Linear Regression: Definition, Applications and Benefits
By Indeed Editorial Team
Published 12 October 2022
The Indeed Editorial Team comprises a diverse and talented team of writers, researchers and subject matter experts equipped with Indeed's data and insights to deliver useful tips to help guide your career journey.
Linear regression is a statistical measure that establishes the relationship between variables that businesses use to develop forecasts and make informed decisions. It has applications in finance, business planning, marketing, health and medicine. Understanding the definition and applications of this type of regression can help you utilise this analysis to improve businesses and promote efficiency. In this article, we define linear regression, outline its applications, provide the equation, explain how to find outliers in data and explain the benefits of using it.
What is linear regression?
Linear regression is a statistical method determining the relationship between two or more variables. It involves using an independent or known variable to predict the dependent or response variable. You can demonstrate the relationship by drawing a straight line on a graph, where the independent variable is on the x-axis and the dependent on the y-axis. An example is determining the effects of different doses of medication, whereby the doses are the independent variable, and the effects are the dependent variable. There are two main types of linear regression. They are:
Simple: This type involves estimating the relationship between two quantitative variables, such as the value of a dependent variable at a particular value of the independent variable. An example is determining the number of sales when customer engagement on social media is at a certain percentage.
Multiple: In this type, you can determine the relationship between several independent variables and a dependent variable. An example is determining how the price of a product, interest rates and competitive prices affect a company's sales.
Applications of linear regression
The following are industries that use this method:
You can use a regression model to determine how products perform in the market by establishing the relationships between several quantitative variables, such as social media engagement, pricing and number of sales. This information allows you to utilise specific marketing strategies to maximise sales and increase revenue. For example, you can use a simple linear model to ascertain how price affects sales and use it to evaluate the strength between the two variables.
Financial analysts use linear models to evaluate a company's operational performance and forecast returns on investment. They also use it in the capital asset pricing model, which studies the relationship between the expected investment returns and the associated market risks. It shows companies if an investment has a fair price and contributes to decisions on whether or not to invest in the asset.
This involves sports analysts using statistics to determine a team's or player's performance in a game. They can use this information to compare teams and players and provide essential information to their followers. They can also use this data to predict game attendance based on the status of the teams playing and the market size, so they can advise team managers on game venues and ticket prices that can maximise profits.
Specialists in this field use this regression model to evaluate the relationship between natural elements, such as soil, water and air. An example is the relationship between the amount of water and plant growth. This can help environmentalists predict the effects of air or water pollution on environmental health.
Medical researchers can use this regression model to determine the relationship between independent characteristics, such as age and body weight, and dependent ones, such as blood pressure. This can help reveal the risk factors associated with diseases. They can use this information to identify high-risk patients and promote healthy lifestyles.
Linear model assumptions
There are particular assumptions to follow to make the data reliable, including:
A linear relationship between the variables: This model assumes that there's a linear relationship between the dependent and independent variables, such that a straight line passes through it.
Normality of the data: This model assumes that the data follows a normal distribution, whereby most data falls within the central region of a bell-shaped curve on a graph.
Homogeneity of the data: This regression model assumes that all the variables have the same characteristics, such that their standard of error is the same.
Linear regression equations
The simple linear regression equation is:
Y = a + bX + u
And the multiple linear regression equation is:
Y = a + (b1 x 1) + (b2 x 2) + (b3 x 3) … + b + u
Y = dependent variable
X = independent variable
a = intercept (where the line intercepts the X or Y axis)
b = slope of the graph
u = regression residual (vertical distance between a data point and the regression line)
This refers to data that is significantly different from the rest of the set. Outliers may occur due to an abnormal variable or an error in calculations. It's important to identify them because it can affect the results significantly. The following are ways to detect these:
Studentised residuals: This involves comparing observed and predicted values in a linear model and determining their standard deviation units to make it easier to find the outliers.
Mahalanobis distance: You can measure the distance between a central point, which can be an overall mean for multivariate data, and another point to find outliers.
Williams plot: This involves using graphic visualisation of data to see outliers for the response variable.
Linear model methods
The following are linear model methods:
This method involves plotting data points on an x- and y-axis graph and drawing a line of best fit to show the relationship between two variables. It aims to make the distance between the line and the data points as small as possible. The line of best fit minimises the sum of squares or the variance. You can use this method both by hand or when using a computer.
This method is an algorithm that involves moving iteratively in the direction of the steepest gradient of a curve to minimise errors. It starts with random values for each coefficient. The next step is calculating the sum of squared errors for each input and output pair and using a learning rate to minimise the error.
Tips on preparing data for linear regression
Consider following these tips when preparing the data:
Find outliers: This regression model assumes that the relationship between variables is linear, so it's important to remove outliers that can affect the results.
Remove collinearity: Collinearity is the correlation between independent variables in this type of regression and can create overfitting of the data, leading to inconsistent results from the model.
Normalise the data: Linear regressions make more accurate predictions when the data follows a normal distribution curve.
Standardise the data: You can do this by subtracting a measure of location, such as the mean, and dividing it by a measure of scale, such as standard deviation, especially when two data sets have different ranges, such as zero to one and zero to 1000.
Input extra data: If there are points with missing values, you can create space for additional imputations. This step may be unnecessary if you're working with large data sets.
Benefits of linear regression
The following are the benefits of using this analysis:
This regression model has applications in predicting outcomes, which can help companies decide whether to take on certain risks or investments. This can facilitate long-term business planning. For example, organisations can use this analysis to determine how many individuals can pass in front of a billboard. They can then use this information to strategically place these billboards to advertise products and get maximum views and sales.
Regression analysis can allow company heads to determine if a decision can lead to unfavourable outcomes and prevent them from occurring. This can enable companies to save on costs and increase revenue. For example, if a manager wants to determine if keeping a retail store open for an extra two hours daily can increase revenue, they can do a regression analysis to predict the outcomes. If the results suggest that this action could lead to higher costs, the company may decide against that option and save money.
Organisations can use this analysis to optimise business processes by determining how a change in a process can affect an outcome. They can use the results to implement new policies and protocols that increase efficiency. For example, a company can assess the relationship between customers' wait time when calling customer service agents and the number of negative reviews or complaints they receive regarding this. They can then use this information to set specific time limits for the agents answering the calls, which in turn may reduce the number of customer complaints.
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