what is regression analysis in research methodology

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

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Regression Analysis – Methods, Types and Examples

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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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Home Market Research

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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Content Index

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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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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Regression Analysis

• Reference work entry
• First Online: 03 December 2021
• Cite this reference work entry

• Bernd Skiera 4 ,
• Jochen Reiner 4 &
• Sönke Albers 5  

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Linear regression analysis is one of the most important statistical methods. It examines the linear relationship between a metric-scaled dependent variable (also called endogenous, explained, response, or predicted variable) and one or more metric-scaled independent variables (also called exogenous, explanatory, control, or predictor variable). We illustrate how regression analysis work and how it supports marketing decisions, e.g., the derivation of an optimal marketing mix. We also outline how to use linear regression analysis to estimate nonlinear functions such as a multiplicative sales response function. Furthermore, we show how to use the results of a regression to calculate elasticities and to identify outliers and discuss in details the problems that occur in case of autocorrelation, multicollinearity and heteroscedasticity. We use a numerical example to illustrate in detail all calculations and use this numerical example to outline the problems that occur in case of endogeneity.

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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.

You May Also Like

A statistical measurement of the dispersion between values in a data collection is a variance. The symbol that is frequently used to represent variation is σ2.

Statistical significance is described as the measure of the null hypothesis being plausible as compared to the acceptable level of vagueness regarding the true answer.

T-distribution describes the probability of data. Although it looks similar to a normal distribution with a bell curve, it has a lower height and a broader curve.

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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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• 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.

Ethics statements

Patient consent for publication.

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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Quantitative Research Methods

• Introduction
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Correlation is the relationship or association between two variables. There are multiple ways to measure correlation, but the most common is Pearson's correlation coefficient (r), which tells you the strength of the linear relationship between two variables. The value of r has a range of -1 to 1 (0 indicates no relationship). Values of r closer to -1 or 1 indicate a stronger relationship and values closer to 0 indicate a weaker relationship.  Because Pearson's coefficient only picks up on linear relationships, and there are many other ways for variables to be associated, it's always best to plot your variables on a scatter plot, so that you can visually inspect them for other types of correlation.

• Correlation Penn State University tutorial
• Correlation and Causation Australian Bureau of Statistics Article

Spurious Relationships

It's important to remember that correlation does not always indicate causation. Two variables can be correlated without either variable causing the other. For instance, ice cream sales and drownings might be correlated, but that doesn't mean that ice cream causes drownings—instead, both ice cream sales and drownings increase when the weather is hot. Relationships like this are called spurious correlations.

• Spuriousness Harvard Business Review article.
• New Evidence for Theory of The Stork A satirical article demonstrating the dangers of confusing correlation with causation.

Regression is a statistical method for estimating the relationship between two or more variables. In theory, regression can be used to predict the value of one variable (the dependent variable) from the value of one or more other variables (the independent variable/s or predictor/s). There are many different types of regression, depending on the number of variables and the properties of the data that one is working with, and each makes assumptions about the relationship between the variables. (For instance, most types of regression assume that the variables have a linear relationship.) Therefore, it is important to understand the assumptions underlying the type of regression that you use and how to properly interpret its results. Because regression will always output a relationship, whether or not the variables are truly causally associated, it is also important to carefully select your predictor variables.

• A Refresher on Regression Analysis Harvard Business Review article.
• Introductory Business Statistics - Regression

Simple Linear Regression

Simple linear regression estimates a linear relationship between one dependent variable and one independent variable.

• Simple Linear Regression Tutorial Penn State University Tutorial
• Statistics 101: Linear Regression, The Very Basics YouTube video from Brandon Foltz.

Multiple Linear Regression

Multiple linear regression estimates a linear relationship between one dependent variable and two or more independent variables.

• Multiple Linear Regression Tutorial Penn State University Tutorial
• Multiple Regression Basics NYU course materials.
• Statistics 101: Multiple Linear Regression, The Very Basics YouTube video from Brandon Foltz.

If you do a subject search for Regression Analysis you'll see that the library has over 200 books about regression.  Select books are listed below.  Also, note that econometrics texts will often include regression analysis and other related methods.  

Search for ebooks using Quicksearch .  Use keywords to search for e-books about Regression .  

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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.

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What Is a Regression?

Understanding regression, calculating regression.

• Regression FAQs
• Macroeconomics

What Is Regression? Definition, Calculation, and Example

Regression is a statistical method used in finance, investing, and other disciplines that attempts to determine the strength and character of the relationship between one dependent variable (usually denoted by Y) and a series of other variables (known as independent variables).

Also called simple regression or ordinary least squares (OLS), linear regression is the most common form of this technique. Linear regression establishes the linear relationship between two variables based on a line of best fit . Linear regression is thus graphically depicted using a straight line with the slope defining how the change in one variable impacts a change in the other. The y-intercept of a linear regression relationship represents the value of one variable when the value of the other is zero. Non-linear regression models also exist, but are far more complex.

Regression analysis is a powerful tool for uncovering the associations between variables observed in data, but cannot easily indicate causation. It is used in several contexts in business, finance, and economics. For instance, it is used to help investment managers value assets and understand the relationships between factors such as commodity prices and the stocks of businesses dealing in those commodities.

Regression as a statistical technique should not be confused with the concept of regression to the mean ( mean reversion ).

Key Takeaways

• A regression is a statistical technique that relates a dependent variable to one or more independent (explanatory) variables.
• A regression model is able to show whether changes observed in the dependent variable are associated with changes in one or more of the explanatory variables.
• It does this by essentially fitting a best-fit line and seeing how the data is dispersed around this line.
• Regression helps economists and financial analysts in things ranging from asset valuation to making predictions.
• In order for regression results to be properly interpreted, several assumptions about the data and the model itself must hold.

Investopedia / Joules Garcia

Regression captures the correlation between variables observed in a data set and quantifies whether those correlations are statistically significant or not.

The two basic types of regression are simple linear regression and  multiple linear regression , although there are non-linear regression methods for more complicated data and analysis. Simple linear regression uses one independent variable to explain or predict the outcome of the dependent variable Y, while multiple linear regression uses two or more independent variables to predict the outcome (while holding all others constant).

Regression can help finance and investment professionals as well as professionals in other businesses. Regression can also help predict sales for a company based on weather, previous sales, GDP growth, or other types of conditions. The capital asset pricing model (CAPM) is an often-used regression model in finance for pricing assets and discovering the costs of capital.

Regression and Econometrics

Econometrics is a set of statistical techniques used to analyze data in finance and economics. An example of the application of econometrics is to study the income effect using observable data. An economist may, for example, hypothesize that as a person increases their income their spending will also increase.

If the data show that such an association is present, a regression analysis can then be conducted to understand the strength of the relationship between income and consumption and whether or not that relationship is statistically significant—that is, it appears to be unlikely that it is due to chance alone.

Note that you can have several explanatory variables in your analysis—for example, changes to GDP and inflation in addition to unemployment in explaining stock market prices. When more than one explanatory variable is used, it is referred to as  multiple linear regression . This is the most commonly used tool in econometrics.

Econometrics is sometimes criticized for relying too heavily on the interpretation of regression output without linking it to economic theory or looking for causal mechanisms. It is crucial that the findings revealed in the data are able to be adequately explained by a theory, even if that means developing your own theory of the underlying processes.

Linear regression models often use a least-squares approach to determine the line of best fit. The least-squares technique is determined by minimizing the sum of squares created by a mathematical function. A square is, in turn, determined by squaring the distance between a data point and the regression line or mean value of the data set.

Once this process has been completed (usually done today with software), a regression model is constructed. The general form of each type of regression model is:

Simple linear regression:

Y = a + b X + u \begin{aligned}&Y = a + bX + u \\\end{aligned} ​ Y = a + b X + u ​

Multiple linear regression:

Y = a + b 1 X 1 + b 2 X 2 + b 3 X 3 + . . . + b t X t + u where: Y = The dependent variable you are trying to predict or explain X = The explanatory (independent) variable(s) you are  using to predict or associate with Y a = The y-intercept b = (beta coefficient) is the slope of the explanatory variable(s) u = The regression residual or error term \begin{aligned}&Y = a + b_1X_1 + b_2X_2 + b_3X_3 + ... + b_tX_t + u \\&\textbf{where:} \\&Y = \text{The dependent variable you are trying to predict} \\&\text{or explain} \\&X = \text{The explanatory (independent) variable(s) you are } \\&\text{using to predict or associate with Y} \\&a = \text{The y-intercept} \\&b = \text{(beta coefficient) is the slope of the explanatory} \\&\text{variable(s)} \\&u = \text{The regression residual or error term} \\\end{aligned} ​ Y = a + b 1 ​ X 1 ​ + b 2 ​ X 2 ​ + b 3 ​ X 3 ​ + ... + b t ​ X t ​ + u where: Y = The dependent variable you are trying to predict or explain X = The explanatory (independent) variable(s) you are  using to predict or associate with Y a = The y-intercept b = (beta coefficient) is the slope of the explanatory variable(s) u = The regression residual or error term ​

Example of How Regression Analysis Is Used in Finance

Regression is often used to determine how many specific factors such as the price of a commodity, interest rates, particular industries, or sectors influence the price movement of an asset. The aforementioned CAPM is based on regression, and it is utilized to project the expected returns for stocks and to generate costs of capital. A stock's returns are regressed against the returns of a broader index, such as the S&P 500, to generate a beta for the particular stock.

Beta is the stock's risk in relation to the market or index and is reflected as the slope in the CAPM model. The return for the stock in question would be the dependent variable Y, while the independent variable X would be the market risk premium.

Additional variables such as the market capitalization of a stock, valuation ratios, and recent returns can be added to the CAPM model to get better estimates for returns. These additional factors are known as the Fama-French factors, named after the professors who developed the multiple linear regression model to better explain asset returns.

Why Is It Called Regression?

Although there is some debate about the origins of the name, the statistical technique described above most likely was termed "regression" by Sir Francis Galton in the 19th century to describe the statistical feature of biological data (such as heights of people in a population) to regress to some mean level. In other words, while there are shorter and taller people, only outliers are very tall or short, and most people cluster somewhere around (or "regress" to) the average.

What Is the Purpose of Regression?

In statistical analysis, regression is used to identify the associations between variables occurring in some data. It can show both the magnitude of such an association and also determine its statistical significance (i.e., whether or not the association is likely due to chance). Regression is a powerful tool for statistical inference and has also been used to try to predict future outcomes based on past observations.

How Do You Interpret a Regression Model?

A regression model output may be in the form of Y = 1.0 + (3.2) X 1 - 2.0( X 2 ) + 0.21.

Here we have a multiple linear regression that relates some variable Y with two explanatory variables X 1 and X 2 . We would interpret the model as the value of Y changes by 3.2x for every one-unit change in X 1 (if X 1 goes up by 2, Y goes up by 6.4, etc.) holding all else constant (all else equal). That means controlling for X 2 , X 1 has this observed relationship. Likewise, holding X1 constant, every one unit increase in X 2 is associated with a 2x decrease in Y. We can also note the y-intercept of 1.0, meaning that Y = 1 when X 1 and X 2 are both zero. The error term (residual) is 0.21.

What Are the Assumptions That Must Hold for Regression Models?

In order to properly interpret the output of a regression model, the following main assumptions about the underlying data process of what you are analyzing must hold:

• The relationship between variables is linear
• Homoskedasticity , or that the variance of the variables and error term must remain constant
• All explanatory variables are independent of one another
• All variables are normally distributed

Margo Bergman. ” Quantitative Analysis for Business: 12. Simple Linear Regression and Correlation .” University of Washington Pressbooks , 2022.

Margo Bergman. ” Quantitative Analysis for Business: 13. Multiple Linear Regression .”  University of Washington Pressbooks , 2022.

Fama, Eugene F. and French, Kenneth R. “ The Cross-Section of Expected Stock Returns .” Journal of Finance , vol. 47, no. 2, June 1992, pp. 427-465.

Stanton, Jeffrey M. “ Galton, Pearson, and the Peas: A Brief History of Linear Regression for Statistics Instructors .” Journal of Statistics Education , vol. 9, no. 3, 2001.

CFA Institute. “ Basics of Multiple Regression and Underlying Assumptions. ”

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• Open access
• Published: 03 May 2024

Multi-objective optimization of wire electrical discharge machining process using multi-attribute decision making techniques and regression analysis

• Masoud Seidi 1 ,
• Saeed Yaghoubi 2 &
• Farshad Rabiei 2  

Scientific Reports volume  14 , Article number:  10234 ( 2024 ) Cite this article

Metrics details

• Mechanical engineering
• Mechanical properties

Wire electrical discharge machining (WEDM) is one of the most important non-traditional machining methods that is widely used in various industries. The present research work is concerned with the influences of process variables on quality of machined specimen obtained from WEDM process. The process parameters to manufacture mold structure included wire feed speed, wire tension and generator power, and in the current research, the effects of these variables on the aim factors, namely dimensional accuracy, hardness and roughness of product surface have been investigated, simultaneously. In order to obtain the optimal experiment, the multi-objective optimization with discrete solution area has been employed. Method based on the removal effects of criteria (MEREC) and weighted aggregates sum product assessment (WASPAS) techniques have been used with the aim of weighting the objective functions and discovering the best practical experiment. In the following, the regression analysis has been employed to study the effects of variables on response factors. A good correlation between the results gained from two analysis methods was observed. Based on MEREC-WASPAS hybrid technique, the weights of roughness, hardness and dimensional accuracy of machined part were calculated to about 89%, 9% and 2%, respectively. In the selected optimal experiment, the amount of wire feed speed, wire tension and generator power variables were considered to, in turn, 2 cm/s, 2.5 kg, and 10%.

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A multi-factorial mathematical model for the selection of electropolishing parameters with a view to reducing the environmental impact

Introduction.

In the manufacturing processes, it is important to have a mold with high precision and quality. This process was done with wood and stone in the past, but nowadays they use metal, plastic, polymers, elastomers, thermoplastic and thermosetting. In general, a mold is a tool for forming any kind of product in a way that helps the product (like a template) takes the desired form. One of the most important types of molds used in industries is a cutting mold. These shearing molds are one of the remarkable widely applied types of press dies, which are used to create cavities, holes, edges and grooves on the workpiece and include many different mechanisms. There are different methods to make the molds. Since the mold must have high dimensional accuracy, it is important to choose its manufacturing method. Electrical discharge machining (EDM) is a machining process in which a generator source is employed to generate a low-voltage and high-ampere spark for machining purposes 1 . One of the most accurate methods of making metal molds is using a wire-cut machine. Wire electrical discharge machining (WEDM) is a device that cuts parts through a thin wire that is inside distilled water or dielectric liquid 2 . By creating a spark between the part and the wire, this device causes momentary melting of that point, where the melting process takes place inside the same dielectric liquid. The material of the workpiece on which the cutting operation is performed can be aluminum 3 , copper 4 , brass, and steel 5 . The sparks produced in this device are completely visible when the water is clean, and the removal of chips can be seen with the naked eye 6 . In the WEDM operation, there is no contact between the wire and the workpiece, and the distance or gap that exists between them greatly increases the accuracy of the manufacturing process 7 . Due to the lack of contact between the wire and specimen, physical pressure similar to grinding and milling processes is not applied to the workpiece 8 . Therefore, this process eliminates mechanical stresses, noise and vibration during machining operation and can produce various materials with a thickness of 300 mm. For this reason, it can be said that wire-cut is a suitable option for making molds and industrial parts 9 due to its high dimensional accuracy 10 and good surface quality 11 . Proper dimensional accuracy leads to improvement of fatigue strength, wear resistance and corrosion resistance. With this regard, any effort to improve dimensional accuracy in this manufacturing process can be a significant step in the direction of producing precise and strong industrial parts 12 . Considering that the cutting dies produce a part in every cycle of a few seconds and are continuously subjected to cyclic loading, it is remarkable to investigate the fatigue phenomenon from this point of view. One of the factors that can increase the fatigue life of cutting molds is enhancing the hardness value 13 .

Several researches have been conducted to improve the quality of parts produced via WEDM process. Zahoor et al. 14 optimized the dimensional accuracy and surface roughness of wire-cut process in Inconel 718 alloy using genetic optimization algorithm. Their results demonstrated that the dimensional accuracy depends on the clear time of the pulse, servo voltage and wire feeding. Chaudhary et al. 15 studied on the effect of changing the input parameters on the optimization of dimensional accuracy in the manufacture of miniature gears by wire-cut operation. A mathematical model was derived using the Taguchi method. TOPSIS and ANOVA showed that the pulse light time is the most significant parameter for good dimensional accuracy. Paturi et al. 16 focused on the influences of control factors such as pulse on time, pulse off time, peak current, voltage and wire feed rate on surface roughness of Inconel 718 steel manufactured part. Their results demonstrated that the peak current has 60.21% effect on the surface roughness. Nair et al. 17 studied on the effect of changing different parameters on dimensional accuracy, material removal rate and surface roughness of Inconel 718 workpiece using wire-cut process. The outcomes gained from their research showed that the dimensional accuracy of machined holes decreases with increasing in wire tension. Arya and Singh 18 optimized the parameters of pulse light time, pulse off time, peak current and server voltage, dimensional deviation and cutting rate in WEDM process. The results of the ANOVA in their research revealed that the pulse light time is the most influential parameter on the dimensional deviation. Ghasempour-Mouziraji et al. 19 tried to minimize the geometrical deviation of parts produced via wire-cut using ANN and non-dominated sorting genetic algorithm (NSGA) methods. Wire speed, pulse time and feed rate were input variables in their study. This research work provided a new perspective to optimize the dimensional accuracy in the wire-cut process. Kiyak et al. 20 investigated on the effect of pulse off time, pulse light time and mold thickness on steel hardness in WEDM operation. They reported the direct influences of these parameters on surface hardness. Ishfaq et al. 21 focused on the optimization of EDM operation in order to reduce the geometric dimensional deviation and electrode wear. According to their findings, the performance of transformer oil in improvement of this process was the best choice in comparison with other dielectrics. An in-depth evaluation on the improvement of wire electrical discharge machining has been carried out by Chatterjee et al. 22 . They employ multi attribute decision making methods in intuitionistic fuzzy environment to achieve proper WEDM condition. Based on their findings, the optimal values of pulse-on time, pulse-off time, wire feed and wire tension were considered to, in turn, 115 μs, 55 μs, 3 m/min, and 7 kg-F. Tiwari et al. 23 employed multi-criteria decision making (MCDM) model to optimize geometrical parameters in EDM process. In their research, the input variables include voltage, tool feed rate, and machining time, and the aim was to study the effects of aforementioned variables on radial overcut, circularity of the machined hole and heat-affected zone.

Based on the researches done, it can be said that checking the quality of the product obtained from the wire electrical discharge machining operation is of great importance. Various criteria have been introduced to investigate the quality of the produced sample, among which the hardness and roughness of the cut specimen surface can be mentioned. Several variables affect the quality of final product. Considering the importance of the three factors of roughness, hardness and dimensional accuracy of the manufactured part, in the current research, the effect of process variables on all three mentioned factors has been simultaneously investigated, which has not been performed in previous studies. The studied variables include wire feed speed, wire tension and generator power, and each of them is defined in three levels. In order to choose the best experimental test, method based on the removal effects of criteria and weighted aggregates sum product assessment technique has been used. Multi-objective optimization with discrete solution area is employed and using multi-attribute decision making techniques and regression analysis, the optimal condition is introduced. Multi-objective optimization with this approach has not been done so far.

Experimental procedures

One of the main steels used in manufacturing of molds is low Molybdenum alloy steel (Mo40), which is called St-4140 in AISI standard. This alloy is fragile due to the presence of chromium, and about 0.2% nickel is added to remove the brittleness. It should be noted that the resistance of these alloys is up to 500–600 °C. Carbon, Chromium, Manganese, Molybdenum and Silicon are alloy elements used in these steels. These materials have a high strength-to-weight ratio, because they are subjected to austenitizing, quenching, and then tempering operations. In the present research work, Mo40 steel was used as the workpiece material. Firstly, in order to achieve uniformity in thickness, the workpiece was completely machined using a shaper machine and brought to final dimensions of 320 mm × 40 mm and thickness of 10 mm. Then, it was fixed on the CHARMILLES 5-axis wire-cut machine. A view of experimental set is shown in Fig.  1 . In the current study, wire feed speed, wire tension and generator power were selected as input variables. Based on previous researches and limitations of the setup, the values of pulse on time (TON), pulse off time (TOFF), current, and voltage have been considered to, in turn, 3 µs, 5 µs, 2 A, and 90 V. The wire material was copper with diameter of 0.2 mm and distilled water was used as the dielectric fluid. The ranges of these parameters are given in Table 1 . It should be noted that the ranges of variables have been considered according to previous studies and setup constrains.

A view of experimental set used in current study.

The purpose of the present research work is to investigate the roughness, hardness and dimensional accuracy of the surface as a result of WEDM operations. In this study, R a , which shows the average deviation of the lows and highs from the middle line, has been chosen as a measure of surface roughness. In order to measure the surface roughness of the final product, Time 3110 portable roughness meter was employed. For each workpiece, the surface roughness test was measured 5 times with 8 mm length of probe, and after eliminating the outlier data, the average data was considered as surface roughness. Leeb Hardness Tester TA—with an accuracy of 0.1 has been used to calculate the surface hardness of the cut specimens. For manufactured parts, the hardness value was measured at four points and after removing the illogical data, the average data was considered as hardness of final product.

Proposed solution approach

Optimization with discrete solution area is a category of optimization problems. Multi-attribute decision making (MADM) techniques are suitable and widely used tools for solving such problems. MADM techniques rank the alternatives employing attributes (criteria) 24 . Attributes are objective functions and alternatives are solutions. Using attributes and alternatives, a matrix is formed, which is called a decision matrix. The decision matrix is the most important input to MADM techniques. There are two categories of attributes: (1) Attribute with a positive aspect (an index whose higher value is more favorable) and (2) An attribute with a negative aspect (an index whose lower value is more favorable). The positive and negative attributes are objective functions whose goal is to maximize and minimize them, respectively. In this research, two techniques, namely WASPAS and MEREC are used. The weighted aggregates sum product assessment (WASPAS) method was introduced by Zavadskas et al. 25 , 26 . This method is a combination of weighted sum model (WSM) and weighted product model (WPM). WASPAS is one of the latest methods of choosing the best alternative in MADM techniques, which is obtained from the combination of previous methods. This method has more accuracy compared to independent ones. WASPAS has two inputs: the decision matrix and the weights of the attributes (criteria). At first, the attribute weights are obtained applying method based on the removal effects of criteria (MEREC) method. In the following, it is provided to WAPAS method as an input. MEREC is a weighting method which is introduced, recently. In multi-attribute decision making (MADM) techniques, before choosing the best alternative, the weight of criteria must be determined, which is often uses the opinions of experts. MEREC is the latest alternative-based method for determining the weight of criteria. The desirability of this method is due to the absence of errors corresponded to the subjective judgments of experts 27 , 28 . The used multi-attribute decision making techniques have a discrete solution space and select one alternative from among the available alternatives and are sensitive to the weight of criteria. All the steps related to determining the weight of criteria and ranking the alternatives with additional explanations are shown in Fig.  2 .

Proposed solution approach.

Results and discussions

In the present section, the findings of current research are presented. For this purpose, the results of experimental tests are given, firstly. In the following, in order to analyze the outcomes obtained from WEDM process, MEREC-WASPAS hybrid technique and regression analysis are implemented and the results of both methods are compared.

Multi-attribute decision making approach

At first, the results of practical experiments obtained from wire electrical discharge machining operation are illustrated in this sub-section. In the following, the outcomes obtained from optimization with discrete solution area are illustrated. As explained earlier, in the current research, the aim is to optimize WEDM process variables. The input variables of wire feed speed, wire tension and generator power are considered in the design range given in Table 1 .

Considering that each variable has three levels, there will be 27 practical experiments in full factorial mode. In order to reduce the number of practical tests in wire-cut operation, the response surface methodology (RSM) was used and its number was decreased to 20 tests considering repeated ones. The experimental specimens obtained from the wire-cut operation are shown in Fig.  3 . As it is known, for producing each product, a cylinder with a base diameter of 10 mm and a height of 10 mm is cut from the workpiece. The objective functions include dimensional accuracy, hardness, and roughness of product surface, and their values have been calculated according to the practical test.

Produced specimens using wire-cut operation.

Table 2 shows the results of practical tests. In the following, the normalized decision matrix using the method mentioned in Fig.  2 is given. This matrix is the input to the MEREC method (Eq.  1 ). Calculations related to the MEREC method are given in Table 3 . The steps of MEREC method are shown in Fig.  2 . As can be seen, the weight of dimensional accuracy, hardness and roughness of machined part are computed to 0.01693, 0.08901 and 0.89406, respectively. The maximum weight is related to surface roughness attribute. Then, the normalized decision matrix using the method mentioned in Fig.  2 is given. This matrix is the input to the WASPAS technique (Eq.  2 ). The outcomes obtained from WASPAS method are given in Tables 4 and 5 . It should be mentioned that the steps of WASPAS method are shown in detail, previously (Fig.  2 ). After implementing WASPAS method, test No. 2 was chosen as the best alternative. According to this, the optimal values of the input variables consisting of wire tension, wire feed speed, and generator power were considered to 2.5 kg, 2 cm/s, and 10%, respectively. The second and third ranks are also highlighted in Tables 4 and 5 . As it is clear, by changing the value of λ-parameter in WASPAS technique, the ranking of options does not change. If the λ-value equals to 0 and 1, WASPAS method is converted into the WSM and WPM techniques, respectively. Therefore, it can be concluded that the ranking obtained from WASPAS, WSM and WPM is the same.

Wire feed speed and wire tension are both parameters of wire movement control. The wire feeding speed indicates the length of the wire that is unrolled from the reel in a unit of time. Unnecessary increase of this parameter leads to growth in wire consumption. On the other hand, being too low will lead to an increase in the possibility of the wire breaking. Considering the irregularities in the wire movement path, choosing the appropriate value for this parameter is most effective in reducing wire vibrations. Wire tension is a longitudinal force that is applied to the wire by machine to keep it in a straight line. Due to the presence of processing forces and the flexibility of the wire, there is always the possibility of the wire bending, which can be reduced by controlling this variable. If proper tension is applied on the wire, the deviation from the path, vibration and deformation will be effectively reduced and as a result, the machining error will be decreased. To create stable machining accuracy, a certain amount of tension in the wire is required to limit deviation of the path, bending and vibration. An increase in wire tension (in considerable ranges) can enhance the dimensional accuracy of final product, significantly. During the sparking, the wire moves at a nearly uniform speed over the workpiece. Due to the impact force of the spark on the wire, it is expected that the wire will vibrate under this impact. By increasing the amount of impact, the wire vibrates with a higher frequency. For this reason, it can be concluded that the roughness of the cut surface will decrease by enhancing the tension of the wire in this operation. With increasing generator power, the energy of sparks enhances, as a result, the volume of craters created on the surface of the workpiece increases. In this situation, the temperature on the surface of the workpiece is very high and due to these major sparks, bigger holes are created due to evaporation and melting of materials. For this reason, it can be said that increasing the generator power increases the cutting speed and decreases the smoothness of the surface in final product. Based on the explanations given above, it can be concluded that the outcomes obtained from practical experiments and multi-objective optimization seem reasonable.

Regression analysis

In the present sub-section, the relationships between three objective functions and three input factors have been extracted via regression analysis. Using Minitab software, these relationships have been derived for three response variables as follows (Eqs. 3 – 5 ). According to the regression equations and without considering the weight for the response variables, the optimal solution has been obtained using Minitab software (Fig.  4 ). As can be seen, the best answer is gained by the Tension = 0.5 kg, Speed = 2 cm/s and Power = 10%. For these values, the amount of response factors, namely dimensional accuracy, hardness, surface roughness were computed to, in turn, 9.981 mm, 71.025 Rockwell B, and 3.020 μm. Based on these findings, an index called desirability has been calculated for the answer. The closer the value of desirability is to one, the more suitable the answer is. The desirability in the answer is obtained to 0.7583. According to regression equations, the contour plots for the three response variables are presented in Fig.  5 .

Response optimization curves for three objective functions.

Contour plots presented for objective functions, namely ( a ) dimensional accuracy, ( b ) hardness, and ( c ) surface roughness.

In this section, the aim is to compare the outcomes obtained from MEREC-WASPAS hybrid technique and regression analysis. Based on the results extracted from both analysis methods, it was observed that the optimal values of the wire feed speed and generator power variables are the same for both analysis methods and the only difference is in the optimal value wire tension variable. As mentioned earlier, the amount of wire tension has a significant effect on surface roughness and dimensional accuracy of the mold produced via WEDM process. Increasing this variable has a significant effect on improving surface quality and dimensional accuracy. Since in the MEREC-WASPAS technique, the weight of the surface roughness factor was much higher than the other two factors, the selection of optimal variables was considered based on the maximum reduction in this response factor. Considering that in the regression analysis, the weight of all objective functions is considered the same, therefore, the maximum value of the wire tension variable (2.5 kg) was obtained for it in order to increase the maximum dimensional accuracy and surface quality, simultaneously. Accordingly, the findings obtained from the regression analysis also seem reasonable.

Conclusions

In the present study, the effect of manufactured part using wire electrical discharge machining operations on the quality of the obtained products has been investigated, experimentally. The material used in this research is Mo40 steel. The considered variables include wire feed speed, wire tension and generator power, and the purpose of the study is to investigate the effects of these variables on the three factors of hardness, roughness and dimensional accuracy of the product surface manufactured by WEDM operation. In order to perform multi-objective optimization and select the best practical test, two techniques, called method based on the removal effects of criteria (MEREC) and weighted aggregates sum product assessment (WASPAS) have been used. In the following, the regression analysis was investigated by considering the same weight for the objective functions. A summary of the results obtained is given below:

Based on MEREC technique, the weight of objective functions consisting of dimensional accuracy, hardness and roughness of product surface were calculated to 0.01693, 0.08901 and 0.89406, respectively.

According to WASPAS technique, test No. 2 was selected as the best choice. In this state, the optimal amounts of wire tension, wire feed speed, and generator power were considered to 2.5 kg, 2 cm/s, and 10%, respectively.

By changing the λ-value in WASPAS technique, the ranking of the superior practical tests did not change, which indicated the robustness of the optimization with discrete solution area in this case study.

In choosing the optimal test, the difference between the two methods of MEREC-WASPAS and regression analysis was related to wire tension variable. Due to the fact that the weight coefficients of the target factors are the same in this method, its maximum value (2.5 kg) was considered as the best experimental test.

Data availability

The datasets generated and analyzed during the current study are available from the corresponding author on reasonable request.

Abbreviations

Artificial neural network

Analysis of variance

• Electrical discharge machining

Multi-attribute decision making

Multi-criteria decision making

• Method based on the removal effects of criteria

Non-dominated sorting genetic algorithm

Response surface methodology

Technique for order of preference by similarity to ideal solution

• Weighted aggregates sum product assessment

Wire electrical discharge machining

Weighted product model

Weighted sum model

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Seidi, M., Yaghoubi, S. & Rabiei, F. Multi-objective optimization of wire electrical discharge machining process using multi-attribute decision making techniques and regression analysis. Sci Rep 14 , 10234 (2024). https://doi.org/10.1038/s41598-024-60825-w

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Towards Sustainable Development Goal of Zero Hunger: Exploring the Dynamic Relationships between Food Pricing, Agriculture, and Food Security

Impact of Zero Tillage Maize Production on Yield, Income, and Resource Utilization in Peninsular India: An Action-Based Quasi-Experimental Research Provisionally Accepted

• 1 Dr. Reddy's Foundation, India
• 2 Indian Institute of Technology Kharagpur, India

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The present study aims to identify the crucial determinants of the widespread adoption of zerotillage (ZT) technology in maize production. The study also measures the impact of ZT adoption on maize yield, income generation, and the expenses associated with different agricultural operations.The study used multi-stage stratified random sampling and conducted a face-to-face questionnaire survey to collect primary data from 1189 maize farmers. Initially, the study employed probit regression analysis to identify the ZT adoption determinants. Subsequently, using the Propensity Score Matching (PSM) approach, the study measures the impact of ZT adoption over conventional tillage in terms of yield, income, and cost management. Finally, the Endogenous Switch Regression (ESR) method was implemented to mitigate unobserved heterogeneity and sample selection bias. Additionally, ESR assessed the robustness of PSM results.The probit model identifies that variables like education, institutional credit adoption, crop insurance, visit of extension agent, landholding size, and prior experience of new technology adoption positively influence ZT adoption. The PSM and ESR approach results suggest that ZT adoption positively impacts farmers' yield and net income while reducing the cultivation cost and labor use. Results show that ZT adoption decreases the cost of land preparation, weed, pest management, and harvesting, thereby decreasing the cultivation cost by INR8376 acre -1 . Moreover, adopting ZT improves maize yield by 2.53quintal acre -1 and minimizes 9.56 persondays acre -1 .The study findings may support policymakers in designing suitable agricultural policies to improve technology adoption and motivate farmers for sustainable production.

Keywords: Conservation tillage, impact assessment, economic analysis, Propensity score matching (PSM) approach, Endogenous Switch Regression, Peninsular India

Received: 28 Dec 2023; Accepted: 02 May 2024.

Copyright: © 2024 Dey, Abbhishek, Saraswathibatla, Singh, Sreedhar, Bommaraboyina, Raj, Kaliki, Choubey, Rongali and Upamaka. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY) . The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

* Correspondence: Dr. Kumar Abbhishek, Dr. Reddy's Foundation, Hyderabad, India

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