exactly do you do it?

Assuming you have your data collection done already, the first and foremost thing you need to do is plot your results on a graph. Doing this makes interpreting regression analysis results much easier as you can clearly see the correlations between dependent and independent variables.

Let’s say you want to carry out a regression analysis to understand the relationship between the number of ads placed and revenue generated.

On the Y-axis, you place the revenue generated. On the X-axis, the number of digital ads. By plotting the information on the graph, and drawing a line (called the regression line) through the middle of the data, you can see the relationship between the number of digital ads placed and revenue generated.

This regression line is the line that provides the best description of the relationship between your independent variables and your dependent variable. In this example, we’ve used a simple linear regression model.

Statistical analysis software can draw this line for you and precisely calculate the regression line. The software then provides a formula for the slope of the line, adding further context to the relationship between your dependent and independent variables.

Simple linear regression analysis

A simple linear model uses a single straight line to determine the relationship between a single independent variable and a dependent variable.

This regression model is mostly used when you want to determine the relationship between two variables (like price increases and sales) or the value of the dependent variable at certain points of the independent variable (for example the sales levels at a certain price rise).

While linear regression is useful, it does require you to make some assumptions.

For example, it requires you to assume that:

• the data was collected using a statistically valid sample collection method that is representative of the target population
• The observed relationship between the variables can’t be explained by a ‘hidden’ third variable – in other words, there are no spurious correlations.
• the relationship between the independent variable and dependent variable is linear – meaning that the best fit along the data points is a straight line and not a curved one

Multiple regression analysis

As the name suggests, multiple regression analysis is a type of regression that uses multiple variables. It uses multiple independent variables to predict the outcome of a single dependent variable. Of the various kinds of multiple regression, multiple linear regression is one of the best-known.

Multiple linear regression is a close relative of the simple linear regression model in that it looks at the impact of several independent variables on one dependent variable. However, like simple linear regression, multiple regression analysis also requires you to make some basic assumptions.

For example, you will be assuming that:

• there is a linear relationship between the dependent and independent variables (it creates a straight line and not a curve through the data points)
• the independent variables aren’t highly correlated in their own right

An example of multiple linear regression would be an analysis of how marketing spend, revenue growth, and general market sentiment affect the share price of a company.

With multiple linear regression models you can estimate how these variables will influence the share price, and to what extent.

Multivariate linear regression

Multivariate linear regression involves more than one dependent variable as well as multiple independent variables, making it more complicated than linear or multiple linear regressions. However, this also makes it much more powerful and capable of making predictions about complex real-world situations.

For example, if an organization wants to establish or estimate how the COVID-19 pandemic has affected employees in its different markets, it can use multivariate linear regression, with the different geographical regions as dependent variables and the different facets of the pandemic as independent variables (such as mental health self-rating scores, proportion of employees working at home, lockdown durations and employee sick days).

Through multivariate linear regression, you can look at relationships between variables in a holistic way and quantify the relationships between them. As you can clearly visualize those relationships, you can make adjustments to dependent and independent variables to see which conditions influence them. Overall, multivariate linear regression provides a more realistic picture than looking at a single variable.

However, because multivariate techniques are complex, they involve high-level mathematics that require a statistical program to analyze the data.

Logistic regression

Logistic regression models the probability of a binary outcome based on independent variables.

So, what is a binary outcome? It’s when there are only two possible scenarios, either the event happens (1) or it doesn’t (0). e.g. yes/no outcomes, pass/fail outcomes, and so on. In other words, if the outcome can be described as being in either one of two categories.

Logistic regression makes predictions based on independent variables that are assumed or known to have an influence on the outcome. For example, the probability of a sports team winning their game might be affected by independent variables like weather, day of the week, whether they are playing at home or away and how they fared in previous matches.

What are some common mistakes with regression analysis?

Across the globe, businesses are increasingly relying on quality data and insights to drive decision-making — but to make accurate decisions, it’s important that the data collected and statistical methods used to analyze it are reliable and accurate.

Using the wrong data or the wrong assumptions can result in poor decision-making, lead to missed opportunities to improve efficiency and savings, and — ultimately — damage your business long term.

• Assumptions

When running regression analysis, be it a simple linear or multiple regression, it’s really important to check that the assumptions your chosen method requires have been met. If your data points don’t conform to a straight line of best fit, for example, you need to apply additional statistical modifications to accommodate the non-linear data. For example, if you are looking at income data, which scales on a logarithmic distribution, you should take the Natural Log of Income as your variable then adjust the outcome after the model is created.

• Correlation vs. causation

It’s a well-worn phrase that bears repeating – correlation does not equal causation. While variables that are linked by causality will always show correlation, the reverse is not always true. Moreover, there is no statistic that can determine causality (although the design of your study overall can).

If you observe a correlation in your results, such as in the first example we gave in this article where there was a correlation between leads and sales, you can’t assume that one thing has influenced the other. Instead, you should use it as a starting point for investigating the relationship between the variables in more depth.

• Choosing the wrong variables to analyze

Before you use any kind of statistical method, it’s important to understand the subject you’re researching in detail. Doing so means you’re making informed choices of variables and you’re not overlooking something important that might have a significant bearing on your dependent variable.

• Model building The variables you include in your analysis are just as important as the variables you choose to exclude. That’s because the strength of each independent variable is influenced by the other variables in the model. Other techniques, such as Key Drivers Analysis, are able to account for these variable interdependencies.

Benefits of using regression analysis

There are several benefits to using regression analysis to judge how changing variables will affect your business and to ensure you focus on the right things when forecasting.

Here are just a few of those benefits:

Make accurate predictions

Regression analysis is commonly used when forecasting and forward planning for a business. For example, when predicting sales for the year ahead, a number of different variables will come into play to determine the eventual result.

Regression analysis can help you determine which of these variables are likely to have the biggest impact based on previous events and help you make more accurate forecasts and predictions.

Identify inefficiencies

Using a regression equation a business can identify areas for improvement when it comes to efficiency, either in terms of people, processes, or equipment.

For example, regression analysis can help a car manufacturer determine order numbers based on external factors like the economy or environment.

Using the initial regression equation, they can use it to determine how many members of staff and how much equipment they need to meet orders.

Drive better decisions

Improving processes or business outcomes is always on the minds of owners and business leaders, but without actionable data, they’re simply relying on instinct, and this doesn’t always work out.

This is particularly true when it comes to issues of price. For example, to what extent will raising the price (and to what level) affect next quarter’s sales?

There’s no way to know this without data analysis. Regression analysis can help provide insights into the correlation between price rises and sales based on historical data.

How do businesses use regression? A real-life example

Marketing and advertising spending are common topics for regression analysis. Companies use regression when trying to assess the value of ad spend and marketing spend on revenue.

A typical example is using a regression equation to assess the correlation between ad costs and conversions of new customers. In this instance,

• our dependent variable (the factor we’re trying to assess the outcomes of) will be our conversions
• the independent variable (the factor we’ll change to assess how it changes the outcome) will be the daily ad spend
• the regression equation will try to determine whether an increase in ad spend has a direct correlation with the number of conversions we have

The analysis is relatively straightforward — using historical data from an ad account, we can use daily data to judge ad spend vs conversions and how changes to the spend alter the conversions.

By assessing this data over time, we can make predictions not only on whether increasing ad spend will lead to increased conversions but also what level of spending will lead to what increase in conversions. This can help to optimize campaign spend and ensure marketing delivers good ROI.

This is an example of a simple linear model. If you wanted to carry out a more complex regression equation, we could also factor in other independent variables such as seasonality, GDP, and the current reach of our chosen advertising networks.

By increasing the number of independent variables, we can get a better understanding of whether ad spend is resulting in an increase in conversions, whether it’s exerting an influence in combination with another set of variables, or if we’re dealing with a correlation with no causal impact – which might be useful for predictions anyway, but isn’t a lever we can use to increase sales.

Using this predicted value of each independent variable, we can more accurately predict how spend will change the conversion rate of advertising.

Regression analysis tools

Regression analysis is an important tool when it comes to better decision-making and improved business outcomes. To get the best out of it, you need to invest in the right kind of statistical analysis software.

The best option is likely to be one that sits at the intersection of powerful statistical analysis and intuitive ease of use, as this will empower everyone from beginners to expert analysts to uncover meaning from data, identify hidden trends and produce predictive models without statistical training being required.

To help prevent costly errors, choose a tool that automatically runs the right statistical tests and visualizations and then translates the results into simple language that anyone can put into action.

With software that’s both powerful and user-friendly, you can isolate key experience drivers, understand what influences the business, apply the most appropriate regression methods, identify data issues, and much more.

With Qualtrics’ Stats iQ™, you don’t have to worry about the regression equation because our statistical software will run the appropriate equation for you automatically based on the variable type you want to monitor. You can also use several equations, including linear regression and logistic regression, to gain deeper insights into business outcomes and make more accurate, data-driven decisions.

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• Volume 24, Issue 4
• Understanding and interpreting regression analysis
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• http://orcid.org/0000-0002-7839-8130 Parveen Ali 1 , 2 ,
• http://orcid.org/0000-0003-0157-5319 Ahtisham Younas 3 , 4
• 1 School of Nursing and Midwifery , University of Sheffield , Sheffield , South Yorkshire , UK
• 2 Sheffiled University Interpersonal Violence Research Group , The University of Sheffiled SEAS , Sheffield , UK
• 3 Faculty of Nursing , Memorial University of Newfoundland , St. John's , Newfoundland and Labrador , Canada
• 4 Swat College of Nursing , Mingora, Swat , Pakistan
• Correspondence to Ahtisham Younas, Memorial University of Newfoundland, St. John's, NL A1C 5S7, Canada; ay6133{at}mun.ca

https://doi.org/10.1136/ebnurs-2021-103425

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• statistics & research methods

Introduction

A nurse educator is interested in finding out the academic and non-academic predictors of success in nursing students. Given the complexity of educational and clinical learning environments, demographic, clinical and academic factors (age, gender, previous educational training, personal stressors, learning demands, motivation, assignment workload, etc) influencing nursing students’ success, she was able to list various potential factors contributing towards success relatively easily. Nevertheless, not all of the identified factors will be plausible predictors of increased success. Therefore, she could use a powerful statistical procedure called regression analysis to identify whether the likelihood of increased success is influenced by factors such as age, stressors, learning demands, motivation and education.

What is regression?

Purposes of regression analysis.

Regression analysis has four primary purposes: description, estimation, prediction and control. 1 , 2 By description, regression can explain the relationship between dependent and independent variables. Estimation means that by using the observed values of independent variables, the value of dependent variable can be estimated. 2 Regression analysis can be useful for predicting the outcomes and changes in dependent variables based on the relationships of dependent and independent variables. Finally, regression enables in controlling the effect of one or more independent variables while investigating the relationship of one independent variable with the dependent variable. 1

Types of regression analyses

There are commonly three types of regression analyses, namely, linear, logistic and multiple regression. The differences among these types are outlined in table 1 in terms of their purpose, nature of dependent and independent variables, underlying assumptions, and nature of curve. 1 , 3 However, more detailed discussion for linear regression is presented as follows.

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Comparison of linear, logistic and multiple regression

Linear regression and interpretation

Linear regression analysis involves examining the relationship between one independent and dependent variable. Statistically, the relationship between one independent variable (x) and a dependent variable (y) is expressed as: y= β 0 + β 1 x+ε. In this equation, β 0 is the y intercept and refers to the estimated value of y when x is equal to 0. The coefficient β 1 is the regression coefficient and denotes that the estimated increase in the dependent variable for every unit increase in the independent variable. The symbol ε is a random error component and signifies imprecision of regression indicating that, in actual practice, the independent variables are cannot perfectly predict the change in any dependent variable. 1 Multiple linear regression follows the same logic as univariate linear regression except (a) multiple regression, there are more than one independent variable and (b) there should be non-collinearity among the independent variables.

Factors affecting regression

Linear and multiple regression analyses are affected by factors, namely, sample size, missing data and the nature of sample. 2

Small sample size may only demonstrate connections among variables with strong relationship. Therefore, sample size must be chosen based on the number of independent variables and expect strength of relationship.

Many missing values in the data set may affect the sample size. Therefore, all the missing values should be adequately dealt with before conducting regression analyses.

The subsamples within the larger sample may mask the actual effect of independent and dependent variables. Therefore, if subsamples are predefined, a regression within the sample could be used to detect true relationships. Otherwise, the analysis should be undertaken on the whole sample.

Building on her research interest mentioned in the beginning, let us consider a study by Ali and Naylor. 4 They were interested in identifying the academic and non-academic factors which predict the academic success of nursing diploma students. This purpose is consistent with one of the above-mentioned purposes of regression analysis (ie, prediction). Ali and Naylor’s chosen academic independent variables were preadmission qualification, previous academic performance and school type and the non-academic variables were age, gender, marital status and time gap. To achieve their purpose, they collected data from 628 nursing students between the age range of 15–34 years. They used both linear and multiple regression analyses to identify the predictors of student success. For analysis, they examined the relationship of academic and non-academic variables across different years of study and noted that academic factors accounted for 36.6%, 44.3% and 50.4% variability in academic success of students in year 1, year 2 and year 3, respectively. 4

Ali and Naylor presented the relationship among these variables using scatter plots, which are commonly used graphs for data display in regression analysis—see examples of various scatter plots in figure 1 . 4 In a scatter plot, the clustering of the dots denoted the strength of relationship, whereas the direction indicates the nature of relationships among variables as positive (ie, increase in one variable results in an increase in the other) and negative (ie, increase in one variable results in decrease in the other).

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An Example of Scatter Plot for Regression.

Table 2 presents the results of regression analysis for academic and non-academic variables for year 4 students’ success. The significant predictors of student success are denoted with a significant p value. For every, significant predictor, the beta value indicates the percentage increase in students’ academic success with one unit increase in the variable.

Regression model for the final year students (N=343)

Conclusions

Regression analysis is a powerful and useful statistical procedure with many implications for nursing research. It enables researchers to describe, predict and estimate the relationships and draw plausible conclusions about the interrelated variables in relation to any studied phenomena. Regression also allows for controlling one or more variables when researchers are interested in examining the relationship among specific variables. Some of the key considerations are presented that may be useful for researchers undertaking regression analysis. While planning and conducting regression analysis, researchers should consider the type and number of dependent and independent variables as well as the nature and size of sample. Choosing a wrong type of regression analysis with small sample may result in erroneous conclusions about the studied phenomenon.

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Not required.

• Montgomery DC ,
• Schneider A ,

Twitter @parveenazamali, @@Ahtisham04

Funding The authors have not declared a specific grant for this research from any funding agency in the public, commercial or not-for-profit sectors.

Competing interests None declared.

Provenance and peer review Commissioned; internally peer reviewed.

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• Knowledge Base
• Simple Linear Regression | An Easy Introduction & Examples

Simple Linear Regression | An Easy Introduction & Examples

Published on February 19, 2020 by Rebecca Bevans . Revised on June 22, 2023.

Simple linear regression is used to estimate the relationship between two quantitative variables . You can use simple linear regression when you want to know:

• How strong the relationship is between two variables (e.g., the relationship between rainfall and soil erosion).
• The value of the dependent variable at a certain value of the independent variable (e.g., the amount of soil erosion at a certain level of rainfall).

Regression models describe the relationship between variables by fitting a line to the observed data. Linear regression models use a straight line, while logistic and nonlinear regression models use a curved line. Regression allows you to estimate how a dependent variable changes as the independent variable(s) change.

If you have more than one independent variable, use multiple linear regression instead.

Table of contents

Assumptions of simple linear regression, how to perform a simple linear regression, interpreting the results, presenting the results, can you predict values outside the range of your data, other interesting articles, frequently asked questions about simple linear regression.

Simple linear regression is a parametric test , meaning that it makes certain assumptions about the data. These assumptions are:

• Homogeneity of variance (homoscedasticity) : the size of the error in our prediction doesn’t change significantly across the values of the independent variable.
• Independence of observations : the observations in the dataset were collected using statistically valid sampling methods , and there are no hidden relationships among observations.
• Normality : The data follows a normal distribution .

Linear regression makes one additional assumption:

• The relationship between the independent and dependent variable is linear : the line of best fit through the data points is a straight line (rather than a curve or some sort of grouping factor).

If your data do not meet the assumptions of homoscedasticity or normality, you may be able to use a nonparametric test instead, such as the Spearman rank test.

If your data violate the assumption of independence of observations (e.g., if observations are repeated over time), you may be able to perform a linear mixed-effects model that accounts for the additional structure in the data.

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Simple linear regression formula

The formula for a simple linear regression is:

• y is the predicted value of the dependent variable ( y ) for any given value of the independent variable ( x ).
• B 0 is the intercept , the predicted value of y when the x is 0.
• B 1 is the regression coefficient – how much we expect y to change as x increases.
• x is the independent variable ( the variable we expect is influencing y ).
• e is the error of the estimate, or how much variation there is in our estimate of the regression coefficient.

Linear regression finds the line of best fit line through your data by searching for the regression coefficient (B 1 ) that minimizes the total error (e) of the model.

While you can perform a linear regression by hand , this is a tedious process, so most people use statistical programs to help them quickly analyze the data.

Simple linear regression in R

R is a free, powerful, and widely-used statistical program. Download the dataset to try it yourself using our income and happiness example.

Dataset for simple linear regression (.csv)

Load the income.data dataset into your R environment, and then run the following command to generate a linear model describing the relationship between income and happiness:

This code takes the data you have collected data = income.data and calculates the effect that the independent variable income has on the dependent variable happiness using the equation for the linear model: lm() .

To learn more, follow our full step-by-step guide to linear regression in R .

To view the results of the model, you can use the summary() function in R:

This function takes the most important parameters from the linear model and puts them into a table, which looks like this:

This output table first repeats the formula that was used to generate the results (‘Call’), then summarizes the model residuals (‘Residuals’), which give an idea of how well the model fits the real data.

Next is the ‘Coefficients’ table. The first row gives the estimates of the y-intercept, and the second row gives the regression coefficient of the model.

Row 1 of the table is labeled (Intercept) . This is the y-intercept of the regression equation, with a value of 0.20. You can plug this into your regression equation if you want to predict happiness values across the range of income that you have observed:

The next row in the ‘Coefficients’ table is income. This is the row that describes the estimated effect of income on reported happiness:

The Estimate column is the estimated effect , also called the regression coefficient or r 2 value. The number in the table (0.713) tells us that for every one unit increase in income (where one unit of income = 10,000) there is a corresponding 0.71-unit increase in reported happiness (where happiness is a scale of 1 to 10).

The Std. Error column displays the standard error of the estimate. This number shows how much variation there is in our estimate of the relationship between income and happiness.

The t value  column displays the test statistic . Unless you specify otherwise, the test statistic used in linear regression is the t value from a two-sided t test . The larger the test statistic, the less likely it is that our results occurred by chance.

The Pr(>| t |)  column shows the p value . This number tells us how likely we are to see the estimated effect of income on happiness if the null hypothesis of no effect were true.

Because the p value is so low ( p < 0.001),  we can reject the null hypothesis and conclude that income has a statistically significant effect on happiness.

The last three lines of the model summary are statistics about the model as a whole. The most important thing to notice here is the p value of the model. Here it is significant ( p < 0.001), which means that this model is a good fit for the observed data.

When reporting your results, include the estimated effect (i.e. the regression coefficient), standard error of the estimate, and the p value. You should also interpret your numbers to make it clear to your readers what your regression coefficient means:

It can also be helpful to include a graph with your results. For a simple linear regression, you can simply plot the observations on the x and y axis and then include the regression line and regression function:

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No! We often say that regression models can be used to predict the value of the dependent variable at certain values of the independent variable. However, this is only true for the range of values where we have actually measured the response.

We can use our income and happiness regression analysis as an example. Between 15,000 and 75,000, we found an r 2 of 0.73 ± 0.0193. But what if we did a second survey of people making between 75,000 and 150,000?

The r 2 for the relationship between income and happiness is now 0.21, or a 0.21-unit increase in reported happiness for every 10,000 increase in income. While the relationship is still statistically significant (p<0.001), the slope is much smaller than before.

What if we hadn’t measured this group, and instead extrapolated the line from the 15–75k incomes to the 70–150k incomes?

You can see that if we simply extrapolated from the 15–75k income data, we would overestimate the happiness of people in the 75–150k income range.

If we instead fit a curve to the data, it seems to fit the actual pattern much better.

It looks as though happiness actually levels off at higher incomes, so we can’t use the same regression line we calculated from our lower-income data to predict happiness at higher levels of income.

Even when you see a strong pattern in your data, you can’t know for certain whether that pattern continues beyond the range of values you have actually measured. Therefore, it’s important to avoid extrapolating beyond what the data actually tell you.

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Chi square test of independence
• Statistical power
• Descriptive statistics
• Degrees of freedom
• Pearson correlation
• Null hypothesis

Methodology

• Double-blind study
• Case-control study
• Research ethics
• Data collection
• Hypothesis testing
• Structured interviews

Research bias

• Hawthorne effect
• Unconscious bias
• Recall bias
• Halo effect
• Self-serving bias
• Information bias

A regression model is a statistical model that estimates the relationship between one dependent variable and one or more independent variables using a line (or a plane in the case of two or more independent variables).

A regression model can be used when the dependent variable is quantitative, except in the case of logistic regression, where the dependent variable is binary.

Simple linear regression is a regression model that estimates the relationship between one independent variable and one dependent variable using a straight line. Both variables should be quantitative.

For example, the relationship between temperature and the expansion of mercury in a thermometer can be modeled using a straight line: as temperature increases, the mercury expands. This linear relationship is so certain that we can use mercury thermometers to measure temperature.

Linear regression most often uses mean-square error (MSE) to calculate the error of the model. MSE is calculated by:

• measuring the distance of the observed y-values from the predicted y-values at each value of x;
• squaring each of these distances;
• calculating the mean of each of the squared distances.

Linear regression fits a line to the data by finding the regression coefficient that results in the smallest MSE.

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What Is Regression Analysis in Business Analytics?

• 14 Dec 2021

Countless factors impact every facet of business. How can you consider those factors and know their true impact?

Imagine you seek to understand the factors that influence people’s decision to buy your company’s product. They range from customers’ physical locations to satisfaction levels among sales representatives to your competitors' Black Friday sales.

Understanding the relationships between each factor and product sales can enable you to pinpoint areas for improvement, helping you drive more sales.

To learn how each factor influences sales, you need to use a statistical analysis method called regression analysis .

If you aren’t a business or data analyst, you may not run regressions yourself, but knowing how analysis works can provide important insight into which factors impact product sales and, thus, which are worth improving.

Access your free e-book today.

Foundational Concepts for Regression Analysis

Before diving into regression analysis, you need to build foundational knowledge of statistical concepts and relationships.

Independent and Dependent Variables

Start with the basics. What relationship are you aiming to explore? Try formatting your answer like this: “I want to understand the impact of [the independent variable] on [the dependent variable].”

The independent variable is the factor that could impact the dependent variable . For example, “I want to understand the impact of employee satisfaction on product sales.”

In this case, employee satisfaction is the independent variable, and product sales is the dependent variable. Identifying the dependent and independent variables is the first step toward regression analysis.

Correlation vs. Causation

One of the cardinal rules of statistically exploring relationships is to never assume correlation implies causation. In other words, just because two variables move in the same direction doesn’t mean one caused the other to occur.

If two or more variables are correlated , their directional movements are related. If two variables are positively correlated , it means that as one goes up or down, so does the other. Alternatively, if two variables are negatively correlated , one goes up while the other goes down.

A correlation’s strength can be quantified by calculating the correlation coefficient , sometimes represented by r . The correlation coefficient falls between negative one and positive one.

r = -1 indicates a perfect negative correlation.

r = 1 indicates a perfect positive correlation.

r = 0 indicates no correlation.

Causation means that one variable caused the other to occur. Proving a causal relationship between variables requires a true experiment with a control group (which doesn’t receive the independent variable) and an experimental group (which receives the independent variable).

While regression analysis provides insights into relationships between variables, it doesn’t prove causation. It can be tempting to assume that one variable caused the other—especially if you want it to be true—which is why you need to keep this in mind any time you run regressions or analyze relationships between variables.

With the basics under your belt, here’s a deeper explanation of regression analysis so you can leverage it to drive strategic planning and decision-making.

Related: How to Learn Business Analytics without a Business Background

What Is Regression Analysis?

Regression analysis is the statistical method used to determine the structure of a relationship between two variables (single linear regression) or three or more variables (multiple regression).

According to the Harvard Business School Online course Business Analytics , regression is used for two primary purposes:

• To study the magnitude and structure of the relationship between variables
• To forecast a variable based on its relationship with another variable

Both of these insights can inform strategic business decisions.

“Regression allows us to gain insights into the structure of that relationship and provides measures of how well the data fit that relationship,” says HBS Professor Jan Hammond, who teaches Business Analytics, one of three courses that comprise the Credential of Readiness (CORe) program . “Such insights can prove extremely valuable for analyzing historical trends and developing forecasts.”

One way to think of regression is by visualizing a scatter plot of your data with the independent variable on the X-axis and the dependent variable on the Y-axis. The regression line is the line that best fits the scatter plot data. The regression equation represents the line’s slope and the relationship between the two variables, along with an estimation of error.

Physically creating this scatter plot can be a natural starting point for parsing out the relationships between variables.

Types of Regression Analysis

There are two types of regression analysis: single variable linear regression and multiple regression.

Single variable linear regression is used to determine the relationship between two variables: the independent and dependent. The equation for a single variable linear regression looks like this:

In the equation:

• ŷ is the expected value of Y (the dependent variable) for a given value of X (the independent variable).
• x is the independent variable.
• α is the Y-intercept, the point at which the regression line intersects with the vertical axis.
• β is the slope of the regression line, or the average change in the dependent variable as the independent variable increases by one.
• ε is the error term, equal to Y – ŷ, or the difference between the actual value of the dependent variable and its expected value.

Multiple regression , on the other hand, is used to determine the relationship between three or more variables: the dependent variable and at least two independent variables. The multiple regression equation looks complex but is similar to the single variable linear regression equation:

Each component of this equation represents the same thing as in the previous equation, with the addition of the subscript k, which is the total number of independent variables being examined. For each independent variable you include in the regression, multiply the slope of the regression line by the value of the independent variable, and add it to the rest of the equation.

How to Run Regressions

You can use a host of statistical programs—such as Microsoft Excel, SPSS, and STATA—to run both single variable linear and multiple regressions. If you’re interested in hands-on practice with this skill, Business Analytics teaches learners how to create scatter plots and run regressions in Microsoft Excel, as well as make sense of the output and use it to drive business decisions.

Calculating Confidence and Accounting for Error

It’s important to note: This overview of regression analysis is introductory and doesn’t delve into calculations of confidence level, significance, variance, and error. When working in a statistical program, these calculations may be provided or require that you implement a function. When conducting regression analysis, these metrics are important for gauging how significant your results are and how much importance to place on them.

Why Use Regression Analysis?

Once you’ve generated a regression equation for a set of variables, you effectively have a roadmap for the relationship between your independent and dependent variables. If you input a specific X value into the equation, you can see the expected Y value.

This can be critical for predicting the outcome of potential changes, allowing you to ask, “What would happen if this factor changed by a specific amount?”

Returning to the earlier example, running a regression analysis could allow you to find the equation representing the relationship between employee satisfaction and product sales. You could input a higher level of employee satisfaction and see how sales might change accordingly. This information could lead to improved working conditions for employees, backed by data that shows the tie between high employee satisfaction and sales.

Whether predicting future outcomes, determining areas for improvement, or identifying relationships between seemingly unconnected variables, understanding regression analysis can enable you to craft data-driven strategies and determine the best course of action with all factors in mind.

Do you want to become a data-driven professional? Explore our eight-week Business Analytics course and our three-course Credential of Readiness (CORe) program to deepen your analytical skills and apply them to real-world business problems.

About the Author

• How it works

A Beginner’s Guide to Regression Analysis

Published by Owen Ingram at September 1st, 2021 , Revised On July 5, 2022

Are you good with data-driven decisions at work? If not, why? What is stopping you from getting on the crest of a wave? There could be just one answer to these questions, and that is “too much data getting in the way.” Do not worry; there is a solution to every problem in this world, and there is definitely one for parsing through tons of data.

Yes, you heard it right! You will not have to get in trouble with the number crunching and counting with this solution. What is the solution?

Well, without further ado, we would like to introduce you to “regression,” which precisely is allowing one to see into the future.

What is Regression Analysis?

Here is a scenario to help you understand what regression is and how it helps you make better strategic decisions in research.

Let’s say you are the CEO of a company and are trying to predict the profit margin for the next month. Now you might have a lot of factors in your mind that can affect the number. Be it the number of sales you get in the month, the number of employees not taking leaves, or the number of hours each worker gives daily. But what if things do not go as planned? The “what if” list here has no stop; it can go on forever.  All these impacting factors here are variables, and regression analysis is the process of mathematically figuring out which of these variables actually have an impact and which are not plausible.

So, we can say that regression analysis helps you find the relationship between a set of dependent and independent variables. There are different ways to find this relationship between variables, which in statistics is named “ regression models .”

We will learn about each in the next heading.

Types of Regression Models

If you are not sure which type of regression model you should use for a particular study, this section might help you.

Though there are numerous types of regression models depending on the type of variables , these are the most common ones.

Linear Regression

Logistic regression, ridge regression, lasso regression, polynomial regression, bayesian linear regression.

Linear regression is the real workhorse of the industry and probably is the first type that comes to mind. It is often known as Linear Least Squares and Ordinary Least Squares . This model consists of a dependent variable and a predictable variable that align with each other. Hence, the name linear regression. If the data you are dealing with contains more than one independent variable , then the linear regression here would be Multi-Linear Regression .

Logistic Regression comes into play when the dependent variable is discrete. This means that the target value will only have one or two values. For instance, a true or false, a yes or no, a 0 or 1, and so on. In this case, a sigmoid curve describes the relationship between the independent and dependent variables .

When using this regression model for the data analysis process , two things should strictly be taken into consideration:

• Make sure there is no multi-linearity (like that in the linear regression model) or correlation between the two variables in the dataset
• Also, ensure that the size of data is big with the equal manifestation of values to come in targeted variables

When there is a high correlation between the independent and dependent variables, this type of regression is used. It is simply because, with multi collinear data, least-square estimates give impartial numbers. However, if the collinearity is high, there might be a slight chance of unfair judgment.

Thus, a bias matrix is brought to the surface in ridge regression. This powerful type of regression is less vulnerable to overfitting. Are you familiar with the ‘overfitting’ word?

Overfitting in statistics is a modeling error that one makes when the function is too closely brought into line with limited data points. When a model in research has been compromised with this error, it might lose its value all at once.

Lasso Regression is best suitable for performing regularization alongside feature selection. This type of regression hinders the absolute size of the regression coefficient. What happens next? The coefficient value will almost come nearer zero, which the complete opposite of what happened in Ridge Regression.

This is why feature selection utilizes this regression model that helps to select a set of features from the dataset. Only required and limited features are used in Lasso Regression, and all the other features are zero. Researchers get rid of the overfitting in the model by doing this. But what if the independent variables are highly collinear?

In that case, this model will only choose one variable and turn the others to zero. We can say that it is somewhat like the Ridge Regression but with variable selection.

This is another type of regression that is almost the same as Multi-Linear Regression but with some changes. In the Polynomial Regression Model, the relationship between the two variables, dependent and independent , is denoted by the nth degree. While in a Multi-Linear Regression Model, the line is linear, here it is the opposite. The best fit line in Polynomial Regression passing through all the points is curved. This curve either depends on the value of n or the value of X.

This model is also prone to overfitting. It is best to assess the curve towards the end as the higher polynomials might give strange and unexpected results on extrapolation.

The last type of regression model we are going to discuss is the Bayesian Linear Regression. Have you heard of the Bayes theorem? Well, this regression type basically uses that to figure out the value of regression coefficients.

It is a lot like both Ridge Regression and Linear Regression, but the stability here is much higher. In this model, we find the value of the posterior distribution of the features instead of working on the least squares.

FAQs About Regression Analysis

What is regression.

It is a technique to find out the relationship between the dependent and independent variables

What is a linear regression model?

Linear Regression Model helps determine the relationship between different continuous variables by fitting a linear equation for dealing with data.

What is the difference between multi-linear regression and polynomial regression?

The only difference between Multi-Linear Regression and polynomial repression is that in the latter relationship between ‘x’ and ‘y’ is denoted by the nth value, so the line here is a curve. While in Multi-Linear, the line is straight.

What is overfitting in statistics?

When a function in statistics corresponds too closely to a particular set of data, some modeling error is possible. This modeling error is called overfitting.

What is ridge regression?

It is a method of finding the coefficients of multiple regression models in which the independent variables are highly correlated. In other words, it is a method to develop a parsimonious model when the number of predictable variables is higher than the observations in a set.

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Analysis of Variance or ANOVA is a statistical test that uses means of two or more groups to analyse the difference. Here is all you need to know about it.

Experimental research refers to the experiments conducted in the laboratory or under observation in controlled conditions. Here is all you need to know about experimental research.

Akaike information criterion (AIC) is a measure of out-of-sample prediction error and, as a result, of statistical model quality for a given set of data.

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Linear Regression in Medical Research

Patrick schober.

From the * Department of Anesthesiology, Amsterdam UMC, Vrije Universiteit Amsterdam, Amsterdam, the Netherlands

Thomas R. Vetter

† Department of Surgery and Perioperative Care, Dell Medical School at the University of Texas at Austin, Austin, Texas.

Related Article, see p 110

Linear regression is used to quantify the relationship between ≥1 independent (predictor) variables and a continuous dependent (outcome) variable.

In this issue of Anesthesia & Analgesia , Müller-Wirtz et al 1 report results of a study in which they used linear regression to assess the relationship in a rat model between tissue propofol concentrations and exhaled propofol concentrations (Figure.

Table 2 given in Müller-Wirtz et al, 1 showing the estimated relationships between tissue (or plasma) propofol concentrations and exhaled propofol concentrations. The authors appropriately report the 95% confidence intervals as a measure of the precision of their estimates, as well as the coefficient of determination ( R 2 ). The presented values indicate, for example, that (1) the exhaled propofol concentrations are estimated to increase on average by 4.6 units, equal to the slope (regression) coefficient, for each 1-unit increase of plasma propofol concentration; (2) the “true” mean increase could plausibly be expected to lie anywhere between 3.6 and 5.7 units as indicated by the slope coefficient’s confidence interval; and (3) the R 2 suggests that about 71% of the variability in the exhaled concentration can be explained by its relationship with plasma propofol concentrations.

Linear regression is used to estimate the association of ≥1 independent (predictor) variables with a continuous dependent (outcome) variable. 2 In the most simple case, thus referred to as “simple linear regression,” there is only one independent variable. Simple linear regression fits a straight line to the data points that best characterizes the relationship

between the dependent ( Y ) variable and the independent ( X ) variable, with the y -axis intercept ( b 0 ), and the regression coefficient being the slope ( b 1 ) of this line:

A model that includes several independent variables is referred to as “multiple linear regression” or “multivariable linear regression.” Even though the term linear regression suggests otherwise, it can also be used to model curved relationships.

Linear regression is an extremely versatile technique that can be used to address a variety of research questions and study aims. Researchers may want to test whether there is evidence for a relationship between a categorical (grouping) variable (eg, treatment group or patient sex) and a quantitative outcome (eg, blood pressure). The 2-sample t test and analysis of variance, 3 which are commonly used for this purpose, are essentially special cases of linear regression. However, linear regression is more flexible, allowing for >1 independent variable and allowing for continuous independent variables. Moreover, when there is >1 independent variable, researchers can also test for the interaction of variables—in other words, whether the effect of 1 independent variable depends on the value or level of another independent variable.

Linear regression not only tests for relationships but also quantifies their direction and strength. The regression coefficient describes the average (expected) change in the dependent variable for each 1-unit change in the independent variable for continuous independent variables or the expected difference versus a reference category for categorical independent variables. The coefficient of determination, commonly referred to as R 2 , describes the proportion of the variability in the outcome variable that can be explained by the independent variables. With simple linear regression, the coefficient of determination is also equal to the square of the Pearson correlation between the x and y values.

When including several independent variables, the regression model estimates the effect of each independent variable while holding the values of all other independent variables constant. 4 Thus, linear regression is useful (1) to distinguish the effects of different variables on the outcome and (2) to control for other variables—like systematic confounding in observational studies or baseline imbalances due to chance in a randomized controlled trial. Ultimately, linear regression can be used to predict the value of the dependent outcome variable based on the value(s) of the independent predictor variable(s).

Valid inferences from linear regression rely on its assumptions being met, including

• the residuals are the differences between the observed values and the values predicted by the regression model, and the residuals must be approximately normally distributed and have approximately the same variance over the range of predicted values;
• the residuals are also assumed to be uncorrelated. In simple language, the observations must be independent of each other; for example, there must not be repeated measurements within the same subjects. Other techniques like linear mixed-effects models are required for correlated data 5 ; and
• the model must be correctly specified, as explained in more detail in the next paragraph.

Whereas Müller-Wirtz et al 1 used simple linear regression to address their research question, researchers often need to specify a multivariable model and make choices on which independent variables to include and on how to model the functional relationship between variables (eg, straight line versus curve; inclusion of interaction terms).

Variable selection is a much-debated topic, and the details are beyond the scope of this Statistical Minute. Basically, variable selection depends on whether the purpose of the model is to understand the relationship between variables or to make predictions. This is also predicated on whether there is informed a priori theory to guide variable selection and on whether the model needs to control for variables that are not of primary interest but are confounders that could distort the relationship between other variables.

Omitting important variables or interactions can lead to biased estimates and a model that poorly describes the true underlying relationships, whereas including too many variables leads to modeling the noise (sampling error) in the data and reduces the precision of the estimates. Various statistics and plots, including adjusted R 2 , Mallows C p , and residual plots are available to assess the goodness of fit of the chosen linear regression model.

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Regression Analysis: Definition, Types, Usage & Advantages

Regression analysis is perhaps one of the most widely used statistical methods for investigating or estimating the relationship between a set of independent and dependent variables. In statistical analysis , distinguishing between categorical data and numerical data is essential, as categorical data involves distinct categories or labels, while numerical data consists of measurable quantities.

It is also used as a blanket term for various data analysis techniques utilized in a qualitative research method for modeling and analyzing numerous variables. In the regression method, the dependent variable is a predictor or an explanatory element, and the dependent variable is the outcome or a response to a specific query.

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Definition of Regression Analysis

Types of regression analysis, regression analysis usage in market research, how regression analysis derives insights from surveys, advantages of using regression analysis in an online survey.

Regression analysis is often used to model or analyze data. Most survey analysts use it to understand the relationship between the variables, which can be further utilized to predict the precise outcome.

For Example – Suppose a soft drink company wants to expand its manufacturing unit to a newer location. Before moving forward, the company wants to analyze its revenue generation model and the various factors that might impact it. Hence, the company conducts an online survey with a specific questionnaire.

After using regression analysis, it becomes easier for the company to analyze the survey results and understand the relationship between different variables like electricity and revenue – here, revenue is the dependent variable.

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In addition, understanding the relationship between different independent variables like pricing, number of workers, and logistics with the revenue helps the company estimate the impact of varied factors on sales and profits.

Survey researchers often use this technique to examine and find a correlation between different variables of interest. It provides an opportunity to gauge the influence of different independent variables on a dependent variable.

Overall, regression analysis saves the survey researchers’ additional efforts in arranging several independent variables in tables and testing or calculating their effect on a dependent variable. Different types of analytical research methods are widely used to evaluate new business ideas and make informed decisions.

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Researchers usually start by learning linear and logistic regression first. Due to the widespread knowledge of these two methods and ease of application, many analysts think there are only two types of models. Each model has its own specialty and ability to perform if specific conditions are met.

This blog explains the commonly used seven types of multiple regression analysis methods that can be used to interpret the enumerated data in various formats.

01. Linear Regression Analysis

It is one of the most widely known modeling techniques, as it is amongst the first elite regression analysis methods picked up by people at the time of learning predictive modeling. Here, the dependent variable is continuous, and the independent variable is more often continuous or discreet with a linear regression line.

Please note that multiple linear regression has more than one independent variable than simple linear regression. Thus, linear regression is best to be used only when there is a linear relationship between the independent and a dependent variable.

A business can use linear regression to measure the effectiveness of the marketing campaigns, pricing, and promotions on sales of a product. Suppose a company selling sports equipment wants to understand if the funds they have invested in the marketing and branding of their products have given them substantial returns or not.

Linear regression is the best statistical method to interpret the results. The best thing about linear regression is it also helps in analyzing the obscure impact of each marketing and branding activity, yet controlling the constituent’s potential to regulate the sales.

If the company is running two or more advertising campaigns simultaneously, one on television and two on radio, then linear regression can easily analyze the independent and combined influence of running these advertisements together.

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02. Logistic Regression Analysis

Logistic regression is commonly used to determine the probability of event success and event failure. Logistic regression is used whenever the dependent variable is binary, like 0/1, True/False, or Yes/No. Thus, it can be said that logistic regression is used to analyze either the close-ended questions in a survey or the questions demanding numeric responses in a survey.

Please note logistic regression does not need a linear relationship between a dependent and an independent variable, just like linear regression. Logistic regression applies a non-linear log transformation for predicting the odds ratio; therefore, it easily handles various types of relationships between a dependent and an independent variable.

Logistic regression is widely used to analyze categorical data, particularly for binary response data in business data modeling. More often, logistic regression is used when the dependent variable is categorical, like to predict whether the health claim made by a person is real(1) or fraudulent, to understand if the tumor is malignant(1) or not.

Businesses use logistic regression to predict whether the consumers in a particular demographic will purchase their product or will buy from the competitors based on age, income, gender, race, state of residence, previous purchase, etc.

03. Polynomial Regression Analysis

Polynomial regression is commonly used to analyze curvilinear data when an independent variable’s power is more than 1. In this regression analysis method, the best-fit line is never a ‘straight line’ but always a ‘curve line’ fitting into the data points.

Please note that polynomial regression is better to use when two or more variables have exponents and a few do not.

Additionally, it can model non-linearly separable data offering the liberty to choose the exact exponent for each variable, and that too with full control over the modeling features available.

When combined with response surface analysis, polynomial regression is considered one of the sophisticated statistical methods commonly used in multisource feedback research. Polynomial regression is used mostly in finance and insurance-related industries where the relationship between dependent and independent variables is curvilinear.

Suppose a person wants to budget expense planning by determining how long it would take to earn a definitive sum. Polynomial regression, by taking into account his/her income and predicting expenses, can easily determine the precise time he/she needs to work to earn that specific sum amount.

04. Stepwise Regression Analysis

This is a semi-automated process with which a statistical model is built either by adding or removing the dependent variable on the t-statistics of their estimated coefficients.

If used properly, the stepwise regression will provide you with more powerful data at your fingertips than any method. It works well when you are working with a large number of independent variables. It just fine-tunes the unit of analysis model by poking variables randomly.

Stepwise regression analysis is recommended to be used when there are multiple independent variables, wherein the selection of independent variables is done automatically without human intervention.

Please note, in stepwise regression modeling, the variable is added or subtracted from the set of explanatory variables. The set of added or removed variables is chosen depending on the test statistics of the estimated coefficient.

Suppose you have a set of independent variables like age, weight, body surface area, duration of hypertension, basal pulse, and stress index based on which you want to analyze its impact on the blood pressure.

In stepwise regression, the best subset of the independent variable is automatically chosen; it either starts by choosing no variable to proceed further (as it adds one variable at a time) or starts with all variables in the model and proceeds backward (removes one variable at a time).

Thus, using regression analysis, you can calculate the impact of each or a group of variables on blood pressure.

05. Ridge Regression Analysis

Ridge regression is based on an ordinary least square method which is used to analyze multicollinearity data (data where independent variables are highly correlated). Collinearity can be explained as a near-linear relationship between variables.

Whenever there is multicollinearity, the estimates of least squares will be unbiased, but if the difference between them is larger, then it may be far away from the true value. However, ridge regression eliminates the standard errors by appending some degree of bias to the regression estimates with a motive to provide more reliable estimates.

If you want, you can also learn about Selection Bias through our blog.

Please note, Assumptions derived through the ridge regression are similar to the least squared regression, the only difference being the normality. Although the value of the coefficient is constricted in the ridge regression, it never reaches zero suggesting the inability to select variables.

Suppose you are crazy about two guitarists performing live at an event near you, and you go to watch their performance with a motive to find out who is a better guitarist. But when the performance starts, you notice that both are playing black-and-blue notes at the same time.

Is it possible to find out the best guitarist having the biggest impact on sound among them when they are both playing loud and fast? As both of them are playing different notes, it is substantially difficult to differentiate them, making it the best case of multicollinearity, which tends to increase the standard errors of the coefficients.

Ridge regression addresses multicollinearity in cases like these and includes bias or a shrinkage estimation to derive results.

06. Lasso Regression Analysis

Lasso (Least Absolute Shrinkage and Selection Operator) is similar to ridge regression; however, it uses an absolute value bias instead of the square bias used in ridge regression.

It was developed way back in 1989 as an alternative to the traditional least-squares estimate with the intention to deduce the majority of problems related to overfitting when the data has a large number of independent variables.

Lasso has the capability to perform both – selecting variables and regularizing them along with a soft threshold. Applying lasso regression makes it easier to derive a subset of predictors from minimizing prediction errors while analyzing a quantitative response.

Please note that regression coefficients reaching zero value after shrinkage are excluded from the lasso model. On the contrary, regression coefficients having more value than zero are strongly associated with the response variables, wherein the explanatory variables can be either quantitative, categorical, or both.

Suppose an automobile company wants to perform a research analysis on average fuel consumption by cars in the US. For samples, they chose 32 models of car and 10 features of automobile design – Number of cylinders, Displacement, Gross horsepower, Rear axle ratio, Weight, ¼ mile time, v/s engine, transmission, number of gears, and number of carburetors.

As you can see a correlation between the response variable mpg (miles per gallon) is extremely correlated to some variables like weight, displacement, number of cylinders, and horsepower. The problem can be analyzed by using the glmnet package in R and lasso regression for feature selection.

07. Elastic Net Regression Analysis

It is a mixture of ridge and lasso regression models trained with L1 and L2 norms. The elastic net brings about a grouping effect wherein strongly correlated predictors tend to be in/out of the model together. Using the elastic net regression model is recommended when the number of predictors is far greater than the number of observations.

Please note that the elastic net regression model came into existence as an option to the lasso regression model as lasso’s variable section was too much dependent on data, making it unstable. By using elastic net regression, statisticians became capable of over-bridging the penalties of ridge and lasso regression only to get the best out of both models.

A clinical research team having access to a microarray data set on leukemia (LEU) was interested in constructing a diagnostic rule based on the expression level of presented gene samples for predicting the type of leukemia. The data set they had, consisted of a large number of genes and a few samples.

Apart from that, they were given a specific set of samples to be used as training samples, out of which some were infected with type 1 leukemia (acute lymphoblastic leukemia) and some with type 2 leukemia (acute myeloid leukemia).

Model fitting and tuning parameter selection by tenfold CV were carried out on the training data. Then they compared the performance of those methods by computing their prediction mean-squared error on the test data to get the necessary results.

A market research survey focuses on three major matrices; Customer Satisfaction , Customer Loyalty , and Customer Advocacy . Remember, although these matrices tell us about customer health and intentions, they fail to tell us ways of improving the position. Therefore, an in-depth survey questionnaire intended to ask consumers the reason behind their dissatisfaction is definitely a way to gain practical insights.

However, it has been found that people often struggle to put forth their motivation or demotivation or describe their satisfaction or dissatisfaction. In addition to that, people always give undue importance to some rational factors, such as price, packaging, etc. Overall, it acts as a predictive analytic and forecasting tool in market research.

When used as a forecasting tool, regression analysis can determine an organization’s sales figures by taking into account external market data. A multinational company conducts a market research survey to understand the impact of various factors such as GDP (Gross Domestic Product), CPI (Consumer Price Index), and other similar factors on its revenue generation model.

Obviously, regression analysis in consideration of forecasted marketing indicators was used to predict a tentative revenue that will be generated in future quarters and even in future years. However, the more forward you go in the future, the data will become more unreliable, leaving a wide margin of error .

Case study of using regression analysis

A water purifier company wanted to understand the factors leading to brand favorability. The survey was the best medium for reaching out to existing and prospective customers. A large-scale consumer survey was planned, and a discreet questionnaire was prepared using the best survey tool .

A number of questions related to the brand, favorability, satisfaction, and probable dissatisfaction were effectively asked in the survey. After getting optimum responses to the survey, regression analysis was used to narrow down the top ten factors responsible for driving brand favorability.

All the ten attributes derived (mentioned in the image below) in one or the other way highlighted their importance in impacting the favorability of that specific water purifier brand.

It is easy to run a regression analysis using Excel or SPSS, but while doing so, the importance of four numbers in interpreting the data must be understood.

The first two numbers out of the four numbers directly relate to the regression model itself.

• F-Value: It helps in measuring the statistical significance of the survey model. Remember, an F-Value significantly less than 0.05 is considered to be more meaningful. Less than 0.05 F-Value ensures survey analysis output is not by chance.
• R-Squared: This is the value wherein the independent variables try to explain the amount of movement by dependent variables. Considering the R-Squared value is 0.7, a tested independent variable can explain 70% of the dependent variable’s movement. It means the survey analysis output we will be getting is highly predictive in nature and can be considered accurate.

The other two numbers relate to each of the independent variables while interpreting regression analysis.

• P-Value: Like F-Value, even the P-Value is statistically significant. Moreover, here it indicates how relevant and statistically significant the independent variable’s effect is. Once again, we are looking for a value of less than 0.05.
• Interpretation: The fourth number relates to the coefficient achieved after measuring the impact of variables. For instance, we test multiple independent variables to get a coefficient. It tells us, ‘by what value the dependent variable is expected to increase when independent variables (which we are considering) increase by one when all other independent variables are stagnant at the same value.

In a few cases, the simple coefficient is replaced by a standardized coefficient demonstrating the contribution from each independent variable to move or bring about a change in the dependent variable.

01. Get access to predictive analytics

Do you know utilizing regression analysis to understand the outcome of a business survey is like having the power to unveil future opportunities and risks?

For example, after seeing a particular television advertisement slot, we can predict the exact number of businesses using that data to estimate a maximum bid for that slot. The finance and insurance industry as a whole depends a lot on regression analysis of survey data to identify trends and opportunities for more accurate planning and decision-making.

02. Enhance operational efficiency

Do you know businesses use regression analysis to optimize their business processes?

For example, before launching a new product line, businesses conduct consumer surveys to better understand the impact of various factors on the product’s production, packaging, distribution, and consumption.

A data-driven foresight helps eliminate the guesswork, hypothesis, and internal politics from decision-making. A deeper understanding of the areas impacting operational efficiencies and revenues leads to better business optimization.

03. Quantitative support for decision-making

Business surveys today generate a lot of data related to finance, revenue, operation, purchases, etc., and business owners are heavily dependent on various data analysis models to make informed business decisions.

For example, regression analysis helps enterprises to make informed strategic workforce decisions. Conducting and interpreting the outcome of employee surveys like Employee Engagement Surveys, Employee Satisfaction Surveys, Employer Improvement Surveys, Employee Exit Surveys, etc., boosts the understanding of the relationship between employees and the enterprise.

It also helps get a fair idea of certain issues impacting the organization’s working culture, working environment, and productivity. Furthermore, intelligent business-oriented interpretations reduce the huge pile of raw data into actionable information to make a more informed decision.

04. Prevent mistakes from happening due to intuitions

By knowing how to use regression analysis for interpreting survey results, one can easily provide factual support to management for making informed decisions. ; but do you know that it also helps in keeping out faults in the judgment?

For example, a mall manager thinks if he extends the closing time of the mall, then it will result in more sales. Regression analysis contradicts the belief that predicting increased revenue due to increased sales won’t support the increased operating expenses arising from longer working hours.

Regression analysis is a useful statistical method for modeling and comprehending the relationships between variables. It provides numerous advantages to various data types and interactions. Researchers and analysts may gain useful insights into the factors influencing a dependent variable and use the results to make informed decisions. 

With QuestionPro Research, you can improve the efficiency and accuracy of regression analysis by streamlining the data gathering, analysis, and reporting processes. The platform’s user-friendly interface and wide range of features make it a valuable tool for researchers and analysts conducting regression analysis as part of their research projects.

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• Published: 26 March 2024

Predicting and improving complex beer flavor through machine learning

• Michiel Schreurs   ORCID: orcid.org/0000-0002-9449-5619 1 , 2 , 3   na1 ,
• Supinya Piampongsant 1 , 2 , 3   na1 ,
• Miguel Roncoroni   ORCID: orcid.org/0000-0001-7461-1427 1 , 2 , 3   na1 ,
• Lloyd Cool   ORCID: orcid.org/0000-0001-9936-3124 1 , 2 , 3 , 4 ,
• Beatriz Herrera-Malaver   ORCID: orcid.org/0000-0002-5096-9974 1 , 2 , 3 ,
• Christophe Vanderaa   ORCID: orcid.org/0000-0001-7443-5427 4 ,
• Florian A. Theßeling 1 , 2 , 3 ,
• Łukasz Kreft   ORCID: orcid.org/0000-0001-7620-4657 5 ,
• Alexander Botzki   ORCID: orcid.org/0000-0001-6691-4233 5 ,
• Philippe Malcorps 6 ,
• Luk Daenen 6 ,
• Tom Wenseleers   ORCID: orcid.org/0000-0002-1434-861X 4 &
• Kevin J. Verstrepen   ORCID: orcid.org/0000-0002-3077-6219 1 , 2 , 3  

Nature Communications volume  15 , Article number:  2368 ( 2024 ) Cite this article

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• Chemical engineering
• Gas chromatography
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• Metabolomics
• Taste receptors

The perception and appreciation of food flavor depends on many interacting chemical compounds and external factors, and therefore proves challenging to understand and predict. Here, we combine extensive chemical and sensory analyses of 250 different beers to train machine learning models that allow predicting flavor and consumer appreciation. For each beer, we measure over 200 chemical properties, perform quantitative descriptive sensory analysis with a trained tasting panel and map data from over 180,000 consumer reviews to train 10 different machine learning models. The best-performing algorithm, Gradient Boosting, yields models that significantly outperform predictions based on conventional statistics and accurately predict complex food features and consumer appreciation from chemical profiles. Model dissection allows identifying specific and unexpected compounds as drivers of beer flavor and appreciation. Adding these compounds results in variants of commercial alcoholic and non-alcoholic beers with improved consumer appreciation. Together, our study reveals how big data and machine learning uncover complex links between food chemistry, flavor and consumer perception, and lays the foundation to develop novel, tailored foods with superior flavors.

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Introduction

Predicting and understanding food perception and appreciation is one of the major challenges in food science. Accurate modeling of food flavor and appreciation could yield important opportunities for both producers and consumers, including quality control, product fingerprinting, counterfeit detection, spoilage detection, and the development of new products and product combinations (food pairing) 1 , 2 , 3 , 4 , 5 , 6 . Accurate models for flavor and consumer appreciation would contribute greatly to our scientific understanding of how humans perceive and appreciate flavor. Moreover, accurate predictive models would also facilitate and standardize existing food assessment methods and could supplement or replace assessments by trained and consumer tasting panels, which are variable, expensive and time-consuming 7 , 8 , 9 . Lastly, apart from providing objective, quantitative, accurate and contextual information that can help producers, models can also guide consumers in understanding their personal preferences 10 .

Despite the myriad of applications, predicting food flavor and appreciation from its chemical properties remains a largely elusive goal in sensory science, especially for complex food and beverages 11 , 12 . A key obstacle is the immense number of flavor-active chemicals underlying food flavor. Flavor compounds can vary widely in chemical structure and concentration, making them technically challenging and labor-intensive to quantify, even in the face of innovations in metabolomics, such as non-targeted metabolic fingerprinting 13 , 14 . Moreover, sensory analysis is perhaps even more complicated. Flavor perception is highly complex, resulting from hundreds of different molecules interacting at the physiochemical and sensorial level. Sensory perception is often non-linear, characterized by complex and concentration-dependent synergistic and antagonistic effects 15 , 16 , 17 , 18 , 19 , 20 , 21 that are further convoluted by the genetics, environment, culture and psychology of consumers 22 , 23 , 24 . Perceived flavor is therefore difficult to measure, with problems of sensitivity, accuracy, and reproducibility that can only be resolved by gathering sufficiently large datasets 25 . Trained tasting panels are considered the prime source of quality sensory data, but require meticulous training, are low throughput and high cost. Public databases containing consumer reviews of food products could provide a valuable alternative, especially for studying appreciation scores, which do not require formal training 25 . Public databases offer the advantage of amassing large amounts of data, increasing the statistical power to identify potential drivers of appreciation. However, public datasets suffer from biases, including a bias in the volunteers that contribute to the database, as well as confounding factors such as price, cult status and psychological conformity towards previous ratings of the product.

Classical multivariate statistics and machine learning methods have been used to predict flavor of specific compounds by, for example, linking structural properties of a compound to its potential biological activities or linking concentrations of specific compounds to sensory profiles 1 , 26 . Importantly, most previous studies focused on predicting organoleptic properties of single compounds (often based on their chemical structure) 27 , 28 , 29 , 30 , 31 , 32 , 33 , thus ignoring the fact that these compounds are present in a complex matrix in food or beverages and excluding complex interactions between compounds. Moreover, the classical statistics commonly used in sensory science 34 , 35 , 36 , 37 , 38 , 39 require a large sample size and sufficient variance amongst predictors to create accurate models. They are not fit for studying an extensive set of hundreds of interacting flavor compounds, since they are sensitive to outliers, have a high tendency to overfit and are less suited for non-linear and discontinuous relationships 40 .

In this study, we combine extensive chemical analyses and sensory data of a set of different commercial beers with machine learning approaches to develop models that predict taste, smell, mouthfeel and appreciation from compound concentrations. Beer is particularly suited to model the relationship between chemistry, flavor and appreciation. First, beer is a complex product, consisting of thousands of flavor compounds that partake in complex sensory interactions 41 , 42 , 43 . This chemical diversity arises from the raw materials (malt, yeast, hops, water and spices) and biochemical conversions during the brewing process (kilning, mashing, boiling, fermentation, maturation and aging) 44 , 45 . Second, the advent of the internet saw beer consumers embrace online review platforms, such as RateBeer (ZX Ventures, Anheuser-Busch InBev SA/NV) and BeerAdvocate (Next Glass, inc.). In this way, the beer community provides massive data sets of beer flavor and appreciation scores, creating extraordinarily large sensory databases to complement the analyses of our professional sensory panel. Specifically, we characterize over 200 chemical properties of 250 commercial beers, spread across 22 beer styles, and link these to the descriptive sensory profiling data of a 16-person in-house trained tasting panel and data acquired from over 180,000 public consumer reviews. These unique and extensive datasets enable us to train a suite of machine learning models to predict flavor and appreciation from a beer’s chemical profile. Dissection of the best-performing models allows us to pinpoint specific compounds as potential drivers of beer flavor and appreciation. Follow-up experiments confirm the importance of these compounds and ultimately allow us to significantly improve the flavor and appreciation of selected commercial beers. Together, our study represents a significant step towards understanding complex flavors and reinforces the value of machine learning to develop and refine complex foods. In this way, it represents a stepping stone for further computer-aided food engineering applications 46 .

To generate a comprehensive dataset on beer flavor, we selected 250 commercial Belgian beers across 22 different beer styles (Supplementary Fig.  S1 ). Beers with ≤ 4.2% alcohol by volume (ABV) were classified as non-alcoholic and low-alcoholic. Blonds and Tripels constitute a significant portion of the dataset (12.4% and 11.2%, respectively) reflecting their presence on the Belgian beer market and the heterogeneity of beers within these styles. By contrast, lager beers are less diverse and dominated by a handful of brands. Rare styles such as Brut or Faro make up only a small fraction of the dataset (2% and 1%, respectively) because fewer of these beers are produced and because they are dominated by distinct characteristics in terms of flavor and chemical composition.

Extensive analysis identifies relationships between chemical compounds in beer

For each beer, we measured 226 different chemical properties, including common brewing parameters such as alcohol content, iso-alpha acids, pH, sugar concentration 47 , and over 200 flavor compounds (Methods, Supplementary Table  S1 ). A large portion (37.2%) are terpenoids arising from hopping, responsible for herbal and fruity flavors 16 , 48 . A second major category are yeast metabolites, such as esters and alcohols, that result in fruity and solvent notes 48 , 49 , 50 . Other measured compounds are primarily derived from malt, or other microbes such as non- Saccharomyces yeasts and bacteria (‘wild flora’). Compounds that arise from spices or staling are labeled under ‘Others’. Five attributes (caloric value, total acids and total ester, hop aroma and sulfur compounds) are calculated from multiple individually measured compounds.

As a first step in identifying relationships between chemical properties, we determined correlations between the concentrations of the compounds (Fig.  1 , upper panel, Supplementary Data  1 and 2 , and Supplementary Fig.  S2 . For the sake of clarity, only a subset of the measured compounds is shown in Fig.  1 ). Compounds of the same origin typically show a positive correlation, while absence of correlation hints at parameters varying independently. For example, the hop aroma compounds citronellol, and alpha-terpineol show moderate correlations with each other (Spearman’s rho=0.39 and 0.57), but not with the bittering hop component iso-alpha acids (Spearman’s rho=0.16 and −0.07). This illustrates how brewers can independently modify hop aroma and bitterness by selecting hop varieties and dosage time. If hops are added early in the boiling phase, chemical conversions increase bitterness while aromas evaporate, conversely, late addition of hops preserves aroma but limits bitterness 51 . Similarly, hop-derived iso-alpha acids show a strong anti-correlation with lactic acid and acetic acid, likely reflecting growth inhibition of lactic acid and acetic acid bacteria, or the consequent use of fewer hops in sour beer styles, such as West Flanders ales and Fruit beers, that rely on these bacteria for their distinct flavors 52 . Finally, yeast-derived esters (ethyl acetate, ethyl decanoate, ethyl hexanoate, ethyl octanoate) and alcohols (ethanol, isoamyl alcohol, isobutanol, and glycerol), correlate with Spearman coefficients above 0.5, suggesting that these secondary metabolites are correlated with the yeast genetic background and/or fermentation parameters and may be difficult to influence individually, although the choice of yeast strain may offer some control 53 .

Spearman rank correlations are shown. Descriptors are grouped according to their origin (malt (blue), hops (green), yeast (red), wild flora (yellow), Others (black)), and sensory aspect (aroma, taste, palate, and overall appreciation). Please note that for the chemical compounds, for the sake of clarity, only a subset of the total number of measured compounds is shown, with an emphasis on the key compounds for each source. For more details, see the main text and Methods section. Chemical data can be found in Supplementary Data  1 , correlations between all chemical compounds are depicted in Supplementary Fig.  S2 and correlation values can be found in Supplementary Data  2 . See Supplementary Data  4 for sensory panel assessments and Supplementary Data  5 for correlation values between all sensory descriptors.

Interestingly, different beer styles show distinct patterns for some flavor compounds (Supplementary Fig.  S3 ). These observations agree with expectations for key beer styles, and serve as a control for our measurements. For instance, Stouts generally show high values for color (darker), while hoppy beers contain elevated levels of iso-alpha acids, compounds associated with bitter hop taste. Acetic and lactic acid are not prevalent in most beers, with notable exceptions such as Kriek, Lambic, Faro, West Flanders ales and Flanders Old Brown, which use acid-producing bacteria ( Lactobacillus and Pediococcus ) or unconventional yeast ( Brettanomyces ) 54 , 55 . Glycerol, ethanol and esters show similar distributions across all beer styles, reflecting their common origin as products of yeast metabolism during fermentation 45 , 53 . Finally, low/no-alcohol beers contain low concentrations of glycerol and esters. This is in line with the production process for most of the low/no-alcohol beers in our dataset, which are produced through limiting fermentation or by stripping away alcohol via evaporation or dialysis, with both methods having the unintended side-effect of reducing the amount of flavor compounds in the final beer 56 , 57 .

Besides expected associations, our data also reveals less trivial associations between beer styles and specific parameters. For example, geraniol and citronellol, two monoterpenoids responsible for citrus, floral and rose flavors and characteristic of Citra hops, are found in relatively high amounts in Christmas, Saison, and Brett/co-fermented beers, where they may originate from terpenoid-rich spices such as coriander seeds instead of hops 58 .

Tasting panel assessments reveal sensorial relationships in beer

To assess the sensory profile of each beer, a trained tasting panel evaluated each of the 250 beers for 50 sensory attributes, including different hop, malt and yeast flavors, off-flavors and spices. Panelists used a tasting sheet (Supplementary Data  3 ) to score the different attributes. Panel consistency was evaluated by repeating 12 samples across different sessions and performing ANOVA. In 95% of cases no significant difference was found across sessions ( p  > 0.05), indicating good panel consistency (Supplementary Table  S2 ).

Aroma and taste perception reported by the trained panel are often linked (Fig.  1 , bottom left panel and Supplementary Data  4 and 5 ), with high correlations between hops aroma and taste (Spearman’s rho=0.83). Bitter taste was found to correlate with hop aroma and taste in general (Spearman’s rho=0.80 and 0.69), and particularly with “grassy” noble hops (Spearman’s rho=0.75). Barnyard flavor, most often associated with sour beers, is identified together with stale hops (Spearman’s rho=0.97) that are used in these beers. Lactic and acetic acid, which often co-occur, are correlated (Spearman’s rho=0.66). Interestingly, sweetness and bitterness are anti-correlated (Spearman’s rho = −0.48), confirming the hypothesis that they mask each other 59 , 60 . Beer body is highly correlated with alcohol (Spearman’s rho = 0.79), and overall appreciation is found to correlate with multiple aspects that describe beer mouthfeel (alcohol, carbonation; Spearman’s rho= 0.32, 0.39), as well as with hop and ester aroma intensity (Spearman’s rho=0.39 and 0.35).

Similar to the chemical analyses, sensorial analyses confirmed typical features of specific beer styles (Supplementary Fig.  S4 ). For example, sour beers (Faro, Flanders Old Brown, Fruit beer, Kriek, Lambic, West Flanders ale) were rated acidic, with flavors of both acetic and lactic acid. Hoppy beers were found to be bitter and showed hop-associated aromas like citrus and tropical fruit. Malt taste is most detected among scotch, stout/porters, and strong ales, while low/no-alcohol beers, which often have a reputation for being ‘worty’ (reminiscent of unfermented, sweet malt extract) appear in the middle. Unsurprisingly, hop aromas are most strongly detected among hoppy beers. Like its chemical counterpart (Supplementary Fig.  S3 ), acidity shows a right-skewed distribution, with the most acidic beers being Krieks, Lambics, and West Flanders ales.

Tasting panel assessments of specific flavors correlate with chemical composition

We find that the concentrations of several chemical compounds strongly correlate with specific aroma or taste, as evaluated by the tasting panel (Fig.  2 , Supplementary Fig.  S5 , Supplementary Data  6 ). In some cases, these correlations confirm expectations and serve as a useful control for data quality. For example, iso-alpha acids, the bittering compounds in hops, strongly correlate with bitterness (Spearman’s rho=0.68), while ethanol and glycerol correlate with tasters’ perceptions of alcohol and body, the mouthfeel sensation of fullness (Spearman’s rho=0.82/0.62 and 0.72/0.57 respectively) and darker color from roasted malts is a good indication of malt perception (Spearman’s rho=0.54).

Heatmap colors indicate Spearman’s Rho. Axes are organized according to sensory categories (aroma, taste, mouthfeel, overall), chemical categories and chemical sources in beer (malt (blue), hops (green), yeast (red), wild flora (yellow), Others (black)). See Supplementary Data  6 for all correlation values.

Interestingly, for some relationships between chemical compounds and perceived flavor, correlations are weaker than expected. For example, the rose-smelling phenethyl acetate only weakly correlates with floral aroma. This hints at more complex relationships and interactions between compounds and suggests a need for a more complex model than simple correlations. Lastly, we uncovered unexpected correlations. For instance, the esters ethyl decanoate and ethyl octanoate appear to correlate slightly with hop perception and bitterness, possibly due to their fruity flavor. Iron is anti-correlated with hop aromas and bitterness, most likely because it is also anti-correlated with iso-alpha acids. This could be a sign of metal chelation of hop acids 61 , given that our analyses measure unbound hop acids and total iron content, or could result from the higher iron content in dark and Fruit beers, which typically have less hoppy and bitter flavors 62 .

Public consumer reviews complement expert panel data

To complement and expand the sensory data of our trained tasting panel, we collected 180,000 reviews of our 250 beers from the online consumer review platform RateBeer. This provided numerical scores for beer appearance, aroma, taste, palate, overall quality as well as the average overall score.

Public datasets are known to suffer from biases, such as price, cult status and psychological conformity towards previous ratings of a product. For example, prices correlate with appreciation scores for these online consumer reviews (rho=0.49, Supplementary Fig.  S6 ), but not for our trained tasting panel (rho=0.19). This suggests that prices affect consumer appreciation, which has been reported in wine 63 , while blind tastings are unaffected. Moreover, we observe that some beer styles, like lagers and non-alcoholic beers, generally receive lower scores, reflecting that online reviewers are mostly beer aficionados with a preference for specialty beers over lager beers. In general, we find a modest correlation between our trained panel’s overall appreciation score and the online consumer appreciation scores (Fig.  3 , rho=0.29). Apart from the aforementioned biases in the online datasets, serving temperature, sample freshness and surroundings, which are all tightly controlled during the tasting panel sessions, can vary tremendously across online consumers and can further contribute to (among others, appreciation) differences between the two categories of tasters. Importantly, in contrast to the overall appreciation scores, for many sensory aspects the results from the professional panel correlated well with results obtained from RateBeer reviews. Correlations were highest for features that are relatively easy to recognize even for untrained tasters, like bitterness, sweetness, alcohol and malt aroma (Fig.  3 and below).

RateBeer text mining results can be found in Supplementary Data  7 . Rho values shown are Spearman correlation values, with asterisks indicating significant correlations ( p  < 0.05, two-sided). All p values were smaller than 0.001, except for Esters aroma (0.0553), Esters taste (0.3275), Esters aroma—banana (0.0019), Coriander (0.0508) and Diacetyl (0.0134).

Besides collecting consumer appreciation from these online reviews, we developed automated text analysis tools to gather additional data from review texts (Supplementary Data  7 ). Processing review texts on the RateBeer database yielded comparable results to the scores given by the trained panel for many common sensory aspects, including acidity, bitterness, sweetness, alcohol, malt, and hop tastes (Fig.  3 ). This is in line with what would be expected, since these attributes require less training for accurate assessment and are less influenced by environmental factors such as temperature, serving glass and odors in the environment. Consumer reviews also correlate well with our trained panel for 4-vinyl guaiacol, a compound associated with a very characteristic aroma. By contrast, correlations for more specific aromas like ester, coriander or diacetyl are underrepresented in the online reviews, underscoring the importance of using a trained tasting panel and standardized tasting sheets with explicit factors to be scored for evaluating specific aspects of a beer. Taken together, our results suggest that public reviews are trustworthy for some, but not all, flavor features and can complement or substitute taste panel data for these sensory aspects.

Models can predict beer sensory profiles from chemical data

The rich datasets of chemical analyses, tasting panel assessments and public reviews gathered in the first part of this study provided us with a unique opportunity to develop predictive models that link chemical data to sensorial features. Given the complexity of beer flavor, basic statistical tools such as correlations or linear regression may not always be the most suitable for making accurate predictions. Instead, we applied different machine learning models that can model both simple linear and complex interactive relationships. Specifically, we constructed a set of regression models to predict (a) trained panel scores for beer flavor and quality and (b) public reviews’ appreciation scores from beer chemical profiles. We trained and tested 10 different models (Methods), 3 linear regression-based models (simple linear regression with first-order interactions (LR), lasso regression with first-order interactions (Lasso), partial least squares regressor (PLSR)), 5 decision tree models (AdaBoost regressor (ABR), extra trees (ET), gradient boosting regressor (GBR), random forest (RF) and XGBoost regressor (XGBR)), 1 support vector regression (SVR), and 1 artificial neural network (ANN) model.

To compare the performance of our machine learning models, the dataset was randomly split into a training and test set, stratified by beer style. After a model was trained on data in the training set, its performance was evaluated on its ability to predict the test dataset obtained from multi-output models (based on the coefficient of determination, see Methods). Additionally, individual-attribute models were ranked per descriptor and the average rank was calculated, as proposed by Korneva et al. 64 . Importantly, both ways of evaluating the models’ performance agreed in general. Performance of the different models varied (Table  1 ). It should be noted that all models perform better at predicting RateBeer results than results from our trained tasting panel. One reason could be that sensory data is inherently variable, and this variability is averaged out with the large number of public reviews from RateBeer. Additionally, all tree-based models perform better at predicting taste than aroma. Linear models (LR) performed particularly poorly, with negative R 2 values, due to severe overfitting (training set R 2  = 1). Overfitting is a common issue in linear models with many parameters and limited samples, especially with interaction terms further amplifying the number of parameters. L1 regularization (Lasso) successfully overcomes this overfitting, out-competing multiple tree-based models on the RateBeer dataset. Similarly, the dimensionality reduction of PLSR avoids overfitting and improves performance, to some extent. Still, tree-based models (ABR, ET, GBR, RF and XGBR) show the best performance, out-competing the linear models (LR, Lasso, PLSR) commonly used in sensory science 65 .

GBR models showed the best overall performance in predicting sensory responses from chemical information, with R 2 values up to 0.75 depending on the predicted sensory feature (Supplementary Table  S4 ). The GBR models predict consumer appreciation (RateBeer) better than our trained panel’s appreciation (R 2 value of 0.67 compared to R 2 value of 0.09) (Supplementary Table  S3 and Supplementary Table  S4 ). ANN models showed intermediate performance, likely because neural networks typically perform best with larger datasets 66 . The SVR shows intermediate performance, mostly due to the weak predictions of specific attributes that lower the overall performance (Supplementary Table  S4 ).

Model dissection identifies specific, unexpected compounds as drivers of consumer appreciation

Next, we leveraged our models to infer important contributors to sensory perception and consumer appreciation. Consumer preference is a crucial sensory aspects, because a product that shows low consumer appreciation scores often does not succeed commercially 25 . Additionally, the requirement for a large number of representative evaluators makes consumer trials one of the more costly and time-consuming aspects of product development. Hence, a model for predicting chemical drivers of overall appreciation would be a welcome addition to the available toolbox for food development and optimization.

Since GBR models on our RateBeer dataset showed the best overall performance, we focused on these models. Specifically, we used two approaches to identify important contributors. First, rankings of the most important predictors for each sensorial trait in the GBR models were obtained based on impurity-based feature importance (mean decrease in impurity). High-ranked parameters were hypothesized to be either the true causal chemical properties underlying the trait, to correlate with the actual causal properties, or to take part in sensory interactions affecting the trait 67 (Fig.  4A ). In a second approach, we used SHAP 68 to determine which parameters contributed most to the model for making predictions of consumer appreciation (Fig.  4B ). SHAP calculates parameter contributions to model predictions on a per-sample basis, which can be aggregated into an importance score.

A The impurity-based feature importance (mean deviance in impurity, MDI) calculated from the Gradient Boosting Regression (GBR) model predicting RateBeer appreciation scores. The top 15 highest ranked chemical properties are shown. B SHAP summary plot for the top 15 parameters contributing to our GBR model. Each point on the graph represents a sample from our dataset. The color represents the concentration of that parameter, with bluer colors representing low values and redder colors representing higher values. Greater absolute values on the horizontal axis indicate a higher impact of the parameter on the prediction of the model. C Spearman correlations between the 15 most important chemical properties and consumer overall appreciation. Numbers indicate the Spearman Rho correlation coefficient, and the rank of this correlation compared to all other correlations. The top 15 important compounds were determined using SHAP (panel B).

Both approaches identified ethyl acetate as the most predictive parameter for beer appreciation (Fig.  4 ). Ethyl acetate is the most abundant ester in beer with a typical ‘fruity’, ‘solvent’ and ‘alcoholic’ flavor, but is often considered less important than other esters like isoamyl acetate. The second most important parameter identified by SHAP is ethanol, the most abundant beer compound after water. Apart from directly contributing to beer flavor and mouthfeel, ethanol drastically influences the physical properties of beer, dictating how easily volatile compounds escape the beer matrix to contribute to beer aroma 69 . Importantly, it should also be noted that the importance of ethanol for appreciation is likely inflated by the very low appreciation scores of non-alcoholic beers (Supplementary Fig.  S4 ). Despite not often being considered a driver of beer appreciation, protein level also ranks highly in both approaches, possibly due to its effect on mouthfeel and body 70 . Lactic acid, which contributes to the tart taste of sour beers, is the fourth most important parameter identified by SHAP, possibly due to the generally high appreciation of sour beers in our dataset.

Interestingly, some of the most important predictive parameters for our model are not well-established as beer flavors or are even commonly regarded as being negative for beer quality. For example, our models identify methanethiol and ethyl phenyl acetate, an ester commonly linked to beer staling 71 , as a key factor contributing to beer appreciation. Although there is no doubt that high concentrations of these compounds are considered unpleasant, the positive effects of modest concentrations are not yet known 72 , 73 .

To compare our approach to conventional statistics, we evaluated how well the 15 most important SHAP-derived parameters correlate with consumer appreciation (Fig.  4C ). Interestingly, only 6 of the properties derived by SHAP rank amongst the top 15 most correlated parameters. For some chemical compounds, the correlations are so low that they would have likely been considered unimportant. For example, lactic acid, the fourth most important parameter, shows a bimodal distribution for appreciation, with sour beers forming a separate cluster, that is missed entirely by the Spearman correlation. Additionally, the correlation plots reveal outliers, emphasizing the need for robust analysis tools. Together, this highlights the need for alternative models, like the Gradient Boosting model, that better grasp the complexity of (beer) flavor.

Finally, to observe the relationships between these chemical properties and their predicted targets, partial dependence plots were constructed for the six most important predictors of consumer appreciation 74 , 75 , 76 (Supplementary Fig.  S7 ). One-way partial dependence plots show how a change in concentration affects the predicted appreciation. These plots reveal an important limitation of our models: appreciation predictions remain constant at ever-increasing concentrations. This implies that once a threshold concentration is reached, further increasing the concentration does not affect appreciation. This is false, as it is well-documented that certain compounds become unpleasant at high concentrations, including ethyl acetate (‘nail polish’) 77 and methanethiol (‘sulfury’ and ‘rotten cabbage’) 78 . The inability of our models to grasp that flavor compounds have optimal levels, above which they become negative, is a consequence of working with commercial beer brands where (off-)flavors are rarely too high to negatively impact the product. The two-way partial dependence plots show how changing the concentration of two compounds influences predicted appreciation, visualizing their interactions (Supplementary Fig.  S7 ). In our case, the top 5 parameters are dominated by additive or synergistic interactions, with high concentrations for both compounds resulting in the highest predicted appreciation.

To assess the robustness of our best-performing models and model predictions, we performed 100 iterations of the GBR, RF and ET models. In general, all iterations of the models yielded similar performance (Supplementary Fig.  S8 ). Moreover, the main predictors (including the top predictors ethanol and ethyl acetate) remained virtually the same, especially for GBR and RF. For the iterations of the ET model, we did observe more variation in the top predictors, which is likely a consequence of the model’s inherent random architecture in combination with co-correlations between certain predictors. However, even in this case, several of the top predictors (ethanol and ethyl acetate) remain unchanged, although their rank in importance changes (Supplementary Fig.  S8 ).

Next, we investigated if a combination of RateBeer and trained panel data into one consolidated dataset would lead to stronger models, under the hypothesis that such a model would suffer less from bias in the datasets. A GBR model was trained to predict appreciation on the combined dataset. This model underperformed compared to the RateBeer model, both in the native case and when including a dataset identifier (R 2  = 0.67, 0.26 and 0.42 respectively). For the latter, the dataset identifier is the most important feature (Supplementary Fig.  S9 ), while most of the feature importance remains unchanged, with ethyl acetate and ethanol ranking highest, like in the original model trained only on RateBeer data. It seems that the large variation in the panel dataset introduces noise, weakening the models’ performances and reliability. In addition, it seems reasonable to assume that both datasets are fundamentally different, with the panel dataset obtained by blind tastings by a trained professional panel.

Lastly, we evaluated whether beer style identifiers would further enhance the model’s performance. A GBR model was trained with parameters that explicitly encoded the styles of the samples. This did not improve model performance (R2 = 0.66 with style information vs R2 = 0.67). The most important chemical features are consistent with the model trained without style information (eg. ethanol and ethyl acetate), and with the exception of the most preferred (strong ale) and least preferred (low/no-alcohol) styles, none of the styles were among the most important features (Supplementary Fig.  S9 , Supplementary Table  S5 and S6 ). This is likely due to a combination of style-specific chemical signatures, such as iso-alpha acids and lactic acid, that implicitly convey style information to the original models, as well as the low number of samples belonging to some styles, making it difficult for the model to learn style-specific patterns. Moreover, beer styles are not rigorously defined, with some styles overlapping in features and some beers being misattributed to a specific style, all of which leads to more noise in models that use style parameters.

Model validation

To test if our predictive models give insight into beer appreciation, we set up experiments aimed at improving existing commercial beers. We specifically selected overall appreciation as the trait to be examined because of its complexity and commercial relevance. Beer flavor comprises a complex bouquet rather than single aromas and tastes 53 . Hence, adding a single compound to the extent that a difference is noticeable may lead to an unbalanced, artificial flavor. Therefore, we evaluated the effect of combinations of compounds. Because Blond beers represent the most extensive style in our dataset, we selected a beer from this style as the starting material for these experiments (Beer 64 in Supplementary Data  1 ).

In the first set of experiments, we adjusted the concentrations of compounds that made up the most important predictors of overall appreciation (ethyl acetate, ethanol, lactic acid, ethyl phenyl acetate) together with correlated compounds (ethyl hexanoate, isoamyl acetate, glycerol), bringing them up to 95 th percentile ethanol-normalized concentrations (Methods) within the Blond group (‘Spiked’ concentration in Fig.  5A ). Compared to controls, the spiked beers were found to have significantly improved overall appreciation among trained panelists, with panelist noting increased intensity of ester flavors, sweetness, alcohol, and body fullness (Fig.  5B ). To disentangle the contribution of ethanol to these results, a second experiment was performed without the addition of ethanol. This resulted in a similar outcome, including increased perception of alcohol and overall appreciation.

Adding the top chemical compounds, identified as best predictors of appreciation by our model, into poorly appreciated beers results in increased appreciation from our trained panel. Results of sensory tests between base beers and those spiked with compounds identified as the best predictors by the model. A Blond and Non/Low-alcohol (0.0% ABV) base beers were brought up to 95th-percentile ethanol-normalized concentrations within each style. B For each sensory attribute, tasters indicated the more intense sample and selected the sample they preferred. The numbers above the bars correspond to the p values that indicate significant changes in perceived flavor (two-sided binomial test: alpha 0.05, n  = 20 or 13).

In a last experiment, we tested whether using the model’s predictions can boost the appreciation of a non-alcoholic beer (beer 223 in Supplementary Data  1 ). Again, the addition of a mixture of predicted compounds (omitting ethanol, in this case) resulted in a significant increase in appreciation, body, ester flavor and sweetness.

Predicting flavor and consumer appreciation from chemical composition is one of the ultimate goals of sensory science. A reliable, systematic and unbiased way to link chemical profiles to flavor and food appreciation would be a significant asset to the food and beverage industry. Such tools would substantially aid in quality control and recipe development, offer an efficient and cost-effective alternative to pilot studies and consumer trials and would ultimately allow food manufacturers to produce superior, tailor-made products that better meet the demands of specific consumer groups more efficiently.

A limited set of studies have previously tried, to varying degrees of success, to predict beer flavor and beer popularity based on (a limited set of) chemical compounds and flavors 79 , 80 . Current sensitive, high-throughput technologies allow measuring an unprecedented number of chemical compounds and properties in a large set of samples, yielding a dataset that can train models that help close the gaps between chemistry and flavor, even for a complex natural product like beer. To our knowledge, no previous research gathered data at this scale (250 samples, 226 chemical parameters, 50 sensory attributes and 5 consumer scores) to disentangle and validate the chemical aspects driving beer preference using various machine-learning techniques. We find that modern machine learning models outperform conventional statistical tools, such as correlations and linear models, and can successfully predict flavor appreciation from chemical composition. This could be attributed to the natural incorporation of interactions and non-linear or discontinuous effects in machine learning models, which are not easily grasped by the linear model architecture. While linear models and partial least squares regression represent the most widespread statistical approaches in sensory science, in part because they allow interpretation 65 , 81 , 82 , modern machine learning methods allow for building better predictive models while preserving the possibility to dissect and exploit the underlying patterns. Of the 10 different models we trained, tree-based models, such as our best performing GBR, showed the best overall performance in predicting sensory responses from chemical information, outcompeting artificial neural networks. This agrees with previous reports for models trained on tabular data 83 . Our results are in line with the findings of Colantonio et al. who also identified the gradient boosting architecture as performing best at predicting appreciation and flavor (of tomatoes and blueberries, in their specific study) 26 . Importantly, besides our larger experimental scale, we were able to directly confirm our models’ predictions in vivo.

Our study confirms that flavor compound concentration does not always correlate with perception, suggesting complex interactions that are often missed by more conventional statistics and simple models. Specifically, we find that tree-based algorithms may perform best in developing models that link complex food chemistry with aroma. Furthermore, we show that massive datasets of untrained consumer reviews provide a valuable source of data, that can complement or even replace trained tasting panels, especially for appreciation and basic flavors, such as sweetness and bitterness. This holds despite biases that are known to occur in such datasets, such as price or conformity bias. Moreover, GBR models predict taste better than aroma. This is likely because taste (e.g. bitterness) often directly relates to the corresponding chemical measurements (e.g., iso-alpha acids), whereas such a link is less clear for aromas, which often result from the interplay between multiple volatile compounds. We also find that our models are best at predicting acidity and alcohol, likely because there is a direct relation between the measured chemical compounds (acids and ethanol) and the corresponding perceived sensorial attribute (acidity and alcohol), and because even untrained consumers are generally able to recognize these flavors and aromas.

The predictions of our final models, trained on review data, hold even for blind tastings with small groups of trained tasters, as demonstrated by our ability to validate specific compounds as drivers of beer flavor and appreciation. Since adding a single compound to the extent of a noticeable difference may result in an unbalanced flavor profile, we specifically tested our identified key drivers as a combination of compounds. While this approach does not allow us to validate if a particular single compound would affect flavor and/or appreciation, our experiments do show that this combination of compounds increases consumer appreciation.

It is important to stress that, while it represents an important step forward, our approach still has several major limitations. A key weakness of the GBR model architecture is that amongst co-correlating variables, the largest main effect is consistently preferred for model building. As a result, co-correlating variables often have artificially low importance scores, both for impurity and SHAP-based methods, like we observed in the comparison to the more randomized Extra Trees models. This implies that chemicals identified as key drivers of a specific sensory feature by GBR might not be the true causative compounds, but rather co-correlate with the actual causative chemical. For example, the high importance of ethyl acetate could be (partially) attributed to the total ester content, ethanol or ethyl hexanoate (rho=0.77, rho=0.72 and rho=0.68), while ethyl phenylacetate could hide the importance of prenyl isobutyrate and ethyl benzoate (rho=0.77 and rho=0.76). Expanding our GBR model to include beer style as a parameter did not yield additional power or insight. This is likely due to style-specific chemical signatures, such as iso-alpha acids and lactic acid, that implicitly convey style information to the original model, as well as the smaller sample size per style, limiting the power to uncover style-specific patterns. This can be partly attributed to the curse of dimensionality, where the high number of parameters results in the models mainly incorporating single parameter effects, rather than complex interactions such as style-dependent effects 67 . A larger number of samples may overcome some of these limitations and offer more insight into style-specific effects. On the other hand, beer style is not a rigid scientific classification, and beers within one style often differ a lot, which further complicates the analysis of style as a model factor.

Our study is limited to beers from Belgian breweries. Although these beers cover a large portion of the beer styles available globally, some beer styles and consumer patterns may be missing, while other features might be overrepresented. For example, many Belgian ales exhibit yeast-driven flavor profiles, which is reflected in the chemical drivers of appreciation discovered by this study. In future work, expanding the scope to include diverse markets and beer styles could lead to the identification of even more drivers of appreciation and better models for special niche products that were not present in our beer set.

In addition to inherent limitations of GBR models, there are also some limitations associated with studying food aroma. Even if our chemical analyses measured most of the known aroma compounds, the total number of flavor compounds in complex foods like beer is still larger than the subset we were able to measure in this study. For example, hop-derived thiols, that influence flavor at very low concentrations, are notoriously difficult to measure in a high-throughput experiment. Moreover, consumer perception remains subjective and prone to biases that are difficult to avoid. It is also important to stress that the models are still immature and that more extensive datasets will be crucial for developing more complete models in the future. Besides more samples and parameters, our dataset does not include any demographic information about the tasters. Including such data could lead to better models that grasp external factors like age and culture. Another limitation is that our set of beers consists of high-quality end-products and lacks beers that are unfit for sale, which limits the current model in accurately predicting products that are appreciated very badly. Finally, while models could be readily applied in quality control, their use in sensory science and product development is restrained by their inability to discern causal relationships. Given that the models cannot distinguish compounds that genuinely drive consumer perception from those that merely correlate, validation experiments are essential to identify true causative compounds.

Despite the inherent limitations, dissection of our models enabled us to pinpoint specific molecules as potential drivers of beer aroma and consumer appreciation, including compounds that were unexpected and would not have been identified using standard approaches. Important drivers of beer appreciation uncovered by our models include protein levels, ethyl acetate, ethyl phenyl acetate and lactic acid. Currently, many brewers already use lactic acid to acidify their brewing water and ensure optimal pH for enzymatic activity during the mashing process. Our results suggest that adding lactic acid can also improve beer appreciation, although its individual effect remains to be tested. Interestingly, ethanol appears to be unnecessary to improve beer appreciation, both for blond beer and alcohol-free beer. Given the growing consumer interest in alcohol-free beer, with a predicted annual market growth of >7% 84 , it is relevant for brewers to know what compounds can further increase consumer appreciation of these beers. Hence, our model may readily provide avenues to further improve the flavor and consumer appreciation of both alcoholic and non-alcoholic beers, which is generally considered one of the key challenges for future beer production.

Whereas we see a direct implementation of our results for the development of superior alcohol-free beverages and other food products, our study can also serve as a stepping stone for the development of novel alcohol-containing beverages. We want to echo the growing body of scientific evidence for the negative effects of alcohol consumption, both on the individual level by the mutagenic, teratogenic and carcinogenic effects of ethanol 85 , 86 , as well as the burden on society caused by alcohol abuse and addiction. We encourage the use of our results for the production of healthier, tastier products, including novel and improved beverages with lower alcohol contents. Furthermore, we strongly discourage the use of these technologies to improve the appreciation or addictive properties of harmful substances.

The present work demonstrates that despite some important remaining hurdles, combining the latest developments in chemical analyses, sensory analysis and modern machine learning methods offers exciting avenues for food chemistry and engineering. Soon, these tools may provide solutions in quality control and recipe development, as well as new approaches to sensory science and flavor research.

Beer selection

250 commercial Belgian beers were selected to cover the broad diversity of beer styles and corresponding diversity in chemical composition and aroma. See Supplementary Fig.  S1 .

Chemical dataset

Sample preparation.

Beers within their expiration date were purchased from commercial retailers. Samples were prepared in biological duplicates at room temperature, unless explicitly stated otherwise. Bottle pressure was measured with a manual pressure device (Steinfurth Mess-Systeme GmbH) and used to calculate CO 2 concentration. The beer was poured through two filter papers (Macherey-Nagel, 500713032 MN 713 ¼) to remove carbon dioxide and prevent spontaneous foaming. Samples were then prepared for measurements by targeted Headspace-Gas Chromatography-Flame Ionization Detector/Flame Photometric Detector (HS-GC-FID/FPD), Headspace-Solid Phase Microextraction-Gas Chromatography-Mass Spectrometry (HS-SPME-GC-MS), colorimetric analysis, enzymatic analysis, Near-Infrared (NIR) analysis, as described in the sections below. The mean values of biological duplicates are reported for each compound.

HS-GC-FID/FPD

HS-GC-FID/FPD (Shimadzu GC 2010 Plus) was used to measure higher alcohols, acetaldehyde, esters, 4-vinyl guaicol, and sulfur compounds. Each measurement comprised 5 ml of sample pipetted into a 20 ml glass vial containing 1.75 g NaCl (VWR, 27810.295). 100 µl of 2-heptanol (Sigma-Aldrich, H3003) (internal standard) solution in ethanol (Fisher Chemical, E/0650DF/C17) was added for a final concentration of 2.44 mg/L. Samples were flushed with nitrogen for 10 s, sealed with a silicone septum, stored at −80 °C and analyzed in batches of 20.

The GC was equipped with a DB-WAXetr column (length, 30 m; internal diameter, 0.32 mm; layer thickness, 0.50 µm; Agilent Technologies, Santa Clara, CA, USA) to the FID and an HP-5 column (length, 30 m; internal diameter, 0.25 mm; layer thickness, 0.25 µm; Agilent Technologies, Santa Clara, CA, USA) to the FPD. N 2 was used as the carrier gas. Samples were incubated for 20 min at 70 °C in the headspace autosampler (Flow rate, 35 cm/s; Injection volume, 1000 µL; Injection mode, split; Combi PAL autosampler, CTC analytics, Switzerland). The injector, FID and FPD temperatures were kept at 250 °C. The GC oven temperature was first held at 50 °C for 5 min and then allowed to rise to 80 °C at a rate of 5 °C/min, followed by a second ramp of 4 °C/min until 200 °C kept for 3 min and a final ramp of (4 °C/min) until 230 °C for 1 min. Results were analyzed with the GCSolution software version 2.4 (Shimadzu, Kyoto, Japan). The GC was calibrated with a 5% EtOH solution (VWR International) containing the volatiles under study (Supplementary Table  S7 ).

HS-SPME-GC-MS

HS-SPME-GC-MS (Shimadzu GCMS-QP-2010 Ultra) was used to measure additional volatile compounds, mainly comprising terpenoids and esters. Samples were analyzed by HS-SPME using a triphase DVB/Carboxen/PDMS 50/30 μm SPME fiber (Supelco Co., Bellefonte, PA, USA) followed by gas chromatography (Thermo Fisher Scientific Trace 1300 series, USA) coupled to a mass spectrometer (Thermo Fisher Scientific ISQ series MS) equipped with a TriPlus RSH autosampler. 5 ml of degassed beer sample was placed in 20 ml vials containing 1.75 g NaCl (VWR, 27810.295). 5 µl internal standard mix was added, containing 2-heptanol (1 g/L) (Sigma-Aldrich, H3003), 4-fluorobenzaldehyde (1 g/L) (Sigma-Aldrich, 128376), 2,3-hexanedione (1 g/L) (Sigma-Aldrich, 144169) and guaiacol (1 g/L) (Sigma-Aldrich, W253200) in ethanol (Fisher Chemical, E/0650DF/C17). Each sample was incubated at 60 °C in the autosampler oven with constant agitation. After 5 min equilibration, the SPME fiber was exposed to the sample headspace for 30 min. The compounds trapped on the fiber were thermally desorbed in the injection port of the chromatograph by heating the fiber for 15 min at 270 °C.

The GC-MS was equipped with a low polarity RXi-5Sil MS column (length, 20 m; internal diameter, 0.18 mm; layer thickness, 0.18 µm; Restek, Bellefonte, PA, USA). Injection was performed in splitless mode at 320 °C, a split flow of 9 ml/min, a purge flow of 5 ml/min and an open valve time of 3 min. To obtain a pulsed injection, a programmed gas flow was used whereby the helium gas flow was set at 2.7 mL/min for 0.1 min, followed by a decrease in flow of 20 ml/min to the normal 0.9 mL/min. The temperature was first held at 30 °C for 3 min and then allowed to rise to 80 °C at a rate of 7 °C/min, followed by a second ramp of 2 °C/min till 125 °C and a final ramp of 8 °C/min with a final temperature of 270 °C.

Mass acquisition range was 33 to 550 amu at a scan rate of 5 scans/s. Electron impact ionization energy was 70 eV. The interface and ion source were kept at 275 °C and 250 °C, respectively. A mix of linear n-alkanes (from C7 to C40, Supelco Co.) was injected into the GC-MS under identical conditions to serve as external retention index markers. Identification and quantification of the compounds were performed using an in-house developed R script as described in Goelen et al. and Reher et al. 87 , 88 (for package information, see Supplementary Table  S8 ). Briefly, chromatograms were analyzed using AMDIS (v2.71) 89 to separate overlapping peaks and obtain pure compound spectra. The NIST MS Search software (v2.0 g) in combination with the NIST2017, FFNSC3 and Adams4 libraries were used to manually identify the empirical spectra, taking into account the expected retention time. After background subtraction and correcting for retention time shifts between samples run on different days based on alkane ladders, compound elution profiles were extracted and integrated using a file with 284 target compounds of interest, which were either recovered in our identified AMDIS list of spectra or were known to occur in beer. Compound elution profiles were estimated for every peak in every chromatogram over a time-restricted window using weighted non-negative least square analysis after which peak areas were integrated 87 , 88 . Batch effect correction was performed by normalizing against the most stable internal standard compound, 4-fluorobenzaldehyde. Out of all 284 target compounds that were analyzed, 167 were visually judged to have reliable elution profiles and were used for final analysis.

Discrete photometric and enzymatic analysis

Discrete photometric and enzymatic analysis (Thermo Scientific TM Gallery TM Plus Beermaster Discrete Analyzer) was used to measure acetic acid, ammonia, beta-glucan, iso-alpha acids, color, sugars, glycerol, iron, pH, protein, and sulfite. 2 ml of sample volume was used for the analyses. Information regarding the reagents and standard solutions used for analyses and calibrations is included in Supplementary Table  S7 and Supplementary Table  S9 .

NIR analyses

NIR analysis (Anton Paar Alcolyzer Beer ME System) was used to measure ethanol. Measurements comprised 50 ml of sample, and a 10% EtOH solution was used for calibration.

Correlation calculations

Pairwise Spearman Rank correlations were calculated between all chemical properties.

Sensory dataset

Trained panel.

Our trained tasting panel consisted of volunteers who gave prior verbal informed consent. All compounds used for the validation experiment were of food-grade quality. The tasting sessions were approved by the Social and Societal Ethics Committee of the KU Leuven (G-2022-5677-R2(MAR)). All online reviewers agreed to the Terms and Conditions of the RateBeer website.

Sensory analysis was performed according to the American Society of Brewing Chemists (ASBC) Sensory Analysis Methods 90 . 30 volunteers were screened through a series of triangle tests. The sixteen most sensitive and consistent tasters were retained as taste panel members. The resulting panel was diverse in age [22–42, mean: 29], sex [56% male] and nationality [7 different countries]. The panel developed a consensus vocabulary to describe beer aroma, taste and mouthfeel. Panelists were trained to identify and score 50 different attributes, using a 7-point scale to rate attributes’ intensity. The scoring sheet is included as Supplementary Data  3 . Sensory assessments took place between 10–12 a.m. The beers were served in black-colored glasses. Per session, between 5 and 12 beers of the same style were tasted at 12 °C to 16 °C. Two reference beers were added to each set and indicated as ‘Reference 1 & 2’, allowing panel members to calibrate their ratings. Not all panelists were present at every tasting. Scores were scaled by standard deviation and mean-centered per taster. Values are represented as z-scores and clustered by Euclidean distance. Pairwise Spearman correlations were calculated between taste and aroma sensory attributes. Panel consistency was evaluated by repeating samples on different sessions and performing ANOVA to identify differences, using the ‘stats’ package (v4.2.2) in R (for package information, see Supplementary Table  S8 ).

Online reviews from a public database

The ‘scrapy’ package in Python (v3.6) (for package information, see Supplementary Table  S8 ). was used to collect 232,288 online reviews (mean=922, min=6, max=5343) from RateBeer, an online beer review database. Each review entry comprised 5 numerical scores (appearance, aroma, taste, palate and overall quality) and an optional review text. The total number of reviews per reviewer was collected separately. Numerical scores were scaled and centered per rater, and mean scores were calculated per beer.

For the review texts, the language was estimated using the packages ‘langdetect’ and ‘langid’ in Python. Reviews that were classified as English by both packages were kept. Reviewers with fewer than 100 entries overall were discarded. 181,025 reviews from >6000 reviewers from >40 countries remained. Text processing was done using the ‘nltk’ package in Python. Texts were corrected for slang and misspellings; proper nouns and rare words that are relevant to the beer context were specified and kept as-is (‘Chimay’,’Lambic’, etc.). A dictionary of semantically similar sensorial terms, for example ‘floral’ and ‘flower’, was created and collapsed together into one term. Words were stemmed and lemmatized to avoid identifying words such as ‘acid’ and ‘acidity’ as separate terms. Numbers and punctuation were removed.

Sentences from up to 50 randomly chosen reviews per beer were manually categorized according to the aspect of beer they describe (appearance, aroma, taste, palate, overall quality—not to be confused with the 5 numerical scores described above) or flagged as irrelevant if they contained no useful information. If a beer contained fewer than 50 reviews, all reviews were manually classified. This labeled data set was used to train a model that classified the rest of the sentences for all beers 91 . Sentences describing taste and aroma were extracted, and term frequency–inverse document frequency (TFIDF) was implemented to calculate enrichment scores for sensorial words per beer.

The sex of the tasting subject was not considered when building our sensory database. Instead, results from different panelists were averaged, both for our trained panel (56% male, 44% female) and the RateBeer reviews (70% male, 30% female for RateBeer as a whole).

Beer price collection and processing

Beer prices were collected from the following stores: Colruyt, Delhaize, Total Wine, BeerHawk, The Belgian Beer Shop, The Belgian Shop, and Beer of Belgium. Where applicable, prices were converted to Euros and normalized per liter. Spearman correlations were calculated between these prices and mean overall appreciation scores from RateBeer and the taste panel, respectively.

Pairwise Spearman Rank correlations were calculated between all sensory properties.

Machine learning models

Predictive modeling of sensory profiles from chemical data.

Regression models were constructed to predict (a) trained panel scores for beer flavors and quality from beer chemical profiles and (b) public reviews’ appreciation scores from beer chemical profiles. Z-scores were used to represent sensory attributes in both data sets. Chemical properties with log-normal distributions (Shapiro-Wilk test, p  <  0.05 ) were log-transformed. Missing chemical measurements (0.1% of all data) were replaced with mean values per attribute. Observations from 250 beers were randomly separated into a training set (70%, 175 beers) and a test set (30%, 75 beers), stratified per beer style. Chemical measurements (p = 231) were normalized based on the training set average and standard deviation. In total, three linear regression-based models: linear regression with first-order interaction terms (LR), lasso regression with first-order interaction terms (Lasso) and partial least squares regression (PLSR); five decision tree models, Adaboost regressor (ABR), Extra Trees (ET), Gradient Boosting regressor (GBR), Random Forest (RF) and XGBoost regressor (XGBR); one support vector machine model (SVR) and one artificial neural network model (ANN) were trained. The models were implemented using the ‘scikit-learn’ package (v1.2.2) and ‘xgboost’ package (v1.7.3) in Python (v3.9.16). Models were trained, and hyperparameters optimized, using five-fold cross-validated grid search with the coefficient of determination (R 2 ) as the evaluation metric. The ANN (scikit-learn’s MLPRegressor) was optimized using Bayesian Tree-Structured Parzen Estimator optimization with the ‘Optuna’ Python package (v3.2.0). Individual models were trained per attribute, and a multi-output model was trained on all attributes simultaneously.

Model dissection

GBR was found to outperform other methods, resulting in models with the highest average R 2 values in both trained panel and public review data sets. Impurity-based rankings of the most important predictors for each predicted sensorial trait were obtained using the ‘scikit-learn’ package. To observe the relationships between these chemical properties and their predicted targets, partial dependence plots (PDP) were constructed for the six most important predictors of consumer appreciation 74 , 75 .

The ‘SHAP’ package in Python (v0.41.0) was implemented to provide an alternative ranking of predictor importance and to visualize the predictors’ effects as a function of their concentration 68 .

Validation of causal chemical properties

To validate the effects of the most important model features on predicted sensory attributes, beers were spiked with the chemical compounds identified by the models and descriptive sensory analyses were carried out according to the American Society of Brewing Chemists (ASBC) protocol 90 .

Compound spiking was done 30 min before tasting. Compounds were spiked into fresh beer bottles, that were immediately resealed and inverted three times. Fresh bottles of beer were opened for the same duration, resealed, and inverted thrice, to serve as controls. Pairs of spiked samples and controls were served simultaneously, chilled and in dark glasses as outlined in the Trained panel section above. Tasters were instructed to select the glass with the higher flavor intensity for each attribute (directional difference test 92 ) and to select the glass they prefer.

The final concentration after spiking was equal to the within-style average, after normalizing by ethanol concentration. This was done to ensure balanced flavor profiles in the final spiked beer. The same methods were applied to improve a non-alcoholic beer. Compounds were the following: ethyl acetate (Merck KGaA, W241415), ethyl hexanoate (Merck KGaA, W243906), isoamyl acetate (Merck KGaA, W205508), phenethyl acetate (Merck KGaA, W285706), ethanol (96%, Colruyt), glycerol (Merck KGaA, W252506), lactic acid (Merck KGaA, 261106).

Significant differences in preference or perceived intensity were determined by performing the two-sided binomial test on each attribute.

Reporting summary

Further information on research design is available in the  Nature Portfolio Reporting Summary linked to this article.

Data availability

The data that support the findings of this work are available in the Supplementary Data files and have been deposited to Zenodo under accession code 10653704 93 . The RateBeer scores data are under restricted access, they are not publicly available as they are property of RateBeer (ZX Ventures, USA). Access can be obtained from the authors upon reasonable request and with permission of RateBeer (ZX Ventures, USA).  Source data are provided with this paper.

Code availability

The code for training the machine learning models, analyzing the models, and generating the figures has been deposited to Zenodo under accession code 10653704 93 .

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Acknowledgements

We thank all lab members for their discussions and thank all tasting panel members for their contributions. Special thanks go out to Dr. Karin Voordeckers for her tremendous help in proofreading and improving the manuscript. M.S. was supported by a Baillet-Latour fellowship, L.C. acknowledges financial support from KU Leuven (C16/17/006), F.A.T. was supported by a PhD fellowship from FWO (1S08821N). Research in the lab of K.J.V. is supported by KU Leuven, FWO, VIB, VLAIO and the Brewing Science Serves Health Fund. Research in the lab of T.W. is supported by FWO (G.0A51.15) and KU Leuven (C16/17/006).

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These authors contributed equally: Michiel Schreurs, Supinya Piampongsant, Miguel Roncoroni.

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VIB—KU Leuven Center for Microbiology, Gaston Geenslaan 1, B-3001, Leuven, Belgium

Michiel Schreurs, Supinya Piampongsant, Miguel Roncoroni, Lloyd Cool, Beatriz Herrera-Malaver, Florian A. Theßeling & Kevin J. Verstrepen

CMPG Laboratory of Genetics and Genomics, KU Leuven, Gaston Geenslaan 1, B-3001, Leuven, Belgium

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Laboratory of Socioecology and Social Evolution, KU Leuven, Naamsestraat 59, B-3000, Leuven, Belgium

Lloyd Cool, Christophe Vanderaa & Tom Wenseleers

VIB Bioinformatics Core, VIB, Rijvisschestraat 120, B-9052, Ghent, Belgium

Łukasz Kreft & Alexander Botzki

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S.P., M.S. and K.J.V. conceived the experiments. S.P., M.S. and K.J.V. designed the experiments. S.P., M.S., M.R., B.H. and F.A.T. performed the experiments. S.P., M.S., L.C., C.V., L.K., A.B., P.M., L.D., T.W. and K.J.V. contributed analysis ideas. S.P., M.S., L.C., C.V., T.W. and K.J.V. analyzed the data. All authors contributed to writing the manuscript.

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Schreurs, M., Piampongsant, S., Roncoroni, M. et al. Predicting and improving complex beer flavor through machine learning. Nat Commun 15 , 2368 (2024). https://doi.org/10.1038/s41467-024-46346-0

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Prediction of malignant esophageal fistula in esophageal cancer using a radiomics-clinical nomogram

• Chao Zhu 1 , 2 , 3 ,
• Wenju Sun 2 ,
• Cunhai Chen 2 ,
• Qingtao Qiu 3 ,
• Shuai Wang 4 ,
• Yang Song 1 &
• Xuezhen Ma 2  

European Journal of Medical Research volume  29 , Article number:  217 ( 2024 ) Cite this article

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Malignant esophageal fistula (MEF), which occurs in 5% to 15% of esophageal cancer (EC) patients, has a poor prognosis. Accurate identification of esophageal cancer patients at high risk of MEF is challenging. The goal of this study was to build and validate a model to predict the occurrence of esophageal fistula in EC patients.

This study retrospectively enrolled 122 esophageal cancer patients treated by chemotherapy or chemoradiotherapy (53 with fistula, 69 without), and all patients were randomly assigned to a training ( n  = 86) and a validation ( n  = 36) cohort. Radiomic features were extracted from pre-treatment CTs, clinically predictors were identified by logistic regression analysis. Lasso regression model was used for feature selection, and radiomics signature building. Multivariable logistic regression analysis was used to develop the clinical nomogram, radiomics-clinical nomogram and radiomics prediction model. The models were validated and compared by discrimination, calibration, reclassification, and clinical benefit.

The radiomic signature consisting of ten selected features, was significantly associated with esophageal fistula ( P  = 0.001). Radiomics-clinical nomogram was created by two predictors including radiomics signature and stenosis, which was identified by logistic regression analysis. The model showed good discrimination with an AUC = 0.782 (95% CI 0.684–0.8796) in the training set and 0.867 (95% CI 0.7461–0.987) in the validation set, with an AIC = 101.1, and good calibration. When compared to the clinical prediction model, the radiomics-clinical nomogram improved NRI by 0.236 (95% CI 0.153, 0.614) and IDI by 0.125 (95% CI 0.040, 0.210), P  = 0.004.

We developed and validated the first radiomics-clinical nomogram for malignant esophageal fistula, which could assist clinicians in identifying patients at high risk of MEF.

Introduction

Malignant esophageal fistula (MEF) is a serious complication of advanced esophageal cancer and is defined as a fistula caused by malignancies [ 1 ]. The two most common types are esophageal mediastinal fistula and esophageal respiratory fistula. Esophageal aortic fistula was rarely reported, because these patients often die of sudden massive hemorrhage and cannot be diagnosed before death [ 2 ]. In untreated EC patients, the incidence of esophageal fistula ranges from 5 to 15% [ 3 ], while radiotherapy patients have a range of 5.6% to 33% [ 4 , 5 , 6 , 7 ]. MEF has a very poor prognosis; patients frequently die within 3 months due to nutritional failure, pneumonia, mediastinal abscess or large vascular injury [ 8 ]. Patients with MEF must be treated with a nutrition tube, gastrostomy, or esophageal stent to prevent digestive fluid leakage into the trachea or mediastinum, all of which significantly reduce quality of life. Therefore, early identification of patients at risk of MEF enables early intervention, which improves patient outcomes.

However, few studies on MEF risk factors have been conducted, the majority of which were retrospective analyses of small samples, and only a few risk factors had been confirmed. T 4 stage (invasion of adjacent organs) had been a known risk factor for MEF. Because the esophagus is a thin-walled organ with a thickness of less than 5 mm, tumors can grow through the entire thickness of the esophageal wall, which is the primary cause of MEF [ 3 ]. It is reported that the incidence of esophageal fistula is 10–22% in T 4 patients with radical concurrent chemoradiotherapy, which is significantly higher than in non-T 4 patients [ 9 , 10 ]. In a retrospective study, ECOG PS, BMI, T 4 , N 2/3 and re-radiotherapy were identified as independent factors, with a C-index of a nomogram incorporating the factors of 0.764 in an external validation cohort [ 11 ]. Another retrospective study revealed quantitative pretreatment CT analysis has excellent performance for predicting fistula formation in esophageal cancer patients [ 12 ].

Recently, radiomics has proven to be a useful tool for identifying the biological behavior of tumors [ 13 ]. Its central concept is to convert images into digital information that can be mined, extracted, and analyzed, as well as to predict target associations and build model [ 14 , 15 ]. Iwashita et al. reported CT radiomics and clinical data might help determine survival outcomes in patients with esophageal cancer treated with radical radiotherapy [ 16 ]. Yang et al. found that radiomics can be used for chemoradiation outcome prediction in esophageal cancer [ 17 ]. Furthermore, some research has demonstrated that radiomics of tumor target areas can be utilized to predict tumor stage, differentiation, and pathological type [ 18 , 19 ]. The present study attempt to develop a joint prediction nomogram by combining clinical factors and pre-treatment CT radiomics to accurately predict MEF. It will help clinicians identify EC patients at high risk of having an esophageal fistula.

Materials and methods

The Shandong Cancer Hospital ethics committee approved this retrospective study (Approval No. 2021003193), and informed consent was waived.

Inclusion criteria were: (a) Esophageal carcinoma proved by pathology; (b) Chest enhanced CT examination before treatment; (c) Patients treated with chemotherapy or chemoradiotherapy. Exclusion criteria: (a) Esophageal fistula appeared prior to treatment; (b) Anastomotic fistula or other medical injuries caused esophageal fistula.

A total of 122 patients treated at Shandong Cancer Hospital from October 2018 to September 2020 were included in the study. Fifty-three patients with MEF were identified, while 69 patients did not have a MEF within their survival period.

Malignant esophageal fistula must be confirmed by endoscopy or esophagography. All cases were divided into a training set and a validation set in a 7:3 ratio using a random algorithm.

Data collection and variable definition

Clinical data such as gender, age, tumor location, TNM stage, radiotherapy, chemoradiotherapy, chemotherapy regimen, and stenosis were obtained using the hospital information system (HIS). To obtain enhanced chest CT images, the picture archiving and communication system (PACS) was used. The image format was digital imaging and communications in medicine (DICOM).

Stenosis was defined as the inability to pass an endoscope through and/or significantly impairing eating (semi-fluid or fluid diet). The chemotherapy regimens mentioned in the article were first-line treatments.

Clinical predictors and clinical nomogram

Logistic univariate and multivariate regression analyses were used to identify clinical independent predictors and create a clinical nomogram using the variables with P  < 0.25 in multivariate analyses.

Tumor segmentation

The region of interest (ROI) was defined as primary tumors, which included lesions with esophageal wall thickening > 5 mm or lumen occlusion diameter > 10 mm but did not include intraluminal gas or oral contrast agent. A radiation oncologist with 10 years of experience manually delineated the ROIs, which were then checked by a radiologist. Both doctors were unaware of the patient’s clinical information. 3D Slicer (version: 4.10.2), an open source software platform for medical image processing and visualization, was used to perform all delineation tasks.

Feature extraction and selection

Radiomic features were extracted automatically from each contoured ROI using Pyradiosity, an open source Python package available at https://pyradiomics.readthedocs.io/en/latest/ .

The Lasso-logistic regression algorithm was used to screen radiomic features related to the presence of MEF. The Pearson correlation test was used to examine the multicollinearity.

Radiomics model construction

Based on the screened features, the logistic regression algorithm generated the radiomics model.

Radiomics-clinical model construction

Two steps were included in radiomics-clinical model construction. First, radiomic signature (radscore) was calculated by adding all filtered eigenvalues multiplied by the corresponding coefficients. The Wilcoxon rank sum (Mann–Whitney) test was applied to the radscores of the training and validation sets to ensure that there was no difference between the two groups. The flowchart for radiomic signature is presented in Fig.  1 . Second, the logistic regression algorithm was used to fit radiomic signature and clinical independent risk factors.

Flowchart of developing a radiomic signature. Following ROI segmentation, 851 radiomic features were extracted from each ROI. The Lasso algorithm selected ten features that had the best correlation with the occurrence of esophageal fistula and a radiomic signature were created. The box chart revealed that there was no significant difference between the training and validation sets, but there was a difference between patients with and without fistula

Validation and comparison

In the validation set, the prediction efficiency of each model was evaluated, including discrimination (AUC/ROC), calibration (calibration curve), goodness of fit (AIC), reclassification ability (NRI, IDI), and clinical benefit (DCA curve). The Delong test was used to compare model discrimination abilities.

Statistical analysis

Stata 15.0 software (Stata Corp, www.stata.com ) was used for statistical analysis. The chi-square test was used to compare categorical variables; the Wilcoxon rank sum test was used to compare continuous variables; all tests were two-sided, and P  < 0.05 was considered statistically significant. R software (version 3.5.3, https://www.r-project.org/ ) was used to select radiomic features and build models. The R program packages used in this study are listed in the table below.

This study enrolled 122 patients with esophageal cancer, including 56 patients with MEF. Squamous cell carcinoma was the pathological type of all patients. The time between the occurrence of an esophageal fistula and diagnosis of EC ranged from 1 to 24 months, with an average of 7.87 ± 8.78 months. Esophageal mediastinal fistula occurred in 22 patients, esophageal tracheal fistula in 30 patients, and esophageal pulmonary fistula in 1 patient. Sixty-nine control cases were hospitalized at the same time and had no esophageal fistula within the survival period or 2 years after diagnosis, with a median follow-up time of 30 months (IQR: 16, 38). Table 1 shows the baseline characteristics of enrolled patients.

In a 7:3 split, all patients were assigned to a training set and a validation set. There were 86 patients in the training set, with 38 (44.2%) having MEF; 36 patients in the validation set, with 15 (41.7%) having MEF. No statistically significant differences existed in baseline clinical characteristics between the two groups (Table 1 ).

The clinical prediction model was built using stenosis ( P  = 0.01), gender ( P  = 0.23), and T stage ( P  = 0.11), all of which were screened by logistic regression algorithm (Table 2 ). The model predicted esophageal fistula with an AUC = 0.691 (95% CI 0.582–0.799) in the training set, and 0.640 (95% CI 0.453–0.827) in the validation set. The AIC value was 115.8. Figure  2 depicts a clinical nomogram and its calibration curve. Each predictor in the nomogram was assigned a score on a point scale. By adding up the total scores projected in the bottom scale, we could estimate the probability of MEF.

A Nomogram developed by clinical risk factors. B Calibration curve plotted for clinical prediction model

Radiomic features

A total of 851 radiomic features were extracted from each ROI, and all features can be divided into 4 categories. The ‘shape’ category had 14 features that indicate the shape and size of regions of interest (ROIs) in 2D and 3D spaces. The ‘first order’ represents statistical eigenvalues of voxel intensity, encompassing mean, maximum, and minimum values. Textural features such as Glcm (GrayLevelCooccurenceMatrix), Glrlm (GrayLevelRunLengthMatrix), Glszm (GrayLevelSizeZoneMatrix), Gldm (GrayLevelDependenceMatrix), and Ngtdm (NeighbouringGrayToneDifferenceMatrix) were computed from multiple statistical matrices and characterized the organization Wavelet-based features were defined as first-order and texture features obtained from eight wavelet decompositions of the original CT images. As a result, the total number of radiomic features can be estimated as 14 + (18 + 24 + 16 + 16 + 14 + 5) + (18 + 24 + 16 + 16 + 14 + 5)8 = 851.

Ten features that were closely related to the presence of an MEF were chosen using Lasso-logistic algorithm (Fig.  1 ). The selected features and their coefficients are shown in Table  3 . There was no significant multicollinearity between the features chosen (Pearson correlation coefficients were all less than 0.9).

Radiomics prediction model

The radiomics prediction model was built based on the selected radiomic features to compare with the clinical and radiomics-clinical models. In validation set, the AUC of radiomics model was 0.692 (95% CI 0.516–0.868) and AIC was 111.4.

Radiomics-clinical nomogram model

The formula for calculating radiomic signature (radscore) was as follows:

In the training set, patients without MEF had a radscore of 0.873 ± 0.372, while patients with MEF had a radscore of 1.275 ± 0.381. Wilcoxon rank sum test revealed a significant difference ( P  = 0.000). In the validation set, patients with MEF had a radscore of 1.125 ± 0.186, while those without had a radscore of 0.857 ± 0.2187, a significant difference ( P  = 0.000). In the training and validation sets, there was no significant difference between patients with MEF ( P  = 0.278), and those without ( P  = 0.6019). The multivariate logistic regression analysis including radscore, gender, T stage and stenosis revealed that stenosis ( P  = 0.023) and radscore ( P  = 0.001) were the independent risk factors (Table  2 ).

A radiomics-clinical nomogram prediction model (Fig.  3 A) was developed using these two variables as inputs, which predicted esophageal fistula with an AUC = 0.782 (95% CI 0.684–0.8796) in the training set and 0.867 (95% CI 0.7461–0.987) in the validation set, with an AIC = 101.1. The calibration curve revealed that the radiomics-clinical nomogram’s predicted results were in good agreement with the actual observations (Fig.  3 B).

A Radiomics-clinical nomogram. B Calibration curve plotted for radiomics-clinical prediction model

Model comparison

Figure  4 A displays the discrimination ability (AUC values) of the clinical model, radiomics model, and radiomics-clinical model, revealing that the Joint model outperformed the other two models (Delong test, P  < 0.05). The net benefit of the radiomics-clinical prediction model under each threshold probability was greater than that of the clinical model and radiomics model, according to the decision curves (Fig.  4 B). In comparison to the clinical model, the NRI of radiomics-clinical model was 23.6% (95% CI 0.153–0.614) (1000 iterations), and IDI was 0.125 (95% CI 0.040–0.210), which was statistically significant ( P  = 0.004).

The model’s performance was compared using the A ROC curve (discrimination), B DCA curve (clinical benefit). Radiomics-clinical prediction model outperformed the other two

Malignant esophageal fistula is a serious complication of esophageal cancer that significantly reduces the patient’s survival time and quality of life. However, there is currently a scarcity of effective predictive methods. Clinical prediction models and quantitative-CT-based prediction models with AUC of 0.805 and 0.841, respectively, have been reported in previous research [ 11 , 12 ]. In present study, we built the radiomics-clinical model, clinical model, and radiomics model all at the same time and compared their predictive abilities. We discovered that the radiomics-clinical model performed significantly better than the other two models.

The radiomics-clinical model developed by fitting stenosis with radiomic signature using logistic regression algorithm had an excellent prediction efficiency with an AUC of 0.867 (95% CI 0.7461–0.987). The discrimination ability was significantly higher than that of the clinical (AUC = 0.640) and radiomics models (AUC = 0.692). In terms of goodness of fit, the AIC of the radiomics-clinical model was 101.1, which was lower than the AICs of the clinical model (AIC = 115.8) and the radiomics model (AIC = 111.4), indicating a better fitting performance.

T 4 indicates that esophageal cancer has infiltrated the entire esophageal layer and has invaded surrounding organs [ 20 ]. Esophageal cancer is prone to spontaneous necrosis, and radiotherapy or chemotherapy can also promote necrosis, resulting in an esophageal fistula, particularly in tumors sensitive to radiation or chemotherapy [ 21 ]. However, whether radiotherapy is the cause of esophageal fistula remains debatable. Some researchers believe there is no evidence that radiotherapy increases the incidence of esophageal fistula because these patients with tracheoesophageal fistula after radiotherapy may have fistula in any case, but radiotherapy causes it to occur earlier [ 3 ]. Other researchers believe that radiotherapy causes tumor tissue to shrink while inhibiting normal tissue repair, which is a major cause of MEF [ 22 , 23 ].

Esophageal stenosis is another clinically independent predictor. Takahiro et al. [ 9 ] discovered that stenosis was the only clinical risk factor for MEF, and that the risk of esophageal fistula in patients with stenosis was twice that of patients without stenosis, which was consistent with our findings. T 4 and stenosis were also independent risk factors for esophageal fistula in patients treated by concurrent radiotherapy and chemotherapy [ 24 ]. However, there is no universally accepted definition of esophageal stenosis, which is usually determined based on the patient’s clinical symptoms and/or the findings of an endoscopic examination [ 25 , 26 ].

In this study, male patients outnumbered female patients by 6.625:1, and female patients were less likely than male patients to develop an esophageal fistula. Guan et al. discovered that gender was an independent risk factor for malignant esophageal fistula [ 27 ]. Previous research has found that gender influenced the prognosis of esophageal cancer [ 28 ]. Estrogen regulates metabolism and organ response after injury [ 29 ]. As a result, women have natural advantages in injury recovery, which may account for gender differences in the occurrence and prognosis of MEF.

The clinical prediction model’s discrimination was unsatisfactory. To improve model performance, we extracted radiomic features from the primary tumor’s pre-treatment CT, screened out the feature set most relevant to the occurrence of MEF using lasso logistic regression and cross validation, and built a combined model with clinical risk factors, which significantly improved prediction efficiency. The biological behavior of esophageal cancer cells is a significant cause of fistula [ 30 ], but clinical risk factors cannot adequately reflect these characteristics. Medical images of tumors are the external manifestation of gene phenotype, and radiomics can convert images into digital data that can be mined, extracted, and analyzed [ 31 ]. Radiomics had been used in tumor studies to predict pathological type, stage, curative effect, and prognosis [ 32 , 33 , 34 ], and the derived radiogenomics can analyze tumor heterogeneity at the gene level [ 35 , 36 ]. In our study, the potential 851 candidate radiomics features were eventually reduced to ten potential predictors by the LASSO method for further integration to form the radiomics signature, which contains useful biological information.

Because previous research has shown that arterial phase can better visualize esophageal tumors, which were chosen as the research object in this study [ 37 ].

This research had some advantages: first, prior to treatment, all of the clinical characteristics and medical images were gathered, which was helpful in developing the treatment strategy. Second, the nomogram prediction model showed the weight of each parameter on the outcome in a more intuitive way, making it more practical for clinical use. There were some limitations: because none of the patients in this group were surgically treated, only a few patients provided data on tumor differentiation, and this variable could not be assessed. To avoid bias, the total dose of radiotherapy was not included in the analysis. This was due to the fact that eight patients developed esophageal fistulas during radiotherapy, and their irradiation dose was 2–40 Gy. Furthermore, the modeling and validation were all completed in a single center. In future work, we will include data from multiple centers for external validation to improve the model’s generalization.

We developed and validated the first radiomics-clinical nomogram which could assist clinicians in identifying patients at high risk of malignant esophageal fistula.

Availability of data and materials

Data and material were available.

Abbreviations

Malignant esophageal fistula

Esophageal cancer

Computed tomography

Least absolute shrinkage and selection operator

American Joint Committee on Cancer

Picture archiving and communication system

Digital imaging and communications in medicine

Region of interest

Receiver operating characteristic curve

Area under curve

Akaike information criterion

Net reclassification improvement

Integrated discrimination improvement

Decision curve

Gray level co-occurrence matrix

Gray level run length matrix

Gray level size zone matrix

Gray level dependence matrix

Neighbouring gray tone difference matrix

Gray level run length

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Chao Zhu & Yang Song

Department of Oncology, Affiliated Qingdao Central Hospital of Qingdao University, Qingdao Cancer Hospital, Qingdao, 266042, China

Chao Zhu, Wenju Sun, Cunhai Chen & Xuezhen Ma

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Zhu, C., Sun, W., Chen, C. et al. Prediction of malignant esophageal fistula in esophageal cancer using a radiomics-clinical nomogram. Eur J Med Res 29 , 217 (2024). https://doi.org/10.1186/s40001-024-01746-2

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Regression Method in Data Mining: A Systematic Literature Review

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• Mohammad Vahid Sebt   ORCID: orcid.org/0000-0001-6402-9214 1 ,
• Yaser Sadati-Keneti 1 ,
• Misagh Rahbari 1 ,
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Regression is one of the most important supervised learning methods in data mining that is used to predict and discover knowledge in data mining science. After reviewing the studies conducted in the field of regression, it has been found that the tendency to use this method is increasing day by day among researchers. This study reviews 500 articles from about 230 reputable journals under one framework over the twenty-first century and also discusses the status and use of regression in data mining research. The systematic framework presented in this study includes the following steps: 1—Examining the position of regression in research conducted in data mining and determining the trend of different journals to conduct research in the field of regression in different years 2—Examining different study areas in the field of regression and determining the trend to conduct research in various areas of study in different years 3—Examining the algorithms used in the field of regression and determining the most widely used and trend to use algorithms by researchers in different years 4—Examining the keywords used in regression research in data mining and determining the strongest and most attractive rules obtained from the relationships of these keywords with each other using the Apriori algorithm.

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Prediction Modeling Methodology

Abbreviations.

Regression tree

Classification and regression tree

Multiple logistic regression

Random forest

Support vector regression

Harmful algal bloom expert system

Artificial neural network

Boosted regression tree

Zero-or-one inflated beta

Logit leaf model

Tree-based pipeline optimization tool

Gradient boosting tree

Multilayer perceptron neural network

Three gorges dam

Least squares support vector machine

Decision tree

Dynamic neural network

Principal component analysis

Customer satisfaction survey

Hybrid particle swarm optimization

CART using house holders matrices

Fourier transform and SVR

Receiver operating characteristic

Chi-square automatic interaction detection

Non-linear regression analysis

K-nearest neighbor

Hierarchical tree-based regression

Interpretative structural modeling with regression tree

Gradient boosted regression

Long-short term memory network

Artificial bee colony with logistic regression

Gross domestic product

X12-auto regressive integrated moving average

Multilayer perceptron

MARS and gradient boosting machine

Logistic regression

Linear regression

Multivariate regression tree

Support vector machine

Group method of data handling

Smooth transition regression

Safety performance functions

Linear discriminant analysis

Classification tree

Multivariate adaptive regression spline

Adaptive neuro-fuzzy inference systems

Interval-valued time series

Extreme gradient boosting

Gradient boosted decision tree

Least squares support vector regression

Quantitative structure activity relationship

Smooth transition regression and CART

Radial basis function network

HPSO, RT and RBFN

Species distribution model

Multidimensional scaling and hierarchical cluster analysis

Area under the curve

Internet of things

Decision tree regression

Gradient boosting regression tree

Extreme learning machine

Filtered logistic single-hidden layer feedforward neural network

Random forest regression

Empirical mode decomposition-based LSTM

International classification of disease 9th—revision clinical modification

Sulfur dioxide

Genetic programming

Gradient boosting decision tree with logistic regression

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Department of Industrial Engineering, Faculty of Engineering, Kharazmi University, Tehran, Iran

Mohammad Vahid Sebt, Yaser Sadati-Keneti, Misagh Rahbari, Zohreh Gholipour & Hamid Mehri

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Sebt, M.V., Sadati-Keneti, Y., Rahbari, M. et al. Regression Method in Data Mining: A Systematic Literature Review. Arch Computat Methods Eng (2024). https://doi.org/10.1007/s11831-024-10088-5

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Received : 22 December 2020

Accepted : 04 February 2024

Published : 27 March 2024

DOI : https://doi.org/10.1007/s11831-024-10088-5

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